Theorem islmhmd 19364

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	. . 3 ⊢ (𝜑 → 𝑆 ∈ LMod)
2		islmhmd.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ LMod)
3	1, 2	jca 508	. 2 ⊢ (𝜑 → (𝑆 ∈ LMod ∧ 𝑇 ∈ LMod))
4		islmhmd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇))
5		islmhmd.c	. . 3 ⊢ (𝜑 → 𝐽 = 𝐾)
6		islmhmd.l	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
7	6	ralrimivva 3156	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
8	4, 5, 7	3jca 1159	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐽 = 𝐾 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))
9		islmhmd.k	. . 3 ⊢ 𝐾 = (Scalar‘𝑆)
10		islmhmd.j	. . 3 ⊢ 𝐽 = (Scalar‘𝑇)
11		islmhmd.n	. . 3 ⊢ 𝑁 = (Base‘𝐾)
12		islmhmd.x	. . 3 ⊢ 𝑋 = (Base‘𝑆)
13		islmhmd.a	. . 3 ⊢ · = ( ·𝑠 ‘𝑆)
14		islmhmd.b	. . 3 ⊢ × = ( ·𝑠 ‘𝑇)
15	9, 10, 11, 12, 13, 14	islmhm 19352	. 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)))))
16	3, 8, 15	sylanbrc 579	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  ∀wral 3093  ‘cfv 6105  (class class class)co 6882  Basecbs 16188  Scalarcsca 16274   ·_𝑠 cvsca 16275   GrpHom cghm 17974  LModclmod 19185   LMHom clmhm 19344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ral 3098  df-rex 3099  df-rab 3102  df-v 3391  df-sbc 3638  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-br 4848  df-opab 4910  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-iota 6068  df-fun 6107  df-fv 6113  df-ov 6885  df-oprab 6886  df-mpt2 6887  df-lmhm 19347

This theorem is referenced by:  0lmhm  19365  idlmhm  19366  invlmhm  19367  lmhmco  19368  lmhmplusg  19369  lmhmvsca  19370  lmhmf1o  19371  reslmhm2  19378  reslmhm2b  19379  pwsdiaglmhm  19382  pwssplit3  19386  frlmup1  20466