Theorem islmhm2 19038

Description: A one-equation proof of linearity of a left module homomorphism, similar to df-lss 18933. (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 19034	. . . 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 19030	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐿 = 𝐾)
7		lmghm 19031	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
8	7	adantr 481	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9		lmhmlmod1 19033	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
10	9	adantr 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑆 ∈ LMod)
11		simpr1 1067	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐸)
12		simpr2 1068	. . . . . . . 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 18880	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
16	10, 11, 12, 15	syl3anc 1326	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
17		simpr3 1069	. . . . . . 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 17665	. . . . . . 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 1326	. . . . . 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 19035	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
24	23	3adant3r3 1276	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))
25	24	oveq1d 6665	. . . . . 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 2656	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))
27	26	ralrimivvva 2972	. . . 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 1242	. . 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 482	. 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 18870	. . . . . 6 ⊢ (𝑆 ∈ LMod → 𝑆 ∈ Grp)
31		lmodgrp 18870	. . . . . 6 ⊢ (𝑇 ∈ LMod → 𝑇 ∈ Grp)
32	30, 31	anim12i 590	. . . . 5 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
33	32	adantr 481	. . . 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 1067	. . . . 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 18871	. . . . . . . . . 10 ⊢ (𝑆 ∈ LMod → 𝐾 ∈ Ring)
36	35	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) → 𝐾 ∈ Ring)
37		eqid 2622	. . . . . . . . . 10 ⊢ (1r‘𝐾) = (1r‘𝐾)
38	14, 37	ringidcl 18568	. . . . . . . . 9 ⊢ (𝐾 ∈ Ring → (1r‘𝐾) ∈ 𝐸)
39		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (1r‘𝐾) → (𝑥 · 𝑦) = ((1r‘𝐾) · 𝑦))
40	39	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1r‘𝐾) → ((𝑥 · 𝑦) + 𝑧) = (((1r‘𝐾) · 𝑦) + 𝑧))
41	40	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑥 = (1r‘𝐾) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = (𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)))
42		oveq1 6657	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1r‘𝐾) → (𝑥 × (𝐹‘𝑦)) = ((1r‘𝐾) × (𝐹‘𝑦)))
43	42	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (𝑥 = (1r‘𝐾) → ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) = (((1r‘𝐾) × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)))
44	41, 43	eqeq12d 2637	. . . . . . . . . . 11 ⊢ (𝑥 = (1r‘𝐾) → ((𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) ↔ (𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)) = (((1r‘𝐾) × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧))))
45	44	2ralbidv 2989	. . . . . . . . . 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))))
46	45	rspcv 3305	. . . . . . . . 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))))
47	36, 38, 46	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))))
48		simplll 798	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑆 ∈ LMod)
49		simprl 794	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
50	1, 4, 13, 37	lmodvs1 18891	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ LMod ∧ 𝑦 ∈ 𝐵) → ((1r‘𝐾) · 𝑦) = 𝑦)
51	48, 49, 50	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((1r‘𝐾) · 𝑦) = 𝑦)
52	51	oveq1d 6665	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (((1r‘𝐾) · 𝑦) + 𝑧) = (𝑦 + 𝑧))
53	52	fveq2d 6195	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘(((1r‘𝐾) · 𝑦) + 𝑧)) = (𝐹‘(𝑦 + 𝑧)))
54		simplrr 801	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐿 = 𝐾)
55	54	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (1r‘𝐿) = (1r‘𝐾))
56	55	oveq1d 6665	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((1r‘𝐿) × (𝐹‘𝑦)) = ((1r‘𝐾) × (𝐹‘𝑦)))
57		simpllr 799	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑇 ∈ LMod)
58		simplrl 800	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐹:𝐵⟶𝐶)
59	58, 49	ffvelrnd 6360	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐹‘𝑦) ∈ 𝐶)
60		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (1r‘𝐿) = (1r‘𝐿)
61	2, 5, 22, 60	lmodvs1 18891	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ LMod ∧ (𝐹‘𝑦) ∈ 𝐶) → ((1r‘𝐿) × (𝐹‘𝑦)) = (𝐹‘𝑦))
62	57, 59, 61	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((1r‘𝐿) × (𝐹‘𝑦)) = (𝐹‘𝑦))
63	56, 62	eqtr3d 2658	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((1r‘𝐾) × (𝐹‘𝑦)) = (𝐹‘𝑦))
64	63	oveq1d 6665	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (((1r‘𝐾) × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) = ((𝐹‘𝑦) ⨣ (𝐹‘𝑧)))
65	53, 64	eqeq12d 2637	. . . . . . . . 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))))
66	65	2ralbidva 2988	. . . . . . . 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))))
67	47, 66	sylibd 229	. . . . . . 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))))
