Theorem islmhmd 19039

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 554	. 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 2971	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
8	4, 5, 7	3jca 1242	. 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 19027	. 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 698	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945   GrpHom cghm 17657  LModclmod 18863   LMHom clmhm 19019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-lmhm 19022

This theorem is referenced by:  0lmhm  19040  idlmhm  19041  invlmhm  19042  lmhmco  19043  lmhmplusg  19044  lmhmvsca  19045  lmhmf1o  19046  reslmhm2  19053  reslmhm2b  19054  pwsdiaglmhm  19057  pwssplit3  19061  frlmup1  20137