Theorem lmhmpropd 20335

Description: Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.)

Hypotheses

Ref	Expression
lmhmpropd.a	⊢ (𝜑 → 𝐵 = (Base‘𝐽))
lmhmpropd.b	⊢ (𝜑 → 𝐶 = (Base‘𝐾))
lmhmpropd.c	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
lmhmpropd.d	⊢ (𝜑 → 𝐶 = (Base‘𝑀))
lmhmpropd.1	⊢ (𝜑 → 𝐹 = (Scalar‘𝐽))
lmhmpropd.2	⊢ (𝜑 → 𝐺 = (Scalar‘𝐾))
lmhmpropd.3	⊢ (𝜑 → 𝐹 = (Scalar‘𝐿))
lmhmpropd.4	⊢ (𝜑 → 𝐺 = (Scalar‘𝑀))
lmhmpropd.p	⊢ 𝑃 = (Base‘𝐹)
lmhmpropd.q	⊢ 𝑄 = (Base‘𝐺)
lmhmpropd.e	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))
lmhmpropd.f	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
lmhmpropd.g	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐽)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
lmhmpropd.h	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝐶)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝑀)𝑦))

Assertion

Ref	Expression
lmhmpropd	⊢ (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))

Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑀,𝑦   𝑥,𝑃,𝑦   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝑄,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem lmhmpropd

