Theorem pwsco2mhm 17371

Description: Left composition with a monoid homomorphism yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
pwsco2mhm.y	⊢ 𝑌 = (𝑅 ↑s 𝐴)
pwsco2mhm.z	⊢ 𝑍 = (𝑆 ↑s 𝐴)
pwsco2mhm.b	⊢ 𝐵 = (Base‘𝑌)
pwsco2mhm.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
pwsco2mhm.f	⊢ (𝜑 → 𝐹 ∈ (𝑅 MndHom 𝑆))

Assertion

Ref	Expression
pwsco2mhm	⊢ (𝜑 → (𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔)) ∈ (𝑌 MndHom 𝑍))

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

Allowed substitution hints:   𝐴(𝑔)   𝑅(𝑔)   𝑆(𝑔)   𝑉(𝑔)

Proof of Theorem pwsco2mhm

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

Step	Hyp	Ref	Expression
1		pwsco2mhm.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑅 MndHom 𝑆))
2		mhmrcl1 17338	. . . . 5 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd)
4		pwsco2mhm.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		pwsco2mhm.y	. . . . 5 ⊢ 𝑌 = (𝑅 ↑s 𝐴)
6	5	pwsmnd 17325	. . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → 𝑌 ∈ Mnd)
7	3, 4, 6	syl2anc 693	. . 3 ⊢ (𝜑 → 𝑌 ∈ Mnd)
8		mhmrcl2 17339	. . . . 5 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑆 ∈ Mnd)
9	1, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Mnd)
10		pwsco2mhm.z	. . . . 5 ⊢ 𝑍 = (𝑆 ↑s 𝐴)
11	10	pwsmnd 17325	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → 𝑍 ∈ Mnd)
12	9, 4, 11	syl2anc 693	. . 3 ⊢ (𝜑 → 𝑍 ∈ Mnd)
13	7, 12	jca 554	. 2 ⊢ (𝜑 → (𝑌 ∈ Mnd ∧ 𝑍 ∈ Mnd))
14		eqid 2622	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
15		eqid 2622	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
16	14, 15	mhmf 17340	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
17	1, 16	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
18	17	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
19		pwsco2mhm.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑌)
20	3	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝑅 ∈ Mnd)
21	4	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝐴 ∈ 𝑉)
22		simpr 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ 𝐵)
23	5, 14, 19, 20, 21, 22	pwselbas 16149	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝑔:𝐴⟶(Base‘𝑅))
24		fco 6058	. . . . . 6 ⊢ ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ 𝑔:𝐴⟶(Base‘𝑅)) → (𝐹 ∘ 𝑔):𝐴⟶(Base‘𝑆))
25	18, 23, 24	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → (𝐹 ∘ 𝑔):𝐴⟶(Base‘𝑆))
26	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝑆 ∈ Mnd)
27		eqid 2622	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
28	10, 15, 27	pwselbasb 16148	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → ((𝐹 ∘ 𝑔) ∈ (Base‘𝑍) ↔ (𝐹 ∘ 𝑔):𝐴⟶(Base‘𝑆)))
29	26, 21, 28	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → ((𝐹 ∘ 𝑔) ∈ (Base‘𝑍) ↔ (𝐹 ∘ 𝑔):𝐴⟶(Base‘𝑆)))
30	25, 29	mpbird 247	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → (𝐹 ∘ 𝑔) ∈ (Base‘𝑍))
31		eqid 2622	. . . 4 ⊢ (𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔)) = (𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))
32	30, 31	fmptd 6385	. . 3 ⊢ (𝜑 → (𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔)):𝐵⟶(Base‘𝑍))
33	1	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐹 ∈ (𝑅 MndHom 𝑆))
34	33	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐴) → 𝐹 ∈ (𝑅 MndHom 𝑆))
35	33, 2	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ Mnd)
36	4	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐴 ∈ 𝑉)
37		simprl 794	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
38	5, 14, 19, 35, 36, 37	pwselbas 16149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥:𝐴⟶(Base‘𝑅))
39	38	ffvelrnda 6359	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) ∈ (Base‘𝑅))
40		simprr 796	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
41	5, 14, 19, 35, 36, 40	pwselbas 16149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦:𝐴⟶(Base‘𝑅))
