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Theorem lmhmlin 19237
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 2760 . . . . . 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 19229 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = 𝐾 ∧ ∀𝑎𝐵𝑏𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)))))
87simprbi 483 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = 𝐾 ∧ ∀𝑎𝐵𝑏𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏))))
98simp3d 1139 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → ∀𝑎𝐵𝑏𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)))
10 oveq1 6820 . . . . . 6 (𝑎 = 𝑋 → (𝑎 · 𝑏) = (𝑋 · 𝑏))
1110fveq2d 6356 . . . . 5 (𝑎 = 𝑋 → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑋 · 𝑏)))
12 oveq1 6820 . . . . 5 (𝑎 = 𝑋 → (𝑎 × (𝐹𝑏)) = (𝑋 × (𝐹𝑏)))
1311, 12eqeq12d 2775 . . . 4 (𝑎 = 𝑋 → ((𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)) ↔ (𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹𝑏))))
14 oveq2 6821 . . . . . 6 (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
1514fveq2d 6356 . . . . 5 (𝑏 = 𝑌 → (𝐹‘(𝑋 · 𝑏)) = (𝐹‘(𝑋 · 𝑌)))
16 fveq2 6352 . . . . . 6 (𝑏 = 𝑌 → (𝐹𝑏) = (𝐹𝑌))
1716oveq2d 6829 . . . . 5 (𝑏 = 𝑌 → (𝑋 × (𝐹𝑏)) = (𝑋 × (𝐹𝑌)))
1815, 17eqeq12d 2775 . . . 4 (𝑏 = 𝑌 → ((𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹𝑏)) ↔ (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌))))
1913, 18rspc2v 3461 . . 3 ((𝑋𝐵𝑌𝐸) → (∀𝑎𝐵𝑏𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌))))
209, 19syl5com 31 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑋𝐵𝑌𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌))))
21203impib 1109 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝐵𝑌𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  cfv 6049  (class class class)co 6813  Basecbs 16059  Scalarcsca 16146   ·𝑠 cvsca 16147   GrpHom cghm 17858  LModclmod 19065   LMHom clmhm 19221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-lmhm 19224
This theorem is referenced by:  islmhm2  19240  lmhmco  19245  lmhmplusg  19246  lmhmvsca  19247  lmhmf1o  19248  lmhmima  19249  lmhmpreima  19250  reslmhm  19254  reslmhm2  19255  reslmhm2b  19256  lmhmeql  19257  ipass  20192  lindfmm  20368  nmoleub2lem3  23115  nmoleub3  23119  mendassa  38266
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