Theorem mhmpropd 13485

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 5627	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓‘(𝑥(+g‘𝐽)𝑦)) = (𝑓‘(𝑥(+g‘𝐿)𝑦)))
3	2	adantlr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓:𝐵⟶𝐶) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓‘(𝑥(+g‘𝐽)𝑦)) = (𝑓‘(𝑥(+g‘𝐿)𝑦)))
4		ffvelcdm 5761	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐵⟶𝐶 ∧ 𝑥 ∈ 𝐵) → (𝑓‘𝑥) ∈ 𝐶)
5		ffvelcdm 5761	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐵⟶𝐶 ∧ 𝑦 ∈ 𝐵) → (𝑓‘𝑦) ∈ 𝐶)
6	4, 5	anim12dan 602	. . . . . . . . . . . . . . . 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 2612	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
9		oveq1 6001	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (𝑥(+g‘𝐾)𝑦) = (𝑤(+g‘𝐾)𝑦))
10		oveq1 6001	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (𝑥(+g‘𝑀)𝑦) = (𝑤(+g‘𝑀)𝑦))
11	9, 10	eqeq12d 2244	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑤 → ((𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦) ↔ (𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝑀)𝑦)))
12		oveq2 6002	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝐾)𝑧))
13		oveq2 6002	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝑤(+g‘𝑀)𝑦) = (𝑤(+g‘𝑀)𝑧))
14	12, 13	eqeq12d 2244	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → ((𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝑀)𝑦) ↔ (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧)))
15	11, 14	cbvral2vw 2776	. . . . . . . . . . . . . . . . 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 122	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑤 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧))
17		oveq1 6001	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝐾)𝑧))
18		oveq1 6001	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤(+g‘𝑀)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧))
19	17, 18	eqeq12d 2244	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑓‘𝑥) → ((𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧) ↔ ((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧)))
20		oveq2 6002	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑓‘𝑦) → ((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))
21		oveq2 6002	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑓‘𝑦) → ((𝑓‘𝑥)(+g‘𝑀)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))
22	20, 21	eqeq12d 2244	. . . . . . . . . . . . . . . . 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 2921	. . . . . . . . . . . . . . . 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 290	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓:𝐵⟶𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))
25	24	anassrs 400	. . . . . . . . . . . . . 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 2244	. . . . . . . . . . . . 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 2552	. . . . . . . . . . . 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 478	. . . . . . . . . . 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 2728	. . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . 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 14	. . . . . . . . . . . 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 276	. . . . . . . . . . 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 2728	. . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . 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 14	. . . . . . . . . . . 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 276	. . . . . . . . . . 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 218	. . . . . . . . . 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 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐵 = (Base‘𝐽))
41	34	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐵 = (Base‘𝐿))
42		simprll 537	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐽 ∈ Mnd)
43	29, 34, 1	mndpropd 13459	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐽 ∈ Mnd ↔ 𝐿 ∈ Mnd))
44	43	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → (𝐽 ∈ Mnd ↔ 𝐿 ∈ Mnd))
45	42, 44	mpbid 147	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐿 ∈ Mnd)
46	1	adantlr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))
47	40, 41, 42, 45, 46	grpidpropdg 13393	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → (0g‘𝐽) = (0g‘𝐿))
48	47	fveq2d 5627	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → (𝑓‘(0g‘𝐽)) = (𝑓‘(0g‘𝐿)))
49		mhmpropd.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 = (Base‘𝐾))
50	49	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐶 = (Base‘𝐾))
51		mhmpropd.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 = (Base‘𝑀))
52	51	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐶 = (Base‘𝑀))
53		simprlr 538	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝐾 ∈ Mnd)
54	49, 51, 7	mndpropd 13459	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝑀 ∈ Mnd))
55	54	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → (𝐾 ∈ Mnd ↔ 𝑀 ∈ Mnd))
56	53, 55	mpbid 147	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → 𝑀 ∈ Mnd)
57	7	adantlr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
58	50, 52, 53, 56, 57	grpidpropdg 13393	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → (0g‘𝐾) = (0g‘𝑀))
59	48, 58	eqeq12d 2244	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → ((𝑓‘(0g‘𝐽)) = (0g‘𝐾) ↔ (𝑓‘(0g‘𝐿)) = (0g‘𝑀)))
60	39, 59	anbi12d 473	. . . . . . . . 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))))
61	60	anassrs 400	. . . . . . . 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))))
62	61	pm5.32da 452	. . . . . . 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)))))
63	29, 49	feq23d 5465	. . . . . . . . 9 ⊢ (𝜑 → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐽)⟶(Base‘𝐾)))
64	63	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐽)⟶(Base‘𝐾)))
65	64	anbi1d 465	. . . . . . 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)))))
66	34, 51	feq23d 5465	. . . . . . . . 9 ⊢ (𝜑 → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐿)⟶(Base‘𝑀)))
67	66	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐿)⟶(Base‘𝑀)))
68	67	anbi1d 465	. . . . . . 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)))))
69	62, 65, 68	3bitr3d 218	. . . . . 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)))))
70		3anass 1006	. . . . . 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))))
71		3anass 1006	. . . . . 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))))
72	69, 70, 71	3bitr4g 223	. . . . 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))))
73	72	pm5.32da 452	. . . 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)))))
74	43, 54	anbi12d 473	. . . . 5 ⊢ (𝜑 → ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ↔ (𝐿 ∈ Mnd ∧ 𝑀 ∈ Mnd)))
75	74	anbi1d 465	. . . 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)))))
76	73, 75	bitrd 188	. . 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)))))
77		eqid 2229	. . . 4 ⊢ (Base‘𝐽) = (Base‘𝐽)
78		eqid 2229	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
79		eqid 2229	. . . 4 ⊢ (+g‘𝐽) = (+g‘𝐽)
80		eqid 2229	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
81		eqid 2229	. . . 4 ⊢ (0g‘𝐽) = (0g‘𝐽)
82		eqid 2229	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
83	77, 78, 79, 80, 81, 82	ismhm 13480	. . 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))))
84		eqid 2229	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
85		eqid 2229	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
86		eqid 2229	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
87		eqid 2229	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
88		eqid 2229	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
89		eqid 2229	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
90	84, 85, 86, 87, 88, 89	ismhm 13480	. . 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))))
91	76, 83, 90	3bitr4g 223	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
92	91	eqrdv 2227	1 ⊢ (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ⟶wf 5310  ‘cfv 5314  (class class class)co 5994  Basecbs 13018  +_gcplusg 13096  0_gc0g 13275  Mndcmnd 13435   MndHom cmhm 13476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-map 6787  df-inn 9099  df-2 9157  df-ndx 13021  df-slot 13022  df-base 13024  df-plusg 13109  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436  df-mhm 13478

This theorem is referenced by:  ghmpropd  13806  rhmpropd  14203