Theorem pwsco1mhm 18713

Description: Right composition with a function on the index sets yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
pwsco1mhm.y	⊢ 𝑌 = (𝑅 ↑s 𝐴)
pwsco1mhm.z	⊢ 𝑍 = (𝑅 ↑s 𝐵)
pwsco1mhm.c	⊢ 𝐶 = (Base‘𝑍)
pwsco1mhm.r	⊢ (𝜑 → 𝑅 ∈ Mnd)
pwsco1mhm.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
pwsco1mhm.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
pwsco1mhm.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
pwsco1mhm	⊢ (𝜑 → (𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹)) ∈ (𝑍 MndHom 𝑌))

Distinct variable groups:   𝐶,𝑔   𝑔,𝑌   𝑔,𝑍   𝑔,𝐹   𝜑,𝑔

Allowed substitution hints:   𝐴(𝑔)   𝐵(𝑔)   𝑅(𝑔)   𝑉(𝑔)   𝑊(𝑔)

Proof of Theorem pwsco1mhm

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

Step	Hyp	Ref	Expression
1		pwsco1mhm.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Mnd)
2		pwsco1mhm.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		pwsco1mhm.z	. . . 4 ⊢ 𝑍 = (𝑅 ↑s 𝐵)
4	3	pwsmnd 18660	. . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝐵 ∈ 𝑊) → 𝑍 ∈ Mnd)
5	1, 2, 4	syl2anc 585	. 2 ⊢ (𝜑 → 𝑍 ∈ Mnd)
6		pwsco1mhm.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		pwsco1mhm.y	. . . 4 ⊢ 𝑌 = (𝑅 ↑s 𝐴)
8	7	pwsmnd 18660	. . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → 𝑌 ∈ Mnd)
9	1, 6, 8	syl2anc 585	. 2 ⊢ (𝜑 → 𝑌 ∈ Mnd)
10		eqid 2733	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		pwsco1mhm.c	. . . . . . . . 9 ⊢ 𝐶 = (Base‘𝑍)
12	3, 10, 11	pwselbasb 17434	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ 𝐵 ∈ 𝑊) → (𝑔 ∈ 𝐶 ↔ 𝑔:𝐵⟶(Base‘𝑅)))
13	1, 2, 12	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ 𝐶 ↔ 𝑔:𝐵⟶(Base‘𝑅)))
14	13	biimpa 478	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔:𝐵⟶(Base‘𝑅))
15		pwsco1mhm.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
16	15	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
17		fco 6742	. . . . . 6 ⊢ ((𝑔:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴⟶𝐵) → (𝑔 ∘ 𝐹):𝐴⟶(Base‘𝑅))
18	14, 16, 17	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → (𝑔 ∘ 𝐹):𝐴⟶(Base‘𝑅))
19		eqid 2733	. . . . . . . 8 ⊢ (Base‘𝑌) = (Base‘𝑌)
20	7, 10, 19	pwselbasb 17434	. . . . . . 7 ⊢ ((𝑅 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → ((𝑔 ∘ 𝐹) ∈ (Base‘𝑌) ↔ (𝑔 ∘ 𝐹):𝐴⟶(Base‘𝑅)))
21	1, 6, 20	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((𝑔 ∘ 𝐹) ∈ (Base‘𝑌) ↔ (𝑔 ∘ 𝐹):𝐴⟶(Base‘𝑅)))
22	21	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → ((𝑔 ∘ 𝐹) ∈ (Base‘𝑌) ↔ (𝑔 ∘ 𝐹):𝐴⟶(Base‘𝑅)))
23	18, 22	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → (𝑔 ∘ 𝐹) ∈ (Base‘𝑌))
24	23	fmpttd 7115	. . 3 ⊢ (𝜑 → (𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹)):𝐶⟶(Base‘𝑌))
25	6	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐴 ∈ 𝑉)
26		fvexd 6907	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ 𝑧 ∈ 𝐴) → (𝑥‘(𝐹‘𝑧)) ∈ V)
27		fvexd 6907	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ 𝑧 ∈ 𝐴) → (𝑦‘(𝐹‘𝑧)) ∈ V)
28	15	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐹:𝐴⟶𝐵)
29	28	ffvelcdmda 7087	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵)
30	28	feqmptd 6961	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
31	1	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑅 ∈ Mnd)
32	2	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐵 ∈ 𝑊)
33		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
34	3, 10, 11, 31, 32, 33	pwselbas 17435	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥:𝐵⟶(Base‘𝑅))
