Theorem mhmpropd 18851

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 6887	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓‘(𝑥(+g‘𝐽)𝑦)) = (𝑓‘(𝑥(+g‘𝐿)𝑦)))
3	2	adantlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓:𝐵⟶𝐶) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓‘(𝑥(+g‘𝐽)𝑦)) = (𝑓‘(𝑥(+g‘𝐿)𝑦)))
4		ffvelcdm 7078	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐵⟶𝐶 ∧ 𝑥 ∈ 𝐵) → (𝑓‘𝑥) ∈ 𝐶)
5		ffvelcdm 7078	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐵⟶𝐶 ∧ 𝑦 ∈ 𝐵) → (𝑓‘𝑦) ∈ 𝐶)
6	4, 5	anim12dan 630	. . . . . . . . . . . . . . . 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 3208	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
9		oveq1 7419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (𝑥(+g‘𝐾)𝑦) = (𝑤(+g‘𝐾)𝑦))
10		oveq1 7419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (𝑥(+g‘𝑀)𝑦) = (𝑤(+g‘𝑀)𝑦))
11	9, 10	eqeq12d 2779	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑤 → ((𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦) ↔ (𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝑀)𝑦)))
12		oveq2 7420	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝐾)𝑧))
13		oveq2 7420	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝑤(+g‘𝑀)𝑦) = (𝑤(+g‘𝑀)𝑧))
14	12, 13	eqeq12d 2779	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → ((𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝑀)𝑦) ↔ (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧)))
15	11, 14	cbvral2vw 3247	. . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑤 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧))
17		oveq1 7419	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝐾)𝑧))
18		oveq1 7419	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤(+g‘𝑀)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧))
19	17, 18	eqeq12d 2779	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑓‘𝑥) → ((𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧) ↔ ((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧)))
20		oveq2 7420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑓‘𝑦) → ((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))
21		oveq2 7420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑓‘𝑦) → ((𝑓‘𝑥)(+g‘𝑀)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))
22	20, 21	eqeq12d 2779	. . . . . . . . . . . . . . . . 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 3594	. . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓:𝐵⟶𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))
25	24	anassrs 472	. . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . 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 3227	. . . . . . . . . . . 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 728	. . . . . . . . . . 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 3320	. . . . . . . . . . . . . 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 3333	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 485	. . . . . . . . . . 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 3320	. . . . . . . . . . . . . 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 3333	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 485	. . . . . . . . . . 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 312	. . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐵 = (Base‘𝐽))
41	34	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐵 = (Base‘𝐿))
42	1	adantlr 727	. . . . . . . . . . . . 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 18721	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → (0g‘𝐽) = (0g‘𝐿))
44	43	fveq2d 6887	. . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐶 = (Base‘𝐾))
47		mhmpropd.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 = (Base‘𝑀))
48	47	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐶 = (Base‘𝑀))
49	7	adantlr 727	. . . . . . . . . . . 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 18721	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → (0g‘𝐾) = (0g‘𝑀))
51	44, 50	eqeq12d 2779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → ((𝑓‘(0g‘𝐽)) = (0g‘𝐾) ↔ (𝑓‘(0g‘𝐿)) = (0g‘𝑀)))
52	39, 51	anbi12d 643	. . . . . . . . 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 472	. . . . . . . 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 589	. . . . . . 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 6702	. . . . . . . . 9 ⊢ (𝜑 → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐽)⟶(Base‘𝐾)))
56	55	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐽)⟶(Base‘𝐾)))
57	56	anbi1d 642	. . . . . . 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 6702	. . . . . . . . 9 ⊢ (𝜑 → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐿)⟶(Base‘𝑀)))
59	58	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐿)⟶(Base‘𝑀)))
60	59	anbi1d 642	. . . . . . 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 312	. . . . . 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 1111	. . . . . 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 1111	. . . . . 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 317	. . . . 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 589	. . . 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 18818	. . . . . 6 ⊢ (𝜑 → (𝐽 ∈ Mnd ↔ 𝐿 ∈ Mnd))
67	45, 47, 7	mndpropd 18818	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝑀 ∈ Mnd))
68	66, 67	anbi12d 643	. . . . 5 ⊢ (𝜑 → ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ↔ (𝐿 ∈ Mnd ∧ 𝑀 ∈ Mnd)))
69	68	anbi1d 642	. . . 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 282	. . 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 2763	. . . 4 ⊢ (Base‘𝐽) = (Base‘𝐽)
72		eqid 2763	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
73		eqid 2763	. . . 4 ⊢ (+g‘𝐽) = (+g‘𝐽)
74		eqid 2763	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
75		eqid 2763	. . . 4 ⊢ (0g‘𝐽) = (0g‘𝐽)
76		eqid 2763	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
77	71, 72, 73, 74, 75, 76	ismhm 18844	. . 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 2763	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
79		eqid 2763	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
80		eqid 2763	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
81		eqid 2763	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
82		eqid 2763	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
83		eqid 2763	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
84	78, 79, 80, 81, 82, 83	ismhm 18844	. . 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 317	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
86	85	eqrdv 2761	1 ⊢ (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  +_gcplusg 17311  0_gc0g 17493  Mndcmnd 18793   MndHom cmhm 18840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8827  df-0g 17495  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842

This theorem is referenced by:  ghmpropd  19327  pwsco1rhm  20585  pwsco2rhm  20586  pwsdiagrhm  20693  rhmpropd  20695