Theorem lmhmpropd 19845

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 19697	. . . . 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 19697	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝑀 ∈ LMod))
17	8, 16	anbi12d 632	. . . 4 ⊢ (𝜑 → ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ↔ (𝐿 ∈ LMod ∧ 𝑀 ∈ LMod)))
18	7	oveqrspc2v 7183	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑧( ·𝑠 ‘𝐽)𝑤) = (𝑧( ·𝑠 ‘𝐿)𝑤))
19	18	adantlr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑧( ·𝑠 ‘𝐽)𝑤) = (𝑧( ·𝑠 ‘𝐿)𝑤))
20	19	fveq2d 6674	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑓‘(𝑧( ·𝑠 ‘𝐽)𝑤)) = (𝑓‘(𝑧( ·𝑠 ‘𝐿)𝑤)))
21		simpll 765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝜑)
22		simprl 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝑃)
23		simplrr 776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝐺 = 𝐹)
24	23	fveq2d 6674	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (Base‘𝐺) = (Base‘𝐹))
25	24, 14, 6	3eqtr4g 2881	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑄 = 𝑃)
26	22, 25	eleqtrrd 2916	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝑄)
27		simplrl 775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑓 ∈ (𝐽 GrpHom 𝐾))
28		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐽) = (Base‘𝐽)
29		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
30	28, 29	ghmf 18362	. . . . . . . . . . . . 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 771	. . . . . . . . . . . . 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 2915	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ (Base‘𝐽))
35	31, 34	ffvelrnd 6852	. . . . . . . . . . 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 2916	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑓‘𝑤) ∈ 𝐶)
38	15	oveqrspc2v 7183	. . . . . . . . . 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 2837	. . . . . . . 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 3198	. . . . . . 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 581	. . . . . 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 1085	. . . . . 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 1085	. . . . . 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 316	. . . . 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 2837	. . . . . 6 ⊢ (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝐾) = (Scalar‘𝐽)))
47	4	fveq2d 6674	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐽)))
48	6, 47	syl5eq 2868	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐽)))
49	1	raleqdv 3415	. . . . . . 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 3401	. . . . . 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 1435	. . . . 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 18396	. . . . . . 7 ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
53	52	eleq2d 2898	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
54	13, 5	eqeq12d 2837	. . . . . 6 ⊢ (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝑀) = (Scalar‘𝐿)))
55	5	fveq2d 6674	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
56	6, 55	syl5eq 2868	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
57	2	raleqdv 3415	. . . . . . 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 3401	. . . . . 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 1432	. . . . 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 311	. . . 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 632	. . 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 2821	. . . 4 ⊢ (Scalar‘𝐽) = (Scalar‘𝐽)
63		eqid 2821	. . . 4 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
64		eqid 2821	. . . 4 ⊢ (Base‘(Scalar‘𝐽)) = (Base‘(Scalar‘𝐽))
65		eqid 2821	. . . 4 ⊢ ( ·𝑠 ‘𝐽) = ( ·𝑠 ‘𝐽)
66		eqid 2821	. . . 4 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
67	62, 63, 64, 28, 65, 66	islmhm 19799	. . 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 2821	. . . 4 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
69		eqid 2821	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
70		eqid 2821	. . . 4 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
71		eqid 2821	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
72		eqid 2821	. . . 4 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
73		eqid 2821	. . . 4 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
74	68, 69, 70, 71, 72, 73	islmhm 19799	. . 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 316	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 LMHom 𝐾) ↔ 𝑓 ∈ (𝐿 LMHom 𝑀)))
76	75	eqrdv 2819	1 ⊢ (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  +_gcplusg 16565  Scalarcsca 16568   ·_𝑠 cvsca 16569   GrpHom cghm 18355  LModclmod 19634   LMHom clmhm 19791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-plusg 16578  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-grp 18106  df-ghm 18356  df-mgp 19240  df-ur 19252  df-ring 19299  df-lmod 19636  df-lmhm 19794

This theorem is referenced by:  phlpropd  20799