Theorem islmhmd 20884

Description: Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)

Hypotheses

Ref	Expression
islmhmd.x	⊢ 𝑋 = (Base‘𝑆)
islmhmd.a	⊢ · = ( ·𝑠 ‘𝑆)
islmhmd.b	⊢ × = ( ·𝑠 ‘𝑇)
islmhmd.k	⊢ 𝐾 = (Scalar‘𝑆)
islmhmd.j	⊢ 𝐽 = (Scalar‘𝑇)
islmhmd.n	⊢ 𝑁 = (Base‘𝐾)
islmhmd.s	⊢ (𝜑 → 𝑆 ∈ LMod)
islmhmd.t	⊢ (𝜑 → 𝑇 ∈ LMod)
islmhmd.c	⊢ (𝜑 → 𝐽 = 𝐾)
islmhmd.f	⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇))
islmhmd.l	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))

Assertion

Ref	Expression
islmhmd	⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))

Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝑋,𝑦   𝑥,𝐽,𝑦   𝑥,𝑁,𝑦   𝑥,𝐾,𝑦

Allowed substitution hints:   · (𝑥,𝑦)   × (𝑥,𝑦)

Proof of Theorem islmhmd

Step	Hyp	Ref	Expression
1		islmhmd.s	. 2 ⊢ (𝜑 → 𝑆 ∈ LMod)
2		islmhmd.t	. 2 ⊢ (𝜑 → 𝑇 ∈ LMod)
3		islmhmd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇))
4		islmhmd.c	. . 3 ⊢ (𝜑 → 𝐽 = 𝐾)
5		islmhmd.l	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
6	5	ralrimivva 3194	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
7	3, 4, 6	3jca 1125	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐽 = 𝐾 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))
8		islmhmd.k	. . 3 ⊢ 𝐾 = (Scalar‘𝑆)
9		islmhmd.j	. . 3 ⊢ 𝐽 = (Scalar‘𝑇)
10		islmhmd.n	. . 3 ⊢ 𝑁 = (Base‘𝐾)
11		islmhmd.x	. . 3 ⊢ 𝑋 = (Base‘𝑆)
12		islmhmd.a	. . 3 ⊢ · = ( ·𝑠 ‘𝑆)
13		islmhmd.b	. . 3 ⊢ × = ( ·𝑠 ‘𝑇)
14	8, 9, 10, 11, 12, 13	islmhm 20872	. 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐽 = 𝐾 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))))
15	1, 2, 7, 14	syl21anbrc 1341	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ‘cfv 6536  (class class class)co 7404  Basecbs 17150  Scalarcsca 17206   ·_𝑠 cvsca 17207   GrpHom cghm 19135  LModclmod 20703   LMHom clmhm 20864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-lmhm 20867

This theorem is referenced by:  0lmhm  20885  idlmhm  20886  invlmhm  20887  lmhmco  20888  lmhmplusg  20889  lmhmvsca  20890  lmhmf1o  20891  reslmhm2  20898  reslmhm2b  20899  pwsdiaglmhm  20902  pwssplit3  20906  frlmup1  21688  imaslmhm  32974  quslmhm  32976  lmhmqusker  33039  frlmsnic  41649