Theorem islmhmd 20931

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 3198	. . 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 20919	. 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3058  ‘cfv 6553  (class class class)co 7426  Basecbs 17187  Scalarcsca 17243   ·_𝑠 cvsca 17244   GrpHom cghm 19174  LModclmod 20750   LMHom clmhm 20911

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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-lmhm 20914

This theorem is referenced by:  0lmhm  20932  idlmhm  20933  invlmhm  20934  lmhmco  20935  lmhmplusg  20936  lmhmvsca  20937  lmhmf1o  20938  reslmhm2  20945  reslmhm2b  20946  pwsdiaglmhm  20949  pwssplit3  20953  frlmup1  21739  imaslmhm  33093  quslmhm  33095  lmhmqusker  33152  frlmsnic  41801