Theorem lmhmpropd 21095

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 20945	. . . . 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 20945	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝑀 ∈ LMod))
17	8, 16	anbi12d 631	. . . 4 ⊢ (𝜑 → ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ↔ (𝐿 ∈ LMod ∧ 𝑀 ∈ LMod)))
18	7	oveqrspc2v 7475	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑧( ·𝑠 ‘𝐽)𝑤) = (𝑧( ·𝑠 ‘𝐿)𝑤))
19	18	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑧( ·𝑠 ‘𝐽)𝑤) = (𝑧( ·𝑠 ‘𝐿)𝑤))
20	19	fveq2d 6924	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)))
21		simpll 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝜑)
22		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝑃)
23		simplrr 777	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝐺 = 𝐹)
24	23	fveq2d 6924	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (Base‘𝐺) = (Base‘𝐹))
25	24, 14, 6	3eqtr4g 2805	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑄 = 𝑃)
26	22, 25	eleqtrrd 2847	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝑄)
27		simplrl 776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑓 ∈ (𝐽 GrpHom 𝐾))
28		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐽) = (Base‘𝐽)
29		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
30	28, 29	ghmf 19260	. . . . . . . . . . . . 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 772	. . . . . . . . . . . . 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 2846	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ (Base‘𝐽))
35	31, 34	ffvelcdmd 7119	. . . . . . . . . . 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 2847	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑓‘𝑤) ∈ 𝐶)
38	15	oveqrspc2v 7475	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑄 ∧ (𝑓‘𝑤) ∈ 𝐶)) → (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤)))
39	21, 26, 37, 38	syl12anc 836	. . . . . . . . 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 2756	. . . . . . . 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 3225	. . . . . . 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 578	. . . . . 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 1089	. . . . . 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 1089	. . . . . 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 2756	. . . . . 6 ⊢ (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝐾) = (Scalar‘𝐽)))
47	4	fveq2d 6924	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐽)))
48	6, 47	eqtrid 2792	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐽)))
49	1	raleqdv 3334	. . . . . . 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 3354	. . . . . 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 1439	. . . . 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 19296	. . . . . . 7 ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
53	52	eleq2d 2830	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
54	13, 5	eqeq12d 2756	. . . . . 6 ⊢ (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝑀) = (Scalar‘𝐿)))
55	5	fveq2d 6924	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
56	6, 55	eqtrid 2792	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
57	2	raleqdv 3334	. . . . . . 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 3354	. . . . . 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 1436	. . . . 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 2740	. . . 4 ⊢ (Scalar‘𝐽) = (Scalar‘𝐽)
63		eqid 2740	. . . 4 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
64		eqid 2740	. . . 4 ⊢ (Base‘(Scalar‘𝐽)) = (Base‘(Scalar‘𝐽))
65		eqid 2740	. . . 4 ⊢ ( ·𝑠 ‘𝐽) = ( ·𝑠 ‘𝐽)
66		eqid 2740	. . . 4 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
67	62, 63, 64, 28, 65, 66	islmhm 21049	. . 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 2740	. . . 4 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
69		eqid 2740	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
70		eqid 2740	. . . 4 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
71		eqid 2740	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
72		eqid 2740	. . . 4 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
73		eqid 2740	. . . 4 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
74	68, 69, 70, 71, 72, 73	islmhm 21049	. . 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 2738	1 ⊢ (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))

Syntax hints:   → wi 4   ∧ 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   GrpHom cghm 19252  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-mhm 18818  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:  phlpropd  21696