42	41	ffvelrnda 6359	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐴) → (𝑦‘𝑤) ∈ (Base‘𝑅))
43		eqid 2622	. . . . . . . . . 10 ⊢ (+g‘𝑅) = (+g‘𝑅)
44		eqid 2622	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
45	14, 43, 44	mhmlin 17342	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (𝑥‘𝑤) ∈ (Base‘𝑅) ∧ (𝑦‘𝑤) ∈ (Base‘𝑅)) → (𝐹‘((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤))) = ((𝐹‘(𝑥‘𝑤))(+g‘𝑆)(𝐹‘(𝑦‘𝑤))))
46	34, 39, 42, 45	syl3anc 1326	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐴) → (𝐹‘((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤))) = ((𝐹‘(𝑥‘𝑤))(+g‘𝑆)(𝐹‘(𝑦‘𝑤))))
47	46	mpteq2dva 4744	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑤 ∈ 𝐴 ↦ (𝐹‘((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤)))) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘(𝑥‘𝑤))(+g‘𝑆)(𝐹‘(𝑦‘𝑤)))))
48		fvexd 6203	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐴) → (𝐹‘(𝑥‘𝑤)) ∈ V)
49		fvexd 6203	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐴) → (𝐹‘(𝑦‘𝑤)) ∈ V)
50	38	feqmptd 6249	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 = (𝑤 ∈ 𝐴 ↦ (𝑥‘𝑤)))
51	33, 16	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
52	51	feqmptd 6249	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐹 = (𝑧 ∈ (Base‘𝑅) ↦ (𝐹‘𝑧)))
53		fveq2 6191	. . . . . . . . 9 ⊢ (𝑧 = (𝑥‘𝑤) → (𝐹‘𝑧) = (𝐹‘(𝑥‘𝑤)))
54	39, 50, 52, 53	fmptco 6396	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹 ∘ 𝑥) = (𝑤 ∈ 𝐴 ↦ (𝐹‘(𝑥‘𝑤))))
55	41	feqmptd 6249	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 = (𝑤 ∈ 𝐴 ↦ (𝑦‘𝑤)))
56		fveq2 6191	. . . . . . . . 9 ⊢ (𝑧 = (𝑦‘𝑤) → (𝐹‘𝑧) = (𝐹‘(𝑦‘𝑤)))
57	42, 55, 52, 56	fmptco 6396	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹 ∘ 𝑦) = (𝑤 ∈ 𝐴 ↦ (𝐹‘(𝑦‘𝑤))))
58	36, 48, 49, 54, 57	offval2 6914	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐹 ∘ 𝑥) ∘𝑓 (+g‘𝑆)(𝐹 ∘ 𝑦)) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘(𝑥‘𝑤))(+g‘𝑆)(𝐹‘(𝑦‘𝑤)))))
59	47, 58	eqtr4d 2659	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑤 ∈ 𝐴 ↦ (𝐹‘((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤)))) = ((𝐹 ∘ 𝑥) ∘𝑓 (+g‘𝑆)(𝐹 ∘ 𝑦)))
60	35	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐴) → 𝑅 ∈ Mnd)
61	14, 43	mndcl 17301	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥‘𝑤) ∈ (Base‘𝑅) ∧ (𝑦‘𝑤) ∈ (Base‘𝑅)) → ((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤)) ∈ (Base‘𝑅))
62	60, 39, 42, 61	syl3anc 1326	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤)) ∈ (Base‘𝑅))
63		eqid 2622	. . . . . . . . 9 ⊢ (+g‘𝑌) = (+g‘𝑌)
64	5, 19, 35, 36, 37, 40, 43, 63	pwsplusgval 16150	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑌)𝑦) = (𝑥 ∘𝑓 (+g‘𝑅)𝑦))
65		fvexd 6203	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) ∈ V)
66		fvexd 6203	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐴) → (𝑦‘𝑤) ∈ V)
67	36, 65, 66, 50, 55	offval2 6914	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∘𝑓 (+g‘𝑅)𝑦) = (𝑤 ∈ 𝐴 ↦ ((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤))))
68	64, 67	eqtrd 2656	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑌)𝑦) = (𝑤 ∈ 𝐴 ↦ ((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤))))
69		fveq2 6191	. . . . . . 7 ⊢ (𝑧 = ((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤)) → (𝐹‘𝑧) = (𝐹‘((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤))))
70	62, 68, 52, 69	fmptco 6396	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹 ∘ (𝑥(+g‘𝑌)𝑦)) = (𝑤 ∈ 𝐴 ↦ (𝐹‘((𝑥‘𝑤)(+g‘𝑅)(𝑦‘𝑤)))))
71	33, 8	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ Mnd)
72		fco 6058	. . . . . . . . 9 ⊢ ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ 𝑥:𝐴⟶(Base‘𝑅)) → (𝐹 ∘ 𝑥):𝐴⟶(Base‘𝑆))
73	51, 38, 72	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹 ∘ 𝑥):𝐴⟶(Base‘𝑆))
74	10, 15, 27	pwselbasb 16148	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → ((𝐹 ∘ 𝑥) ∈ (Base‘𝑍) ↔ (𝐹 ∘ 𝑥):𝐴⟶(Base‘𝑆)))
