Theorem mgmhmpropd 43418

Description: Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.)

Hypotheses

Ref	Expression
mgmhmpropd.a	⊢ (𝜑 → 𝐵 = (Base‘𝐽))
mgmhmpropd.b	⊢ (𝜑 → 𝐶 = (Base‘𝐾))
mgmhmpropd.c	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
mgmhmpropd.d	⊢ (𝜑 → 𝐶 = (Base‘𝑀))
mgmhmpropd.0	⊢ (𝜑 → 𝐵 ≠ ∅)
mgmhmpropd.C	⊢ (𝜑 → 𝐶 ≠ ∅)
mgmhmpropd.e	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))
mgmhmpropd.f	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))

Assertion

Ref	Expression
mgmhmpropd	⊢ (𝜑 → (𝐽 MgmHom 𝐾) = (𝐿 MgmHom 𝑀))

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

Proof of Theorem mgmhmpropd

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

Step	Hyp	Ref	Expression
1		mgmhmpropd.e	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))
2	1	fveq2d 6503	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓‘(𝑥(+g‘𝐽)𝑦)) = (𝑓‘(𝑥(+g‘𝐿)𝑦)))
3	2	adantlr 702	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:𝐵⟶𝐶) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓‘(𝑥(+g‘𝐽)𝑦)) = (𝑓‘(𝑥(+g‘𝐿)𝑦)))
4		ffvelrn 6674	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐵⟶𝐶 ∧ 𝑥 ∈ 𝐵) → (𝑓‘𝑥) ∈ 𝐶)
5		ffvelrn 6674	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐵⟶𝐶 ∧ 𝑦 ∈ 𝐵) → (𝑓‘𝑦) ∈ 𝐶)
6	4, 5	anim12dan 609	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐵⟶𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑓‘𝑥) ∈ 𝐶 ∧ (𝑓‘𝑦) ∈ 𝐶))
7		mgmhmpropd.f	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
8	7	ralrimivva 3142	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
9		oveq1 6983	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → (𝑥(+g‘𝐾)𝑦) = (𝑤(+g‘𝐾)𝑦))
10		oveq1 6983	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → (𝑥(+g‘𝑀)𝑦) = (𝑤(+g‘𝑀)𝑦))
11	9, 10	eqeq12d 2794	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → ((𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦) ↔ (𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝑀)𝑦)))
12		oveq2 6984	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝐾)𝑧))
13		oveq2 6984	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑤(+g‘𝑀)𝑦) = (𝑤(+g‘𝑀)𝑧))
14	12, 13	eqeq12d 2794	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝑀)𝑦) ↔ (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧)))
15	11, 14	cbvral2v 3393	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦) ↔ ∀𝑤 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧))
16	8, 15	sylib 210	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑤 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧))
17		oveq1 6983	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝐾)𝑧))
18		oveq1 6983	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤(+g‘𝑀)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧))
19	17, 18	eqeq12d 2794	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑓‘𝑥) → ((𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧) ↔ ((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧)))
20		oveq2 6984	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑓‘𝑦) → ((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))
21		oveq2 6984	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑓‘𝑦) → ((𝑓‘𝑥)(+g‘𝑀)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))
22	20, 21	eqeq12d 2794	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑓‘𝑦) → (((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧) ↔ ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))
23	19, 22	rspc2va 3550	. . . . . . . . . . . . . 14 ⊢ ((((𝑓‘𝑥) ∈ 𝐶 ∧ (𝑓‘𝑦) ∈ 𝐶) ∧ ∀𝑤 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧)) → ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))
24	6, 16, 23	syl2anr 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓:𝐵⟶𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))
25	24	anassrs 460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:𝐵⟶𝐶) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))
26	3, 25	eqeq12d 2794	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:𝐵⟶𝐶) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))
27	26	2ralbidva 3149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:𝐵⟶𝐶) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))
28	27	adantrl 703	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ 𝑓:𝐵⟶𝐶)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))
29		mgmhmpropd.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (Base‘𝐽))
30		raleq 3346	. . . . . . . . . . . 12 ⊢ (𝐵 = (Base‘𝐽) → (∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))))