68	67	exp32 631	. . . . . 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))))))
69	68	3imp2 1282	. . . . 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)))
70	34, 69	jca 554	. . . 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))))
71	1, 2, 18, 19	isghm 17660	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹‘𝑦) ⨣ (𝐹‘𝑧)))))
72	33, 70, 71	sylanbrc 698	. . 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))
73		simpr2 1068	. . 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)
74		eqid 2622	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
75		eqid 2622	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
76	74, 75	ghmid 17666	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
77	72, 76	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))
78	30	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ Grp)
79	1, 74	grpidcl 17450	. . . . . . . . . 10 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ 𝐵)
80		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑧 = (0g‘𝑆) → ((𝑥 · 𝑦) + 𝑧) = ((𝑥 · 𝑦) + (0g‘𝑆)))
81	80	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑧 = (0g‘𝑆) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = (𝐹‘((𝑥 · 𝑦) + (0g‘𝑆))))
82		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑧 = (0g‘𝑆) → (𝐹‘𝑧) = (𝐹‘(0g‘𝑆)))
83	82	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑧 = (0g‘𝑆) → ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘(0g‘𝑆))))
84	81, 83	eqeq12d 2637	. . . . . . . . . . 11 ⊢ (𝑧 = (0g‘𝑆) → ((𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧)) ↔ (𝐹‘((𝑥 · 𝑦) + (0g‘𝑆))) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘(0g‘𝑆)))))
85	84	rspcv 3305	. . . . . . . . . 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)))))
86	78, 79, 85	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)))))
87		simplll 798	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ LMod)
88		simprl 794	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐸)
89		simprr 796	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
90	87, 88, 89, 15	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
91	1, 18, 74	grprid 17453	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Grp ∧ (𝑥 · 𝑦) ∈ 𝐵) → ((𝑥 · 𝑦) + (0g‘𝑆)) = (𝑥 · 𝑦))
92	78, 90, 91	syl2anc 693	. . . . . . . . . . 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))
93	92	fveq2d 6195	. . . . . . . . . 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)))
94		simplr3 1105	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
95	94	oveq2d 6666	. . . . . . . . . . 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)))
96		simpllr 799	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑇 ∈ LMod)
97	96, 31	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑇 ∈ Grp)
98		simplr2 1104	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝐿 = 𝐾)
99	98	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (Base‘𝐿) = (Base‘𝐾))
100	99, 14	syl6eqr 2674	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (Base‘𝐿) = 𝐸)
101	88, 100	eleqtrrd 2704	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐿))
102		simplr1 1103	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → 𝐹:𝐵⟶𝐶)
103	102, 89	ffvelrnd 6360	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑦) ∈ 𝐶)
104		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐿) = (Base‘𝐿)
105	2, 5, 22, 104	lmodvscl 18880	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐿) ∧ (𝐹‘𝑦) ∈ 𝐶) → (𝑥 × (𝐹‘𝑦)) ∈ 𝐶)
106	96, 101, 103, 105	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) ∧ (𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐵)) → (𝑥 × (𝐹‘𝑦)) ∈ 𝐶)
107	2, 19, 75	grprid 17453	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Grp ∧ (𝑥 × (𝐹‘𝑦)) ∈ 𝐶) → ((𝑥 × (𝐹‘𝑦)) ⨣ (0g‘𝑇)) = (𝑥 × (𝐹‘𝑦)))
108	97, 106, 107	syl2anc 693	. . . . . . . . . . 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)))
109	95, 108	eqtrd 2656	. . . . . . . . . 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)))
110	93, 109	eqeq12d 2637	. . . . . . . . 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))))
111	86, 110	sylibd 229	. . . . . . . 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))))
112	111	ralimdvva 2964	. . . . . . 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))))
113	112	3exp2 1285	. . . . . 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)))))))
114	113	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)))))))
115	114	3imp2 1282	. . . 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))))
116	77, 115	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)))
117	4, 5, 14, 1, 13, 22	islmhm3 19028	. . . 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)))))
118	117	adantr 481	. . 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)))))
119	72, 73, 116, 118	mpbir3and 1245	. 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))
120	29, 119	impbida 877	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  Scalarcsca 15944   ·_𝑠 cvsca 15945  0_gc0g 16100  Grpcgrp 17422   GrpHom cghm 17657  1_rcur 18501  Ringcrg 18547  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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lmhm 19022

This theorem is referenced by:  isphld  19999