Theorem lmhmpropd 20110

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 19962	. . . . 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 19962	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝑀 ∈ LMod))
17	8, 16	anbi12d 634	. . . 4 ⊢ (𝜑 → ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ↔ (𝐿 ∈ LMod ∧ 𝑀 ∈ LMod)))
18	7	oveqrspc2v 7240	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑧( ·𝑠 ‘𝐽)𝑤) = (𝑧( ·𝑠 ‘𝐿)𝑤))
19	18	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑧( ·𝑠 ‘𝐽)𝑤) = (𝑧( ·𝑠 ‘𝐿)𝑤))
20	19	fveq2d 6721	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)))
21		simpll 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝜑)
22		simprl 771	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝑃)
23		simplrr 778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝐺 = 𝐹)
24	23	fveq2d 6721	. . . . . . . . . . . 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 2841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝑄)
27		simplrl 777	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑓 ∈ (𝐽 GrpHom 𝐾))
28		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐽) = (Base‘𝐽)
29		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
30	28, 29	ghmf 18626	. . . . . . . . . . . . 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 773	. . . . . . . . . . . . 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 2840	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ (Base‘𝐽))
35	31, 34	ffvelrnd 6905	. . . . . . . . . . 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 2841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑓‘𝑤) ∈ 𝐶)
38	15	oveqrspc2v 7240	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑄 ∧ (𝑓‘𝑤) ∈ 𝐶)) → (𝑧( ·𝑠 ‘𝐾)(𝑓‘𝑤)) = (𝑧( ·𝑠 ‘𝑀)(𝑓‘𝑤)))
39	21, 26, 37, 38	syl12anc 837	. . . . . . . . 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 2753	. . . . . . . 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 3119	. . . . . . 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 582	. . . . . 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 1091	. . . . . 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 1091	. . . . . 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 317	. . . . 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 2753	. . . . . 6 ⊢ (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝐾) = (Scalar‘𝐽)))
47	4	fveq2d 6721	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐽)))
48	6, 47	syl5eq 2790	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐽)))
49	1	raleqdv 3325	. . . . . . 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 3313	. . . . . 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 1441	. . . . 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 18660	. . . . . . 7 ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
53	52	eleq2d 2823	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
54	13, 5	eqeq12d 2753	. . . . . 6 ⊢ (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝑀) = (Scalar‘𝐿)))
55	5	fveq2d 6721	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
56	6, 55	syl5eq 2790	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
57	2	raleqdv 3325	. . . . . . 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 3313	. . . . . 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 1438	. . . . 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 312	. . . 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 634	. . 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 2737	. . . 4 ⊢ (Scalar‘𝐽) = (Scalar‘𝐽)
63		eqid 2737	. . . 4 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
64		eqid 2737	. . . 4 ⊢ (Base‘(Scalar‘𝐽)) = (Base‘(Scalar‘𝐽))
65		eqid 2737	. . . 4 ⊢ ( ·𝑠 ‘𝐽) = ( ·𝑠 ‘𝐽)
66		eqid 2737	. . . 4 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
67	62, 63, 64, 28, 65, 66	islmhm 20064	. . 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 2737	. . . 4 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
69		eqid 2737	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
70		eqid 2737	. . . 4 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
71		eqid 2737	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
72		eqid 2737	. . . 4 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
73		eqid 2737	. . . 4 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
74	68, 69, 70, 71, 72, 73	islmhm 20064	. . 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 317	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 LMHom 𝐾) ↔ 𝑓 ∈ (𝐿 LMHom 𝑀)))
76	75	eqrdv 2735	1 ⊢ (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  +_gcplusg 16802  Scalarcsca 16805   ·_𝑠 cvsca 16806   GrpHom cghm 18619  LModclmod 19899   LMHom clmhm 20056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-plusg 16815  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-mhm 18218  df-grp 18368  df-ghm 18620  df-mgp 19505  df-ur 19517  df-ring 19564  df-lmod 19901  df-lmhm 20059

This theorem is referenced by:  phlpropd  20617