31	30	raleqbi1dv 3344	. . . . . . . . . . 11 ⊢ (𝐵 = (Base‘𝐽) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))))
32	29, 31	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))))
33	32	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ 𝑓:𝐵⟶𝐶)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))))
34		mgmhmpropd.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
35		raleq 3346	. . . . . . . . . . . 12 ⊢ (𝐵 = (Base‘𝐿) → (∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))
36	35	raleqbi1dv 3344	. . . . . . . . . . 11 ⊢ (𝐵 = (Base‘𝐿) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))
37	34, 36	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))
38	37	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ 𝑓:𝐵⟶𝐶)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))
39	28, 33, 38	3bitr3d 301	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ 𝑓:𝐵⟶𝐶)) → (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))
40	39	anassrs 460	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) ∧ 𝑓:𝐵⟶𝐶) → (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))
41	40	pm5.32da 571	. . . . . 6 ⊢ ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → ((𝑓:𝐵⟶𝐶 ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))) ↔ (𝑓:𝐵⟶𝐶 ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))))
42		mgmhmpropd.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 = (Base‘𝐾))
43	29, 42	feq23d 6339	. . . . . . . 8 ⊢ (𝜑 → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐽)⟶(Base‘𝐾)))
44	43	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐽)⟶(Base‘𝐾)))
45	44	anbi1d 620	. . . . . 6 ⊢ ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → ((𝑓:𝐵⟶𝐶 ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))) ↔ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))))
46		mgmhmpropd.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 = (Base‘𝑀))
47	34, 46	feq23d 6339	. . . . . . . 8 ⊢ (𝜑 → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐿)⟶(Base‘𝑀)))
48	47	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐿)⟶(Base‘𝑀)))
49	48	anbi1d 620	. . . . . 6 ⊢ ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → ((𝑓:𝐵⟶𝐶 ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))))
50	41, 45, 49	3bitr3d 301	. . . . 5 ⊢ ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → ((𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))))
51	50	pm5.32da 571	. . . 4 ⊢ (𝜑 → (((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))) ↔ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))))
52		mgmhmpropd.0	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ ∅)
53	29, 34, 52, 1	mgmpropd 43408	. . . . . 6 ⊢ (𝜑 → (𝐽 ∈ Mgm ↔ 𝐿 ∈ Mgm))
54		mgmhmpropd.C	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ ∅)
55	42, 46, 54, 7	mgmpropd 43408	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ Mgm ↔ 𝑀 ∈ Mgm))
56	53, 55	anbi12d 621	. . . . 5 ⊢ (𝜑 → ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ↔ (𝐿 ∈ Mgm ∧ 𝑀 ∈ Mgm)))
57	56	anbi1d 620	. . . 4 ⊢ (𝜑 → (((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))) ↔ ((𝐿 ∈ Mgm ∧ 𝑀 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))))
58	51, 57	bitrd 271	. . 3 ⊢ (𝜑 → (((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))) ↔ ((𝐿 ∈ Mgm ∧ 𝑀 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))))
59		eqid 2779	. . . 4 ⊢ (Base‘𝐽) = (Base‘𝐽)
60		eqid 2779	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
61		eqid 2779	. . . 4 ⊢ (+g‘𝐽) = (+g‘𝐽)
62		eqid 2779	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
63	59, 60, 61, 62	ismgmhm 43416	. . 3 ⊢ (𝑓 ∈ (𝐽 MgmHom 𝐾) ↔ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))))
64		eqid 2779	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
65		eqid 2779	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
66		eqid 2779	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
67		eqid 2779	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
68	64, 65, 66, 67	ismgmhm 43416	. . 3 ⊢ (𝑓 ∈ (𝐿 MgmHom 𝑀) ↔ ((𝐿 ∈ Mgm ∧ 𝑀 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))))
69	58, 63, 68	3bitr4g 306	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 MgmHom 𝐾) ↔ 𝑓 ∈ (𝐿 MgmHom 𝑀)))
70	69	eqrdv 2777	1 ⊢ (𝜑 → (𝐽 MgmHom 𝐾) = (𝐿 MgmHom 𝑀))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050   ≠ wne 2968  ∀wral 3089  ∅c0 4179  ⟶wf 6184  ‘cfv 6188  (class class class)co 6976  Basecbs 16339  +_gcplusg 16421  Mgmcmgm 17708   MgmHom cmgmhm 43410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-map 8208  df-mgm 17710  df-mgmhm 43412

This theorem is referenced by: (None)