Theorem islmhm2 21293

Description: A one-equation proof of linearity of a left module homomorphism, similar to df-lss 21187. (Contributed by Mario Carneiro, 7-Oct-2015.)

Hypotheses

Ref	Expression
islmhm2.b	⊢ 𝐵 = (Base‘𝑆)
islmhm2.c	⊢ 𝐶 = (Base‘𝑇)
islmhm2.k	⊢ 𝐾 = (Scalar‘𝑆)
islmhm2.l	⊢ 𝐿 = (Scalar‘𝑇)
islmhm2.e	⊢ 𝐸 = (Base‘𝐾)
islmhm2.p	⊢ + = (+g‘𝑆)
islmhm2.q	⊢ ⨣ = (+g‘𝑇)
islmhm2.m	⊢ · = ( ·𝑠 ‘𝑆)
islmhm2.n	⊢ × = ( ·𝑠 ‘𝑇)

Assertion

Ref	Expression
islmhm2	⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))))

Distinct variable groups:   𝑥,𝑦,𝑧, ⨣   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥, + ,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝐿,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧   𝑥, · ,𝑧   𝑥, × ,𝑧

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

Proof of Theorem islmhm2

Step	Hyp	Ref	Expression
1		islmhm2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
2		islmhm2.c	. . . . 5 ⊢ 𝐶 = (Base‘𝑇)
3	1, 2	lmhmf 21289	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝐵⟶𝐶)
4		islmhm2.k	. . . . 5 ⊢ 𝐾 = (Scalar‘𝑆)
5		islmhm2.l	. . . . 5 ⊢ 𝐿 = (Scalar‘𝑇)
6	4, 5	lmhmsca 21285	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐿 = 𝐾)
7		lmghm 21286	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
8	7	adantr 486	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9		lmhmlmod1 21288	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
10	9	adantr 486	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑆 ∈ LMod)
11		simpr1 1213	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐸)
12		simpr2 1214	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
13		islmhm2.m	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑆)
14		islmhm2.e	. . . . . . . . 9 ⊢ 𝐸 = (Base‘𝐾)
15	1, 4, 13, 14	lmodvscl 21133	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
16	10, 11, 12, 15	syl3anc 1398	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
17		simpr3 1215	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
18		islmhm2.p	. . . . . . . 8 ⊢ + = (+g‘𝑆)
19		islmhm2.q	. . . . . . . 8 ⊢ ⨣ = (+g‘𝑇)
20	1, 18, 19	ghmlin 19415	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑥 · 𝑦) ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝐹‘(𝑥 · 𝑦)) ⨣ (𝐹‘𝑧)))
21	8, 16, 17, 20	syl3anc 1398	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝐹‘(𝑥 · 𝑦)) ⨣ (𝐹‘𝑧)))
22		islmhm2.n	. . . . . . . . 9 ⊢ × = ( ·𝑠 ‘𝑇)
23	4, 14, 1, 13, 22	lmhmlin 21290	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
24	23	3adant3r3 1203	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
25	24	oveq1d 7427	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝐹‘(𝑥 · 𝑦)) ⨣ (𝐹‘𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))
26	21, 25	eqtrd 2796	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))
27	26	ralrimivvva 3209	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))
28	3, 6, 27	3jca 1146	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧))))
29	28	adantl 487	. 2 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧))))
30		lmodgrp 21122	. . . . . 6 ⊢ (𝑆 ∈ LMod → 𝑆 ∈ Grp)
31		lmodgrp 21122	. . . . . 6 ⊢ (𝑇 ∈ LMod → 𝑇 ∈ Grp)
32	30, 31	anim12i 625	. . . . 5 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
33	32	adantr 486	. . . 4 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
34		simpr1 1213	. . . . 5 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))) → 𝐹:𝐵⟶𝐶)
35	4	lmodring 21123	. . . . . . . . . 10 ⊢ (𝑆 ∈ LMod → 𝐾 ∈ Ring)
36	35	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) → 𝐾 ∈ Ring)
37		eqid 2761	. . . . . . . . . 10 ⊢ (1r‘𝐾) = (1r‘𝐾)
38	14, 37	ringidcl 20474	. . . . . . . . 9 ⊢ (𝐾 ∈ Ring → (1r‘𝐾) ∈ 𝐸)
39		oveq1 7419	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1r‘𝐾) → (𝑥 · 𝑦) = ((1r‘𝐾) · 𝑦))
40	39	fvoveq1d 7434	. . . . . . . . . . . 12 ⊢ (𝑥 = (1r‘𝐾) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = (𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)))
41		oveq1 7419	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1r‘𝐾) → (𝑥 × (𝐹‘𝑦)) = ((1r‘𝐾) × (𝐹‘𝑦)))
42	41	oveq1d 7427	. . . . . . . . . . . 12 ⊢ (𝑥 = (1r‘𝐾) → ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) = (((1r‘𝐾) × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))
43	40, 42	eqeq12d 2777	. . . . . . . . . . 11 ⊢ (𝑥 = (1r‘𝐾) → ((𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) ↔ (𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)) = (((1r‘𝐾) × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧))))
