  Proton-proton collisions at the LHC generate a high-intensity collimated beam
of neutrinos in the forward (beam) direction, characterised by energies of up
to several TeV. The recent observation of LHC neutrinos by FASER$\nu$ and
SND@LHC signals that this hitherto ignored particle beam is now available for
scientific inquiry. Here we quantify the impact that neutrino deep-inelastic
scattering (DIS) measurements at the LHC would have on the parton distributions
(PDFs) of protons and heavy nuclei. We generate projections for DIS structure
functions for FASER$\nu$ and SND@LHC at Run III, as well as for the
FASER$\nu$2, AdvSND, and FLArE experiments to be hosted at the proposed Forward
Physics Facility (FPF) operating concurrently with the High-Luminosity LHC
(HL-LHC). We determine that up to one million electron- and muon-neutrino DIS
interactions within detector acceptance can be expected by the end of the
HL-LHC, covering a kinematic region in $x$ and $Q^2$ overlapping with that of
the Electron-Ion Collider. Including these DIS projections into global (n)PDF
analyses, specifically PDF4LHC21, NNPDF4.0, and EPPS21, reveals a significant
reduction of PDF uncertainties, in particular for strangeness and the up and
down valence PDFs. We show that LHC neutrino data enables improved theoretical
predictions for core processes at the HL-LHC, such as Higgs and weak gauge
boson production. Our analysis demonstrates that exploiting the LHC neutrino
beam effectively provides CERN with a "Neutrino-Ion Collider" without requiring
modifications in its accelerator infrastructure.
