Theorem mhmpropd 12862

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 5521	. . . . . . . . . . . . . . 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 5651	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐵⟶𝐶 ∧ 𝑥 ∈ 𝐵) → (𝑓‘𝑥) ∈ 𝐶)
5		ffvelcdm 5651	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐵⟶𝐶 ∧ 𝑦 ∈ 𝐵) → (𝑓‘𝑦) ∈ 𝐶)
6	4, 5	anim12dan 600	. . . . . . . . . . . . . . . 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 2559	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
9		oveq1 5884	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (𝑥(+g‘𝐾)𝑦) = (𝑤(+g‘𝐾)𝑦))
10		oveq1 5884	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (𝑥(+g‘𝑀)𝑦) = (𝑤(+g‘𝑀)𝑦))
11	9, 10	eqeq12d 2192	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑤 → ((𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦) ↔ (𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝑀)𝑦)))
12		oveq2 5885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝐾)𝑧))
13		oveq2 5885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝑤(+g‘𝑀)𝑦) = (𝑤(+g‘𝑀)𝑧))
14	12, 13	eqeq12d 2192	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → ((𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝑀)𝑦) ↔ (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧)))
15	11, 14	cbvral2vw 2716	. . . . . . . . . . . . . . . . 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 5884	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝐾)𝑧))
18		oveq1 5884	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤(+g‘𝑀)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧))
19	17, 18	eqeq12d 2192	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑓‘𝑥) → ((𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧) ↔ ((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧)))
20		oveq2 5885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑓‘𝑦) → ((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))
21		oveq2 5885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑓‘𝑦) → ((𝑓‘𝑥)(+g‘𝑀)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))
22	20, 21	eqeq12d 2192	. . . . . . . . . . . . . . . . 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 2857	. . . . . . . . . . . . . . . 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 2192	. . . . . . . . . . . . 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 2499	. . . . . . . . . . . 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 2673	. . . . . . . . . . . . . 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 2681	. . . . . . . . . . . . 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 2673	. . . . . . . . . . . . . 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 2681	. . . . . . . . . . . . 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 12846	. . . . . . . . . . . . . . 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 12798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → (0g‘𝐽) = (0g‘𝐿))
48	47	fveq2d 5521	. . . . . . . . . . 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 12846	. . . . . . . . . . . . . 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 12798	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵⟶𝐶)) → (0g‘𝐾) = (0g‘𝑀))
59	48, 58	eqeq12d 2192	. . . . . . . . . 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 5363	. . . . . . . . 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 5363	. . . . . . . . 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 982	. . . . . 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 982	. . . . . 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 2177	. . . 4 ⊢ (Base‘𝐽) = (Base‘𝐽)
78		eqid 2177	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
79		eqid 2177	. . . 4 ⊢ (+g‘𝐽) = (+g‘𝐽)
80		eqid 2177	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
81		eqid 2177	. . . 4 ⊢ (0g‘𝐽) = (0g‘𝐽)
82		eqid 2177	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
83	77, 78, 79, 80, 81, 82	ismhm 12858	. . 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 2177	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
85		eqid 2177	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
86		eqid 2177	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
87		eqid 2177	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
88		eqid 2177	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
89		eqid 2177	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
90	84, 85, 86, 87, 88, 89	ismhm 12858	. . 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 2175	1 ⊢ (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ⟶wf 5214  ‘cfv 5218  (class class class)co 5877  Basecbs 12464  +_gcplusg 12538  0_gc0g 12710  Mndcmnd 12822   MndHom cmhm 12854

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-mhm 12856

This theorem is referenced by: (None)