44	43	2ralbidv 3227	. . . . . . . . . 10 ⊢ (𝑥 = (1r‘𝐾) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)) = (((1r‘𝐾) × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧))))
45	44	rspcv 3573	. . . . . . . . 9 ⊢ ((1r‘𝐾) ∈ 𝐸 → (∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)) = (((1r‘𝐾) × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧))))
46	36, 38, 45	3syl 19	. . . . . . . 8 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) → (∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)) = (((1r‘𝐾) × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧))))
47		simplll 787	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑆 ∈ LMod)
48		simprl 783	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
49	1, 4, 13, 37	lmodvs1 21145	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ LMod ∧ 𝑦 ∈ 𝐵) → ((1r‘𝐾) · 𝑦) = 𝑦)
50	47, 48, 49	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((1r‘𝐾) · 𝑦) = 𝑦)
51	50	fvoveq1d 7434	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)) = (𝐹‘(𝑦 + 𝑧)))
52		simplrr 790	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐿 = 𝐾)
53	52	fveq2d 6881	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (1r‘𝐿) = (1r‘𝐾))
54	53	oveq1d 7427	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((1r‘𝐿) × (𝐹‘𝑦)) = ((1r‘𝐾) × (𝐹‘𝑦)))
55		simpllr 788	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑇 ∈ LMod)
56		simplrl 789	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐹:𝐵⟶𝐶)
57	56, 48	ffvelcdmd 7077	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘𝑦) ∈ 𝐶)
58		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (1r‘𝐿) = (1r‘𝐿)
59	2, 5, 22, 58	lmodvs1 21145	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ LMod ∧ (𝐹‘𝑦) ∈ 𝐶) → ((1r‘𝐿) × (𝐹‘𝑦)) = (𝐹‘𝑦))
60	55, 57, 59	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((1r‘𝐿) × (𝐹‘𝑦)) = (𝐹‘𝑦))
61	54, 60	eqtr3d 2798	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((1r‘𝐾) × (𝐹‘𝑦)) = (𝐹‘𝑦))
62	61	oveq1d 7427	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (((1r‘𝐾) × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) = ((𝐹‘𝑦) ⨣ (𝐹‘𝑧)))
63	51, 62	eqeq12d 2777	. . . . . . . . 9 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)) = (((1r‘𝐾) × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) ↔ (𝐹‘(𝑦 + 𝑧)) = ((𝐹‘𝑦) ⨣ (𝐹‘𝑧))))
64	63	2ralbidva 3225	. . . . . . . 8 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)) = (((1r‘𝐾) × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹‘𝑦) ⨣ (𝐹‘𝑧))))
65	46, 64	sylibd 242	. . . . . . 7 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) → (∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹‘𝑦) ⨣ (𝐹‘𝑧))))
66	65	exp32 426	. . . . . 6 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹:𝐵⟶𝐶 → (𝐿 = 𝐾 → (∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹‘𝑦) ⨣ (𝐹‘𝑧))))))
67	66	3imp2 1368	. . . . 5 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹‘𝑦) ⨣ (𝐹‘𝑧)))
68	34, 67	jca 521	. . . 4 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))) → (𝐹:𝐵⟶𝐶 ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹‘𝑦) ⨣ (𝐹‘𝑧))))
69	1, 2, 18, 19	isghm 19410	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹‘𝑦) ⨣ (𝐹‘𝑧)))))
70	33, 68, 69	sylanbrc 595	. . 3 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
71		simpr2 1214	. . 3 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))) → 𝐿 = 𝐾)
72		eqid 2761	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
73		eqid 2761	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
74	72, 73	ghmid 19416	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
75	70, 74	syl 18	. . . 4 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
76	30	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ Grp)
77	1, 72	grpidcl 19156	. . . . . . . . . 10 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ 𝐵)
78		oveq2 7420	. . . . . . . . . . . . 13 ⊢ (𝑧 = (0g‘𝑆) → ((𝑥 · 𝑦) + 𝑧) = ((𝑥 · 𝑦) + (0g‘𝑆)))
79	78	fveq2d 6881	. . . . . . . . . . . 12 ⊢ (𝑧 = (0g‘𝑆) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = (𝐹‘((𝑥 · 𝑦) + (0g‘𝑆))))
80		fveq2 6877	. . . . . . . . . . . . 13 ⊢ (𝑧 = (0g‘𝑆) → (𝐹‘𝑧) = (𝐹‘(0g‘𝑆)))
81	80	oveq2d 7428	. . . . . . . . . . . 12 ⊢ (𝑧 = (0g‘𝑆) → ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘(0g‘𝑆))))
82	79, 81	eqeq12d 2777	. . . . . . . . . . 11 ⊢ (𝑧 = (0g‘𝑆) → ((𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) ↔ (𝐹‘((𝑥 · 𝑦) + (0g‘𝑆))) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘(0g‘𝑆)))))
83	82	rspcv 3573	. . . . . . . . . 10 ⊢ ((0g‘𝑆) ∈ 𝐵 → (∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) → (𝐹‘((𝑥 · 𝑦) + (0g‘𝑆))) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘(0g‘𝑆)))))
