Theorem islmhm2 21060

Description: A one-equation proof of linearity of a left module homomorphism, similar to df-lss 20953. (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 21056	. . . 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 21052	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐿 = 𝐾)
7		lmghm 21053	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
8	7	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9		lmhmlmod1 21055	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑆 ∈ LMod)
11		simpr1 1194	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐸)
12		simpr2 1195	. . . . . . . 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 20898	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
16	10, 11, 12, 15	syl3anc 1371	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
17		simpr3 1196	. . . . . . 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 19261	. . . . . . 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 1371	. . . . . 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 21057	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
24	23	3adant3r3 1184	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
25	24	oveq1d 7463	. . . . . 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 2780	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))
27	26	ralrimivvva 3211	. . . 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 1128	. . 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 481	. 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 20887	. . . . . 6 ⊢ (𝑆 ∈ LMod → 𝑆 ∈ Grp)
31		lmodgrp 20887	. . . . . 6 ⊢ (𝑇 ∈ LMod → 𝑇 ∈ Grp)
32	30, 31	anim12i 612	. . . . 5 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
33	32	adantr 480	. . . 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 1194	. . . . 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 20888	. . . . . . . . . 10 ⊢ (𝑆 ∈ LMod → 𝐾 ∈ Ring)
36	35	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) → 𝐾 ∈ Ring)
37		eqid 2740	. . . . . . . . . 10 ⊢ (1r‘𝐾) = (1r‘𝐾)
38	14, 37	ringidcl 20289	. . . . . . . . 9 ⊢ (𝐾 ∈ Ring → (1r‘𝐾) ∈ 𝐸)
39		oveq1 7455	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1r‘𝐾) → (𝑥 · 𝑦) = ((1r‘𝐾) · 𝑦))
40	39	fvoveq1d 7470	. . . . . . . . . . . 12 ⊢ (𝑥 = (1r‘𝐾) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = (𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)))
41		oveq1 7455	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1r‘𝐾) → (𝑥 × (𝐹‘𝑦)) = ((1r‘𝐾) × (𝐹‘𝑦)))
42	41	oveq1d 7463	. . . . . . . . . . . 12 ⊢ (𝑥 = (1r‘𝐾) → ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) = (((1r‘𝐾) × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))
43	40, 42	eqeq12d 2756	. . . . . . . . . . 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 3631	. . . . . . . . 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 18	. . . . . . . 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 774	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑆 ∈ LMod)
48		simprl 770	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
49	1, 4, 13, 37	lmodvs1 20910	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ LMod ∧ 𝑦 ∈ 𝐵) → ((1r‘𝐾) · 𝑦) = 𝑦)
50	47, 48, 49	syl2anc 583	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((1r‘𝐾) · 𝑦) = 𝑦)
51	50	fvoveq1d 7470	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)) = (𝐹‘(𝑦 + 𝑧)))
52		simplrr 777	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐿 = 𝐾)
53	52	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (1r‘𝐿) = (1r‘𝐾))
54	53	oveq1d 7463	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((1r‘𝐿) × (𝐹‘𝑦)) = ((1r‘𝐾) × (𝐹‘𝑦)))
55		simpllr 775	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑇 ∈ LMod)
56		simplrl 776	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐹:𝐵⟶𝐶)
57	56, 48	ffvelcdmd 7119	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘𝑦) ∈ 𝐶)
58		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (1r‘𝐿) = (1r‘𝐿)
59	2, 5, 22, 58	lmodvs1 20910	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ LMod ∧ (𝐹‘𝑦) ∈ 𝐶) → ((1r‘𝐿) × (𝐹‘𝑦)) = (𝐹‘𝑦))
60	55, 57, 59	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((1r‘𝐿) × (𝐹‘𝑦)) = (𝐹‘𝑦))
61	54, 60	eqtr3d 2782	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((1r‘𝐾) × (𝐹‘𝑦)) = (𝐹‘𝑦))
62	61	oveq1d 7463	. . . . . . . . . 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 2756	. . . . . . . . 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 239	. . . . . . 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 420	. . . . . 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 1349	. . . . 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 511	. . . 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 19255	. . . 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 582	. . 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 1195	. . 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 2740	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
73		eqid 2740	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
74	72, 73	ghmid 19262	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
75	70, 74	syl 17	. . . 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 729	. . . . . . . . . 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 19005	. . . . . . . . . 10 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ 𝐵)
78		oveq2 7456	. . . . . . . . . . . . 13 ⊢ (𝑧 = (0g‘𝑆) → ((𝑥 · 𝑦) + 𝑧) = ((𝑥 · 𝑦) + (0g‘𝑆)))
79	78	fveq2d 6924	. . . . . . . . . . . 12 ⊢ (𝑧 = (0g‘𝑆) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = (𝐹‘((𝑥 · 𝑦) + (0g‘𝑆))))
80		fveq2 6920	. . . . . . . . . . . . 13 ⊢ (𝑧 = (0g‘𝑆) → (𝐹‘𝑧) = (𝐹‘(0g‘𝑆)))
81	80	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (𝑧 = (0g‘𝑆) → ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘(0g‘𝑆))))
82	79, 81	eqeq12d 2756	. . . . . . . . . . 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 3631	. . . . . . . . . 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 18	. . . . . . . . 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 774	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ LMod)
86		simprl 770	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐸)
87		simprr 772	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 19008	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Grp ∧ (𝑥 · 𝑦) ∈ 𝐵) → ((𝑥 · 𝑦) + (0g‘𝑆)) = (𝑥 · 𝑦))
90	76, 88, 89	syl2anc 583	. . . . . . . . . . 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 6924	. . . . . . . . . 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 1217	. . . . . . . . . . . 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 7464	. . . . . . . . . . 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 775	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑇 ∈ Grp)
96		simplr2 1216	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝐿 = 𝐾)
97	96	fveq2d 6924	. . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . 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 2847	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐿))
100		simplr1 1215	. . . . . . . . . . . . . 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 7119	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑦) ∈ 𝐶)
102		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐿) = (Base‘𝐿)
103	2, 5, 22, 102	lmodvscl 20898	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐿) ∧ (𝐹‘𝑦) ∈ 𝐶) → (𝑥 × (𝐹‘𝑦)) ∈ 𝐶)
104	94, 99, 101, 103	syl3anc 1371	. . . . . . . . . . . 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 19008	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Grp ∧ (𝑥 × (𝐹‘𝑦)) ∈ 𝐶) → ((𝑥 × (𝐹‘𝑦)) ⨣ (0g‘𝑇)) = (𝑥 × (𝐹‘𝑦)))
106	95, 104, 105	syl2anc 583	. . . . . . . . . . 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 2780	. . . . . . . . . 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 2756	. . . . . . . . 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 239	. . . . . . . 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 3212	. . . . . . 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 1354	. . . . . 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 97	. . . . 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 1349	. . . 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 15	. . 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 21050	. . . 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 480	. . 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 1342	. 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 800	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)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17499  Grpcgrp 18973   GrpHom cghm 19252  1_rcur 20208  Ringcrg 20260  LModclmod 20880   LMHom clmhm 21041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-ghm 19253  df-mgp 20162  df-ur 20209  df-ring 20262  df-lmod 20882  df-lmhm 21044

This theorem is referenced by:  isphld  21695