35	34	feqmptd 6961	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 = (𝑤 ∈ 𝐵 ↦ (𝑥‘𝑤)))
36		fveq2 6892	. . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑧) → (𝑥‘𝑤) = (𝑥‘(𝐹‘𝑧)))
37	29, 30, 35, 36	fmptco 7127	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∘ 𝐹) = (𝑧 ∈ 𝐴 ↦ (𝑥‘(𝐹‘𝑧))))
38		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
39	3, 10, 11, 31, 32, 38	pwselbas 17435	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦:𝐵⟶(Base‘𝑅))
40	39	feqmptd 6961	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 = (𝑤 ∈ 𝐵 ↦ (𝑦‘𝑤)))
41		fveq2 6892	. . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑧) → (𝑦‘𝑤) = (𝑦‘(𝐹‘𝑧)))
42	29, 30, 40, 41	fmptco 7127	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑦 ∘ 𝐹) = (𝑧 ∈ 𝐴 ↦ (𝑦‘(𝐹‘𝑧))))
43	25, 26, 27, 37, 42	offval2 7690	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝑥 ∘ 𝐹) ∘f (+g‘𝑅)(𝑦 ∘ 𝐹)) = (𝑧 ∈ 𝐴 ↦ ((𝑥‘(𝐹‘𝑧))(+g‘𝑅)(𝑦‘(𝐹‘𝑧)))))
44		fco 6742	. . . . . . . . 9 ⊢ ((𝑥:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴⟶𝐵) → (𝑥 ∘ 𝐹):𝐴⟶(Base‘𝑅))
45	34, 28, 44	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∘ 𝐹):𝐴⟶(Base‘𝑅))
46	7, 10, 19	pwselbasb 17434	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → ((𝑥 ∘ 𝐹) ∈ (Base‘𝑌) ↔ (𝑥 ∘ 𝐹):𝐴⟶(Base‘𝑅)))
47	31, 25, 46	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝑥 ∘ 𝐹) ∈ (Base‘𝑌) ↔ (𝑥 ∘ 𝐹):𝐴⟶(Base‘𝑅)))
48	45, 47	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∘ 𝐹) ∈ (Base‘𝑌))
49		fco 6742	. . . . . . . . 9 ⊢ ((𝑦:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴⟶𝐵) → (𝑦 ∘ 𝐹):𝐴⟶(Base‘𝑅))
50	39, 28, 49	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑦 ∘ 𝐹):𝐴⟶(Base‘𝑅))
51	7, 10, 19	pwselbasb 17434	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∘ 𝐹) ∈ (Base‘𝑌) ↔ (𝑦 ∘ 𝐹):𝐴⟶(Base‘𝑅)))
52	31, 25, 51	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝑦 ∘ 𝐹) ∈ (Base‘𝑌) ↔ (𝑦 ∘ 𝐹):𝐴⟶(Base‘𝑅)))
53	50, 52	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑦 ∘ 𝐹) ∈ (Base‘𝑌))
54		eqid 2733	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
55		eqid 2733	. . . . . . 7 ⊢ (+g‘𝑌) = (+g‘𝑌)
56	7, 19, 31, 25, 48, 53, 54, 55	pwsplusgval 17436	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝑥 ∘ 𝐹)(+g‘𝑌)(𝑦 ∘ 𝐹)) = ((𝑥 ∘ 𝐹) ∘f (+g‘𝑅)(𝑦 ∘ 𝐹)))
57		eqid 2733	. . . . . . . . 9 ⊢ (+g‘𝑍) = (+g‘𝑍)
58	3, 11, 31, 32, 33, 38, 54, 57	pwsplusgval 17436	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝑍)𝑦) = (𝑥 ∘f (+g‘𝑅)𝑦))
59		fvexd 6907	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ 𝑤 ∈ 𝐵) → (𝑥‘𝑤) ∈ V)
60		fvexd 6907	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ 𝑤 ∈ 𝐵) → (𝑦‘𝑤) ∈ V)
61	32, 59, 60, 35, 40	offval2 7690	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∘f (+g‘𝑅)𝑦) = (𝑤 ∈ 𝐵 ↦ ((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤))))
62	58, 61	eqtrd 2773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝑍)𝑦) = (𝑤 ∈ 𝐵 ↦ ((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤))))
63	36, 41	oveq12d 7427	. . . . . . 7 ⊢ (𝑤 = (𝐹‘𝑧) → ((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤)) = ((𝑥‘(𝐹‘𝑧))(+g‘𝑅)(𝑦‘(𝐹‘𝑧))))
64	29, 30, 62, 63	fmptco 7127	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝑥(+g‘𝑍)𝑦) ∘ 𝐹) = (𝑧 ∈ 𝐴 ↦ ((𝑥‘(𝐹‘𝑧))(+g‘𝑅)(𝑦‘(𝐹‘𝑧)))))
65	43, 56, 64	3eqtr4rd 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝑥(+g‘𝑍)𝑦) ∘ 𝐹) = ((𝑥 ∘ 𝐹)(+g‘𝑌)(𝑦 ∘ 𝐹)))
66		eqid 2733	. . . . . 6 ⊢ (𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹)) = (𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))
