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Theorem lmhmlin 19258
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 2761 . . . . . 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 19250 . . . . 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 6822 . . . . . 6 (𝑎 = 𝑋 → (𝑎 · 𝑏) = (𝑋 · 𝑏))
1110fveq2d 6358 . . . . 5 (𝑎 = 𝑋 → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑋 · 𝑏)))
12 oveq1 6822 . . . . 5 (𝑎 = 𝑋 → (𝑎 × (𝐹𝑏)) = (𝑋 × (𝐹𝑏)))
1311, 12eqeq12d 2776 . . . 4 (𝑎 = 𝑋 → ((𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)) ↔ (𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹𝑏))))
14 oveq2 6823 . . . . . 6 (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
1514fveq2d 6358 . . . . 5 (𝑏 = 𝑌 → (𝐹‘(𝑋 · 𝑏)) = (𝐹‘(𝑋 · 𝑌)))
16 fveq2 6354 . . . . . 6 (𝑏 = 𝑌 → (𝐹𝑏) = (𝐹𝑌))
1716oveq2d 6831 . . . . 5 (𝑏 = 𝑌 → (𝑋 × (𝐹𝑏)) = (𝑋 × (𝐹𝑌)))
1815, 17eqeq12d 2776 . . . 4 (𝑏 = 𝑌 → ((𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹𝑏)) ↔ (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌))))
1913, 18rspc2v 3462 . . 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 2140  wral 3051  cfv 6050  (class class class)co 6815  Basecbs 16080  Scalarcsca 16167   ·𝑠 cvsca 16168   GrpHom cghm 17879  LModclmod 19086   LMHom clmhm 19242
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 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056
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 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-iota 6013  df-fun 6052  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-lmhm 19245
This theorem is referenced by:  islmhm2  19261  lmhmco  19266  lmhmplusg  19267  lmhmvsca  19268  lmhmf1o  19269  lmhmima  19270  lmhmpreima  19271  reslmhm  19275  reslmhm2  19276  reslmhm2b  19277  lmhmeql  19278  ipass  20213  lindfmm  20389  nmoleub2lem3  23136  nmoleub3  23140  mendassa  38285
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