84	76, 77, 83	3syl 19	. . . . . . . . 9 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) → (𝐹‘((𝑥 · 𝑦) + (0g‘𝑆))) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘(0g‘𝑆)))))
85		simplll 787	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ LMod)
86		simprl 783	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐸)
87		simprr 785	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
88	85, 86, 87, 15	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
89	1, 18, 72	grprid 19159	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Grp ∧ (𝑥 · 𝑦) ∈ 𝐵) → ((𝑥 · 𝑦) + (0g‘𝑆)) = (𝑥 · 𝑦))
90	76, 88, 89	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 · 𝑦) + (0g‘𝑆)) = (𝑥 · 𝑦))
91	90	fveq2d 6881	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘((𝑥 · 𝑦) + (0g‘𝑆))) = (𝐹‘(𝑥 · 𝑦)))
92		simplr3 1236	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
93	92	oveq2d 7428	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘(0g‘𝑆))) = ((𝑥 × (𝐹‘𝑦)) ⨣ (0g‘𝑇)))
94		simpllr 788	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑇 ∈ LMod)
95	94, 31	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑇 ∈ Grp)
96		simplr2 1235	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝐿 = 𝐾)
97	96	fveq2d 6881	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (Base‘𝐿) = (Base‘𝐾))
98	97, 14	eqtr4di 2814	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (Base‘𝐿) = 𝐸)
99	86, 98	eleqtrrd 2864	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐿))
100		simplr1 1234	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝐹:𝐵⟶𝐶)
101	100, 87	ffvelcdmd 7077	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑦) ∈ 𝐶)
102		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐿) = (Base‘𝐿)
103	2, 5, 22, 102	lmodvscl 21133	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐿) ∧ (𝐹‘𝑦) ∈ 𝐶) → (𝑥 × (𝐹‘𝑦)) ∈ 𝐶)
104	94, 99, 101, 103	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (𝑥 × (𝐹‘𝑦)) ∈ 𝐶)
105	2, 19, 73	grprid 19159	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Grp ∧ (𝑥 × (𝐹‘𝑦)) ∈ 𝐶) → ((𝑥 × (𝐹‘𝑦)) ⨣ (0g‘𝑇)) = (𝑥 × (𝐹‘𝑦)))
106	95, 104, 105	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 × (𝐹‘𝑦)) ⨣ (0g‘𝑇)) = (𝑥 × (𝐹‘𝑦)))
107	93, 106	eqtrd 2796	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘(0g‘𝑆))) = (𝑥 × (𝐹‘𝑦)))
108	91, 107	eqeq12d 2777	. . . . . . . . 9 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘((𝑥 · 𝑦) + (0g‘𝑆))) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘(0g‘𝑆))) ↔ (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))
109	84, 108	sylibd 242	. . . . . . . 8 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))
110	109	ralimdvva 3210	. . . . . . 7 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) → (∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) → ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))
111	110	3exp2 1373	. . . . . 6 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹:𝐵⟶𝐶 → (𝐿 = 𝐾 → ((𝐹‘(0g‘𝑆)) = (0g‘𝑇) → (∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) → ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))))))
112	111	com45 98	. . . . 5 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹:𝐵⟶𝐶 → (𝐿 = 𝐾 → (∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) → ((𝐹‘(0g‘𝑆)) = (0g‘𝑇) → ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))))))
113	112	3imp2 1368	. . . 4 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))) → ((𝐹‘(0g‘𝑆)) = (0g‘𝑇) → ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))
114	75, 113	mpd 16	. . 3 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))) → ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
115	4, 5, 14, 1, 13, 22	islmhm3 21283	. . . 4 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))))
116	115	adantr 486	. . 3 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))))
117	70, 71, 114, 116	mpbir3and 1361	. 2 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
118	29, 117	impbida 813	1 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  +_gcplusg 17408  Scalarcsca 17411   ·_𝑠 cvsca 17412  0_gc0g 17590  Grpcgrp 19124   GrpHom cghm 19407  1_rcur 20387  Ringcrg 20439  LModclmod 21115   LMHom clmhm 21274

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-plusg 17421  df-0g 17592  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-grp 19127  df-ghm 19408  df-mgp 20341  df-ur 20388  df-ring 20441  df-lmod 21117  df-lmhm 21277

This theorem is used by:  isphld  21940