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Theorem lmhmlin 20638
Description: A homomorphism of left modules is 𝐾-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lmhmlin.k 𝐾 = (Scalar‘𝑆)
lmhmlin.b 𝐵 = (Base‘𝐾)
lmhmlin.e 𝐸 = (Base‘𝑆)
lmhmlin.m · = ( ·𝑠𝑆)
lmhmlin.n × = ( ·𝑠𝑇)
Assertion
Ref Expression
lmhmlin ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝐵𝑌𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌)))

Proof of Theorem lmhmlin
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmlin.k . . . . . 6 𝐾 = (Scalar‘𝑆)
2 eqid 2732 . . . . . 6 (Scalar‘𝑇) = (Scalar‘𝑇)
3 lmhmlin.b . . . . . 6 𝐵 = (Base‘𝐾)
4 lmhmlin.e . . . . . 6 𝐸 = (Base‘𝑆)
5 lmhmlin.m . . . . . 6 · = ( ·𝑠𝑆)
6 lmhmlin.n . . . . . 6 × = ( ·𝑠𝑇)
71, 2, 3, 4, 5, 6islmhm 20630 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = 𝐾 ∧ ∀𝑎𝐵𝑏𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)))))
87simprbi 497 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = 𝐾 ∧ ∀𝑎𝐵𝑏𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏))))
98simp3d 1144 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → ∀𝑎𝐵𝑏𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)))
10 fvoveq1 7428 . . . . 5 (𝑎 = 𝑋 → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑋 · 𝑏)))
11 oveq1 7412 . . . . 5 (𝑎 = 𝑋 → (𝑎 × (𝐹𝑏)) = (𝑋 × (𝐹𝑏)))
1210, 11eqeq12d 2748 . . . 4 (𝑎 = 𝑋 → ((𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)) ↔ (𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹𝑏))))
13 oveq2 7413 . . . . . 6 (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
1413fveq2d 6892 . . . . 5 (𝑏 = 𝑌 → (𝐹‘(𝑋 · 𝑏)) = (𝐹‘(𝑋 · 𝑌)))
15 fveq2 6888 . . . . . 6 (𝑏 = 𝑌 → (𝐹𝑏) = (𝐹𝑌))
1615oveq2d 7421 . . . . 5 (𝑏 = 𝑌 → (𝑋 × (𝐹𝑏)) = (𝑋 × (𝐹𝑌)))
1714, 16eqeq12d 2748 . . . 4 (𝑏 = 𝑌 → ((𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹𝑏)) ↔ (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌))))
1812, 17rspc2v 3621 . . 3 ((𝑋𝐵𝑌𝐸) → (∀𝑎𝐵𝑏𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌))))
199, 18syl5com 31 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑋𝐵𝑌𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌))))
20193impib 1116 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝐵𝑌𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  cfv 6540  (class class class)co 7405  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197   GrpHom cghm 19083  LModclmod 20463   LMHom clmhm 20622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-lmhm 20625
This theorem is referenced by:  islmhm2  20641  lmhmco  20646  lmhmplusg  20647  lmhmvsca  20648  lmhmf1o  20649  lmhmima  20650  lmhmpreima  20651  reslmhm  20655  reslmhm2  20656  reslmhm2b  20657  lmhmeql  20658  ipass  21189  lindfmm  21373  nmoleub2lem3  24622  nmoleub3  24626  lmhmqusker  32522  mendassa  41921
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