Dummy variables 𝑧 𝑤 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmhmpropd.a	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐽))
2		lmhmpropd.c	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3		lmhmpropd.e	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))
4		lmhmpropd.1	. . . . . 6 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐽))
5		lmhmpropd.3	. . . . . 6 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿))
6		lmhmpropd.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐹)
7		lmhmpropd.g	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐽)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
8	1, 2, 3, 4, 5, 6, 7	lmodpropd 20186	. . . . 5 ⊢ (𝜑 → (𝐽 ∈ LMod ↔ 𝐿 ∈ LMod))
9		lmhmpropd.b	. . . . . 6 ⊢ (𝜑 → 𝐶 = (Base‘𝐾))
10		lmhmpropd.d	. . . . . 6 ⊢ (𝜑 → 𝐶 = (Base‘𝑀))
11		lmhmpropd.f	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
12		lmhmpropd.2	. . . . . 6 ⊢ (𝜑 → 𝐺 = (Scalar‘𝐾))
13		lmhmpropd.4	. . . . . 6 ⊢ (𝜑 → 𝐺 = (Scalar‘𝑀))
14		lmhmpropd.q	. . . . . 6 ⊢ 𝑄 = (Base‘𝐺)
15		lmhmpropd.h	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝐶)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝑀)𝑦))
16	9, 10, 11, 12, 13, 14, 15	lmodpropd 20186	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝑀 ∈ LMod))
17	8, 16	anbi12d 631	. . . 4 ⊢ (𝜑 → ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ↔ (𝐿 ∈ LMod ∧ 𝑀 ∈ LMod)))
18	7	oveqrspc2v 7302	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑧( ·𝑠 ‘𝐽)𝑤) = (𝑧( ·𝑠 ‘𝐿)𝑤))
19	18	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑧( ·𝑠 ‘𝐽)𝑤) = (𝑧( ·𝑠 ‘𝐿)𝑤))
20	19	fveq2d 6778	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)))
21		simpll 764	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝜑)
22		simprl 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝑃)
23		simplrr 775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝐺 = 𝐹)
24	23	fveq2d 6778	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (Base‘𝐺) = (Base‘𝐹))
25	24, 14, 6	3eqtr4g 2803	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑄 = 𝑃)
26	22, 25	eleqtrrd 2842	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝑄)
27		simplrl 774	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑓 ∈ (𝐽 GrpHom 𝐾))
28		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐽) = (Base‘𝐽)
29		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
30	28, 29	ghmf 18838	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝑓:(Base‘𝐽)⟶(Base‘𝐾))
31	27, 30	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑓:(Base‘𝐽)⟶(Base‘𝐾))
32		simprr 770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
33	21, 1	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝐵 = (Base‘𝐽))
34	32, 33	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ (Base‘𝐽))
35	31, 34	ffvelrnd 6962	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑓‘𝑤) ∈ (Base‘𝐾))
36	21, 9	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝐶 = (Base‘𝐾))
37	35, 36	eleqtrrd 2842	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑓‘𝑤) ∈ 𝐶)
38	15	oveqrspc2v 7302	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑄 ∧ (𝑓‘𝑤) ∈ 𝐶)) → (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤)))
39	21, 26, 37, 38	syl12anc 834	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤)))
40	20, 39	eqeq12d 2754	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → ((𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤)) ↔ (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤))))
41	40	2ralbidva 3128	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) → (∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤)) ↔ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤))))
42	41	pm5.32da 579	. . . . . 6 ⊢ (𝜑 → (((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹) ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤))) ↔ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹) ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤)))))
43		df-3an 1088	. . . . . 6 ⊢ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤))) ↔ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹) ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤))))
44		df-3an 1088	. . . . . 6 ⊢ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤))) ↔ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹) ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤))))
45	42, 43, 44	3bitr4g 314	. . . . 5 ⊢ (𝜑 → ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤))) ↔ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤)))))
46	12, 4	eqeq12d 2754	. . . . . 6 ⊢ (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝐾) = (Scalar‘𝐽)))
47	4	fveq2d 6778	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐽)))
48	6, 47	eqtrid 2790	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐽)))
49	1	raleqdv 3348	. . . . . . 7 ⊢ (𝜑 → (∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤))))
50	48, 49	raleqbidv 3336	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤)) ↔ ∀𝑧 ∈ (Base‘(Scalar‘𝐽))∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤))))
51	46, 50	3anbi23d 1438	. . . . 5 ⊢ (𝜑 → ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤))) ↔ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ (Scalar‘𝐾) = (Scalar‘𝐽) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐽))∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤)))))
52	1, 9, 2, 10, 3, 11	ghmpropd 18872	. . . . . . 7 ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
53	52	eleq2d 2824	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
54	13, 5	eqeq12d 2754	. . . . . 6 ⊢ (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝑀) = (Scalar‘𝐿)))
55	5	fveq2d 6778	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
56	6, 55	eqtrid 2790	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
57	2	raleqdv 3348	. . . . . . 7 ⊢ (𝜑 → (∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤))))
58	56, 57	raleqbidv 3336	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤)) ↔ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤))))
59	53, 54, 58	3anbi123d 1435	. . . . 5 ⊢ (𝜑 → ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝐵 (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤))) ↔ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ (Scalar‘𝑀) = (Scalar‘𝐿) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤)))))
60	45, 51, 59	3bitr3d 309	. . . 4 ⊢ (𝜑 → ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ (Scalar‘𝐾) = (Scalar‘𝐽) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐽))∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤))) ↔ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ (Scalar‘𝑀) = (Scalar‘𝐿) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤)))))
61	17, 60	anbi12d 631	. . 3 ⊢ (𝜑 → (((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ (Scalar‘𝐾) = (Scalar‘𝐽) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐽))∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤)))) ↔ ((𝐿 ∈ LMod ∧ 𝑀 ∈ LMod) ∧ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ (Scalar‘𝑀) = (Scalar‘𝐿) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤))))))
62		eqid 2738	. . . 4 ⊢ (Scalar‘𝐽) = (Scalar‘𝐽)
63		eqid 2738	. . . 4 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
64		eqid 2738	. . . 4 ⊢ (Base‘(Scalar‘𝐽)) = (Base‘(Scalar‘𝐽))
65		eqid 2738	. . . 4 ⊢ ( ·𝑠 ‘𝐽) = ( ·𝑠 ‘𝐽)
66		eqid 2738	. . . 4 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
67	62, 63, 64, 28, 65, 66	islmhm 20289	. . 3 ⊢ (𝑓 ∈ (𝐽 LMHom 𝐾) ↔ ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ (Scalar‘𝐾) = (Scalar‘𝐽) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐽))∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤)))))
68		eqid 2738	. . . 4 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
69		eqid 2738	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
70		eqid 2738	. . . 4 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
71		eqid 2738	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
72		eqid 2738	. . . 4 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
73		eqid 2738	. . . 4 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
74	68, 69, 70, 71, 72, 73	islmhm 20289	. . 3 ⊢ (𝑓 ∈ (𝐿 LMHom 𝑀) ↔ ((𝐿 ∈ LMod ∧ 𝑀 ∈ LMod) ∧ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ (Scalar‘𝑀) = (Scalar‘𝐿) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤)))))
75	61, 67, 74	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 LMHom 𝐾) ↔ 𝑓 ∈ (𝐿 LMHom 𝑀)))
76	75	eqrdv 2736	1 ⊢ (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  Scalarcsca 16965   ·_𝑠 cvsca 16966   GrpHom cghm 18831  LModclmod 20123   LMHom clmhm 20281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-plusg 16975  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-grp 18580  df-ghm 18832  df-mgp 19721  df-ur 19738  df-ring 19785  df-lmod 20125  df-lmhm 20284

This theorem is referenced by:  phlpropd  20860