Theorem mhmpropd 18809

Description: Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.)

Hypotheses

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

Assertion

Ref	Expression
mhmpropd	⊢ (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 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 mhmpropd

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

Step	Hyp	Ref	Expression
1		mhmpropd.e	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))
2	1	fveq2d 6867	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓‘(𝑥(+g‘𝐽)𝑦)) = (𝑓‘(𝑥(+g‘𝐿)𝑦)))
3	2	adantlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓:𝐵⟶𝐶) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓‘(𝑥(+g‘𝐽)𝑦)) = (𝑓‘(𝑥(+g‘𝐿)𝑦)))
4		ffvelcdm 7058	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐵⟶𝐶 ∧ 𝑥 ∈ 𝐵) → (𝑓‘𝑥) ∈ 𝐶)
5		ffvelcdm 7058	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐵⟶𝐶 ∧ 𝑦 ∈ 𝐵) → (𝑓‘𝑦) ∈ 𝐶)
6	4, 5	anim12dan 628	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐵⟶𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑓‘𝑥) ∈ 𝐶 ∧ (𝑓‘𝑦) ∈ 𝐶))
7		mhmpropd.f	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
8	7	ralrimivva 3204	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
9		oveq1 7399	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (𝑥(+g‘𝐾)𝑦) = (𝑤(+g‘𝐾)𝑦))
10		oveq1 7399	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (𝑥(+g‘𝑀)𝑦) = (𝑤(+g‘𝑀)𝑦))
11	9, 10	eqeq12d 2777	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑤 → ((𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦) ↔ (𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝑀)𝑦)))
12		oveq2 7400	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝐾)𝑧))
13		oveq2 7400	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝑤(+g‘𝑀)𝑦) = (𝑤(+g‘𝑀)𝑧))
14	12, 13	eqeq12d 2777	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → ((𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝑀)𝑦) ↔ (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧)))
15	11, 14	cbvral2vw 3243	. . . . . . . . . . . . . . . . 17 ⊢ (∀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 220	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑤 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧))
17		oveq1 7399	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝐾)𝑧))
18		oveq1 7399	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤(+g‘𝑀)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧))
19	17, 18	eqeq12d 2777	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑓‘𝑥) → ((𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧) ↔ ((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧)))
20		oveq2 7400	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑓‘𝑦) → ((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))
21		oveq2 7400	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑓‘𝑦) → ((𝑓‘𝑥)(+g‘𝑀)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))
22	20, 21	eqeq12d 2777	. . . . . . . . . . . . . . . . 17 ⊢ (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 3593	. . . . . . . . . . . . . . . 16 ⊢ ((((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 606	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓:𝐵⟶𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))
25	24	anassrs 471	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓:𝐵⟶𝐶) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))
26	3, 25	eqeq12d 2777	. . . . . . . . . . . . 13 ⊢ (((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 3223	. . . . . . . . . . . 12 ⊢ ((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 726	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 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		mhmpropd.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = (Base‘𝐽))
30		raleq 3316	. . . . . . . . . . . . . 14 ⊢ (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 3329	. . . . . . . . . . . . 13 ⊢ (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	. . . . . . . . . . . 12 ⊢ (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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 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		mhmpropd.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
35		raleq 3316	. . . . . . . . . . . . . 14 ⊢ (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 3329	. . . . . . . . . . . . 13 ⊢ (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	. . . . . . . . . . . 12 ⊢ (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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 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 311	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 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	29	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐵 = (Base‘𝐽))
41	34	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐵 = (Base‘𝐿))
42	1	adantlr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))
43	40, 41, 42	grpidpropd 18679	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → (0g‘𝐽) = (0g‘𝐿))
44	43	fveq2d 6867	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → (𝑓‘(0g‘𝐽)) = (𝑓‘(0g‘𝐿)))