67		coeq1 5858	. . . . . 6 ⊢ (𝑔 = (𝑥(+g‘𝑍)𝑦) → (𝑔 ∘ 𝐹) = ((𝑥(+g‘𝑍)𝑦) ∘ 𝐹))
68	11, 57	mndcl 18633	. . . . . . . 8 ⊢ ((𝑍 ∈ Mnd ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥(+g‘𝑍)𝑦) ∈ 𝐶)
69	68	3expb 1121	. . . . . . 7 ⊢ ((𝑍 ∈ Mnd ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝑍)𝑦) ∈ 𝐶)
70	5, 69	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝑍)𝑦) ∈ 𝐶)
71		ovex 7442	. . . . . . 7 ⊢ (𝑥(+g‘𝑍)𝑦) ∈ V
72	15, 6	fexd 7229	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ V)
73	72	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐹 ∈ V)
74		coexg 7920	. . . . . . 7 ⊢ (((𝑥(+g‘𝑍)𝑦) ∈ V ∧ 𝐹 ∈ V) → ((𝑥(+g‘𝑍)𝑦) ∘ 𝐹) ∈ V)
75	71, 73, 74	sylancr 588	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝑥(+g‘𝑍)𝑦) ∘ 𝐹) ∈ V)
76	66, 67, 70, 75	fvmptd3 7022	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘(𝑥(+g‘𝑍)𝑦)) = ((𝑥(+g‘𝑍)𝑦) ∘ 𝐹))
77		coeq1 5858	. . . . . . 7 ⊢ (𝑔 = 𝑥 → (𝑔 ∘ 𝐹) = (𝑥 ∘ 𝐹))
78		coexg 7920	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝐹 ∈ V) → (𝑥 ∘ 𝐹) ∈ V)
79	33, 73, 78	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∘ 𝐹) ∈ V)
80	66, 77, 33, 79	fvmptd3 7022	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘𝑥) = (𝑥 ∘ 𝐹))
81		coeq1 5858	. . . . . . 7 ⊢ (𝑔 = 𝑦 → (𝑔 ∘ 𝐹) = (𝑦 ∘ 𝐹))
82		coexg 7920	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝐹 ∈ V) → (𝑦 ∘ 𝐹) ∈ V)
83	38, 73, 82	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑦 ∘ 𝐹) ∈ V)
84	66, 81, 38, 83	fvmptd3 7022	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘𝑦) = (𝑦 ∘ 𝐹))
85	80, 84	oveq12d 7427	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘𝑥)(+g‘𝑌)((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘𝑦)) = ((𝑥 ∘ 𝐹)(+g‘𝑌)(𝑦 ∘ 𝐹)))
86	65, 76, 85	3eqtr4d 2783	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘(𝑥(+g‘𝑍)𝑦)) = (((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘𝑥)(+g‘𝑌)((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘𝑦)))
87	86	ralrimivva 3201	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘(𝑥(+g‘𝑍)𝑦)) = (((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘𝑥)(+g‘𝑌)((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘𝑦)))
88		coeq1 5858	. . . . 5 ⊢ (𝑔 = (0g‘𝑍) → (𝑔 ∘ 𝐹) = ((0g‘𝑍) ∘ 𝐹))
89		eqid 2733	. . . . . . 7 ⊢ (0g‘𝑍) = (0g‘𝑍)
90	11, 89	mndidcl 18640	. . . . . 6 ⊢ (𝑍 ∈ Mnd → (0g‘𝑍) ∈ 𝐶)
91	5, 90	syl 17	. . . . 5 ⊢ (𝜑 → (0g‘𝑍) ∈ 𝐶)
92		coexg 7920	. . . . . 6 ⊢ (((0g‘𝑍) ∈ 𝐶 ∧ 𝐹 ∈ V) → ((0g‘𝑍) ∘ 𝐹) ∈ V)
93	91, 72, 92	syl2anc 585	. . . . 5 ⊢ (𝜑 → ((0g‘𝑍) ∘ 𝐹) ∈ V)
94	66, 88, 91, 93	fvmptd3 7022	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘(0g‘𝑍)) = ((0g‘𝑍) ∘ 𝐹))
95	3, 10, 11, 1, 2, 91	pwselbas 17435	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑍):𝐵⟶(Base‘𝑅))
96		fco 6742	. . . . . . 7 ⊢ (((0g‘𝑍):𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴⟶𝐵) → ((0g‘𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅))
97	95, 15, 96	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((0g‘𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅))
98	97	ffnd 6719	. . . . 5 ⊢ (𝜑 → ((0g‘𝑍) ∘ 𝐹) Fn 𝐴)
99		fvexd 6907	. . . . . 6 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
100		fnconstg 6780	. . . . . 6 ⊢ ((0g‘𝑅) ∈ V → (𝐴 × {(0g‘𝑅)}) Fn 𝐴)
101	99, 100	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 × {(0g‘𝑅)}) Fn 𝐴)
102		eqid 2733	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
