  We present the average metallicity and star-formation rate of Lya emitters
(LAEs) measured from our large-area survey with three narrow-band (NB) filters
covering the Lya, [OII]3727, and Ha+[NII] lines of LAEs at z=2.2. We select 919
z=2.2 LAEs from Subaru/Suprime-Cam NB data in conjunction with Magellan/IMACS
spectroscopy. Of these LAEs, 561 and 105 are observed with KPNO/NEWFIRM
near-infrared NB filters whose central wavelengths are matched to redshifted
[OII] and Ha nebular lines, respectively. By stacking the near-infrared images
of the LAEs, we successfully obtain average nebular-line fluxes of LAEs, the
majority of which are too faint to be identified individually by narrow-band
imaging or deep spectroscopy. The stacked object has an Ha luminosity of
1.7x10^{42} erg s^{-1} corresponding to a star formation rate (SFR) of 14
M_{sun} yr^{-1}. We place, for the first time, a firm lower limit to the
average metallicity of LAEs of Z>~0.09 Z_{sun} (2sigma) based on the
[OII]/(Ha+[NII]) index together with photo-ionization models and empirical
relations. This lower limit of metallicity rules out the hypothesis that LAEs,
so far observed at z~2, are extremely metal poor (Z<2x10^{-2} Z_{sun}) young
galaxies at the 4sigma level. This limit is higher than a simple extrapolation
of the observed mass-metallicity relation of z~2 UV-selected galaxies toward
lower masses (5x10^{8} M_{sun}), but roughly consistent with a recently
proposed fundamental mass-metallicity relation when the LAEs' relatively low
SFR is taken into account. The Ha and Lya luminosities of our NB-selected LAEs
indicate that the escape fraction of Lya photons is ~12-30 %, much higher than
the values derived for other galaxy populations at z~2.