45		mhmpropd.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 = (Base‘𝐾))
46	45	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐶 = (Base‘𝐾))
47		mhmpropd.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 = (Base‘𝑀))
48	47	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐶 = (Base‘𝑀))
49	7	adantlr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
50	46, 48, 49	grpidpropd 18679	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → (0g‘𝐾) = (0g‘𝑀))
51	44, 50	eqeq12d 2777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → ((𝑓‘(0g‘𝐽)) = (0g‘𝐾) ↔ (𝑓‘(0g‘𝐿)) = (0g‘𝑀)))
52	39, 51	anbi12d 641	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → ((∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐽)) = (0g‘𝐾)) ↔ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀))))
53	52	anassrs 471	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) ∧ 𝑓:𝐵⟶𝐶) → ((∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐽)) = (0g‘𝐾)) ↔ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀))))
54	53	pm5.32da 587	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → ((𝑓:𝐵⟶𝐶 ∧ (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐽)) = (0g‘𝐾))) ↔ (𝑓:𝐵⟶𝐶 ∧ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀)))))
55	29, 45	feq23d 6682	. . . . . . . . 9 ⊢ (𝜑 → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐽)⟶(Base‘𝐾)))
56	55	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐽)⟶(Base‘𝐾)))
57	56	anbi1d 640	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → ((𝑓:𝐵⟶𝐶 ∧ (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐽)) = (0g‘𝐾))) ↔ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐽)) = (0g‘𝐾)))))
58	34, 47	feq23d 6682	. . . . . . . . 9 ⊢ (𝜑 → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐿)⟶(Base‘𝑀)))
59	58	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐿)⟶(Base‘𝑀)))
60	59	anbi1d 640	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → ((𝑓:𝐵⟶𝐶 ∧ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀))) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀)))))
61	54, 57, 60	3bitr3d 311	. . . . . 6 ⊢ ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → ((𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐽)) = (0g‘𝐾))) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀)))))
62		3anass 1105	. . . . . 6 ⊢ ((𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐽)) = (0g‘𝐾)) ↔ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐽)) = (0g‘𝐾))))
63		3anass 1105	. . . . . 6 ⊢ ((𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀)) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀))))
64	61, 62, 63	3bitr4g 316	. . . . 5 ⊢ ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → ((𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐽)) = (0g‘𝐾)) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀))))
65	64	pm5.32da 587	. . . 4 ⊢ (𝜑 → (((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐽)) = (0g‘𝐾))) ↔ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀)))))
66	29, 34, 1	mndpropd 18776	. . . . . 6 ⊢ (𝜑 → (𝐽 ∈ Mnd ↔ 𝐿 ∈ Mnd))
67	45, 47, 7	mndpropd 18776	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝑀 ∈ Mnd))
68	66, 67	anbi12d 641	. . . . 5 ⊢ (𝜑 → ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ↔ (𝐿 ∈ Mnd ∧ 𝑀 ∈ Mnd)))
69	68	anbi1d 640	. . . 4 ⊢ (𝜑 → (((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀))) ↔ ((𝐿 ∈ Mnd ∧ 𝑀 ∈ Mnd) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀)))))
70	65, 69	bitrd 281	. . 3 ⊢ (𝜑 → (((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐽)) = (0g‘𝐾))) ↔ ((𝐿 ∈ Mnd ∧ 𝑀 ∈ Mnd) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀)))))
71		eqid 2761	. . . 4 ⊢ (Base‘𝐽) = (Base‘𝐽)
72		eqid 2761	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
73		eqid 2761	. . . 4 ⊢ (+g‘𝐽) = (+g‘𝐽)
74		eqid 2761	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
75		eqid 2761	. . . 4 ⊢ (0g‘𝐽) = (0g‘𝐽)
76		eqid 2761	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
77	71, 72, 73, 74, 75, 76	ismhm 18802	. . 3 ⊢ (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐽)) = (0g‘𝐾))))
78		eqid 2761	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
79		eqid 2761	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
80		eqid 2761	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
81		eqid 2761	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
82		eqid 2761	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
83		eqid 2761	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
84	78, 79, 80, 81, 82, 83	ismhm 18802	. . 3 ⊢ (𝑓 ∈ (𝐿 MndHom 𝑀) ↔ ((𝐿 ∈ Mnd ∧ 𝑀 ∈ Mnd) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝐿)) = (0g‘𝑀))))
85	70, 77, 84	3bitr4g 316	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
86	85	eqrdv 2759	1 ⊢ (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  0_gc0g 17451  Mndcmnd 18751   MndHom cmhm 18798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-map 8805  df-0g 17453  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-mhm 18800

This theorem is referenced by:  ghmpropd  19279  pwsco1rhm  20530  pwsco2rhm  20531  pwsdiagrhm  20636  rhmpropd  20638