Theorem islmhmd 21029

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 3182	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
7	3, 4, 6	3jca 1134	. 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 21017	. 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 1351	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  Scalarcsca 17214   ·_𝑠 cvsca 17215   GrpHom cghm 19178  LModclmod 20850   LMHom clmhm 21009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-lmhm 21012

This theorem is referenced by:  0lmhm  21030  idlmhm  21031  invlmhm  21032  lmhmco  21033  lmhmplusg  21034  lmhmvsca  21035  lmhmf1o  21036  reslmhm2  21043  reslmhm2b  21044  pwsdiaglmhm  21047  pwssplit3  21051  frlmup1  21773  imaslmhm  33440  quslmhm  33442  lmhmqusker  33500  frlmsnic  43026