103	3, 102	pws0g 18661	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Mnd ∧ 𝐵 ∈ 𝑊) → (𝐵 × {(0g‘𝑅)}) = (0g‘𝑍))
104	1, 2, 103	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 × {(0g‘𝑅)}) = (0g‘𝑍))
105	104	fveq1d 6894	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 × {(0g‘𝑅)})‘(𝐹‘𝑥)) = ((0g‘𝑍)‘(𝐹‘𝑥)))
106	105	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐵 × {(0g‘𝑅)})‘(𝐹‘𝑥)) = ((0g‘𝑍)‘(𝐹‘𝑥)))
107		fvex 6905	. . . . . . . 8 ⊢ (0g‘𝑅) ∈ V
108	15	ffvelcdmda 7087	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
109		fvconst2g 7203	. . . . . . . 8 ⊢ (((0g‘𝑅) ∈ V ∧ (𝐹‘𝑥) ∈ 𝐵) → ((𝐵 × {(0g‘𝑅)})‘(𝐹‘𝑥)) = (0g‘𝑅))
110	107, 108, 109	sylancr 588	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐵 × {(0g‘𝑅)})‘(𝐹‘𝑥)) = (0g‘𝑅))
111	106, 110	eqtr3d 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((0g‘𝑍)‘(𝐹‘𝑥)) = (0g‘𝑅))
112		fvco3 6991	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (((0g‘𝑍) ∘ 𝐹)‘𝑥) = ((0g‘𝑍)‘(𝐹‘𝑥)))
113	15, 112	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((0g‘𝑍) ∘ 𝐹)‘𝑥) = ((0g‘𝑍)‘(𝐹‘𝑥)))
114		fvconst2g 7203	. . . . . . 7 ⊢ (((0g‘𝑅) ∈ V ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {(0g‘𝑅)})‘𝑥) = (0g‘𝑅))
115	99, 114	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {(0g‘𝑅)})‘𝑥) = (0g‘𝑅))
116	111, 113, 115	3eqtr4d 2783	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((0g‘𝑍) ∘ 𝐹)‘𝑥) = ((𝐴 × {(0g‘𝑅)})‘𝑥))
117	98, 101, 116	eqfnfvd 7036	. . . 4 ⊢ (𝜑 → ((0g‘𝑍) ∘ 𝐹) = (𝐴 × {(0g‘𝑅)}))
118	7, 102	pws0g 18661	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐴 × {(0g‘𝑅)}) = (0g‘𝑌))
119	1, 6, 118	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐴 × {(0g‘𝑅)}) = (0g‘𝑌))
120	94, 117, 119	3eqtrd 2777	. . 3 ⊢ (𝜑 → ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘(0g‘𝑍)) = (0g‘𝑌))
121	24, 87, 120	3jca 1129	. 2 ⊢ (𝜑 → ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹)):𝐶⟶(Base‘𝑌) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘(𝑥(+g‘𝑍)𝑦)) = (((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘𝑥)(+g‘𝑌)((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘𝑦)) ∧ ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘(0g‘𝑍)) = (0g‘𝑌)))
122		eqid 2733	. . 3 ⊢ (0g‘𝑌) = (0g‘𝑌)
123	11, 19, 57, 55, 89, 122	ismhm 18673	. 2 ⊢ ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹)) ∈ (𝑍 MndHom 𝑌) ↔ ((𝑍 ∈ Mnd ∧ 𝑌 ∈ Mnd) ∧ ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹)):𝐶⟶(Base‘𝑌) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘(𝑥(+g‘𝑍)𝑦)) = (((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘𝑥)(+g‘𝑌)((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘𝑦)) ∧ ((𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹))‘(0g‘𝑍)) = (0g‘𝑌))))
124	5, 9, 121, 123	syl21anbrc 1345	1 ⊢ (𝜑 → (𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹)) ∈ (𝑍 MndHom 𝑌))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475  {csn 4629   ↦ cmpt 5232   × cxp 5675   ∘ ccom 5681   Fn wfn 6539  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409   ∘_f cof 7668  Basecbs 17144  +_gcplusg 17197  0_gc0g 17385   ↑_s cpws 17392  Mndcmnd 18625   MndHom cmhm 18669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-struct 17080  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-hom 17221  df-cco 17222  df-0g 17387  df-prds 17393  df-pws 17395  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671

This theorem is referenced by:  pwsco1rhm  20277