75	71, 36, 74	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐹 ∘ 𝑥) ∈ (Base‘𝑍) ↔ (𝐹 ∘ 𝑥):𝐴⟶(Base‘𝑆)))
76	73, 75	mpbird 247	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹 ∘ 𝑥) ∈ (Base‘𝑍))
77		fco 6058	. . . . . . . . 9 ⊢ ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ 𝑦:𝐴⟶(Base‘𝑅)) → (𝐹 ∘ 𝑦):𝐴⟶(Base‘𝑆))
78	51, 41, 77	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹 ∘ 𝑦):𝐴⟶(Base‘𝑆))
79	10, 15, 27	pwselbasb 16148	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → ((𝐹 ∘ 𝑦) ∈ (Base‘𝑍) ↔ (𝐹 ∘ 𝑦):𝐴⟶(Base‘𝑆)))
80	71, 36, 79	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐹 ∘ 𝑦) ∈ (Base‘𝑍) ↔ (𝐹 ∘ 𝑦):𝐴⟶(Base‘𝑆)))
81	78, 80	mpbird 247	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹 ∘ 𝑦) ∈ (Base‘𝑍))
82		eqid 2622	. . . . . . 7 ⊢ (+g‘𝑍) = (+g‘𝑍)
83	10, 27, 71, 36, 76, 81, 44, 82	pwsplusgval 16150	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐹 ∘ 𝑥)(+g‘𝑍)(𝐹 ∘ 𝑦)) = ((𝐹 ∘ 𝑥) ∘𝑓 (+g‘𝑆)(𝐹 ∘ 𝑦)))
84	59, 70, 83	3eqtr4d 2666	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹 ∘ (𝑥(+g‘𝑌)𝑦)) = ((𝐹 ∘ 𝑥)(+g‘𝑍)(𝐹 ∘ 𝑦)))
85	19, 63	mndcl 17301	. . . . . . . 8 ⊢ ((𝑌 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑌)𝑦) ∈ 𝐵)
86	85	3expb 1266	. . . . . . 7 ⊢ ((𝑌 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑌)𝑦) ∈ 𝐵)
87	7, 86	sylan 488	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑌)𝑦) ∈ 𝐵)
88		coexg 7117	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (𝑥(+g‘𝑌)𝑦) ∈ 𝐵) → (𝐹 ∘ (𝑥(+g‘𝑌)𝑦)) ∈ V)
89	33, 87, 88	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹 ∘ (𝑥(+g‘𝑌)𝑦)) ∈ V)
90		coeq2 5280	. . . . . . 7 ⊢ (𝑔 = (𝑥(+g‘𝑌)𝑦) → (𝐹 ∘ 𝑔) = (𝐹 ∘ (𝑥(+g‘𝑌)𝑦)))
91	90, 31	fvmptg 6280	. . . . . 6 ⊢ (((𝑥(+g‘𝑌)𝑦) ∈ 𝐵 ∧ (𝐹 ∘ (𝑥(+g‘𝑌)𝑦)) ∈ V) → ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘(𝑥(+g‘𝑌)𝑦)) = (𝐹 ∘ (𝑥(+g‘𝑌)𝑦)))
92	87, 89, 91	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘(𝑥(+g‘𝑌)𝑦)) = (𝐹 ∘ (𝑥(+g‘𝑌)𝑦)))
93		coeq2 5280	. . . . . . . 8 ⊢ (𝑔 = 𝑥 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝑥))
94	93, 31	fvmptg 6280	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝐹 ∘ 𝑥) ∈ (Base‘𝑍)) → ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑥) = (𝐹 ∘ 𝑥))
95	37, 76, 94	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑥) = (𝐹 ∘ 𝑥))
96		coeq2 5280	. . . . . . . 8 ⊢ (𝑔 = 𝑦 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝑦))
97	96, 31	fvmptg 6280	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝐹 ∘ 𝑦) ∈ (Base‘𝑍)) → ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑦) = (𝐹 ∘ 𝑦))
98	40, 81, 97	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑦) = (𝐹 ∘ 𝑦))
99	95, 98	oveq12d 6668	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑥)(+g‘𝑍)((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑦)) = ((𝐹 ∘ 𝑥)(+g‘𝑍)(𝐹 ∘ 𝑦)))
100	84, 92, 99	3eqtr4d 2666	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘(𝑥(+g‘𝑌)𝑦)) = (((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑥)(+g‘𝑍)((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑦)))
101	100	ralrimivva 2971	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘(𝑥(+g‘𝑌)𝑦)) = (((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑥)(+g‘𝑍)((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑦)))
102		eqid 2622	. . . . . . 7 ⊢ (0g‘𝑌) = (0g‘𝑌)
103	19, 102	mndidcl 17308	. . . . . 6 ⊢ (𝑌 ∈ Mnd → (0g‘𝑌) ∈ 𝐵)
104	7, 103	syl 17	. . . . 5 ⊢ (𝜑 → (0g‘𝑌) ∈ 𝐵)
105		coexg 7117	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (0g‘𝑌) ∈ 𝐵) → (𝐹 ∘ (0g‘𝑌)) ∈ V)
106	1, 104, 105	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ (0g‘𝑌)) ∈ V)
107		coeq2 5280	. . . . . 6 ⊢ (𝑔 = (0g‘𝑌) → (𝐹 ∘ 𝑔) = (𝐹 ∘ (0g‘𝑌)))
108	107, 31	fvmptg 6280	. . . . 5 ⊢ (((0g‘𝑌) ∈ 𝐵 ∧ (𝐹 ∘ (0g‘𝑌)) ∈ V) → ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘(0g‘𝑌)) = (𝐹 ∘ (0g‘𝑌)))
109	104, 106, 108	syl2anc 693	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘(0g‘𝑌)) = (𝐹 ∘ (0g‘𝑌)))
110		ffn 6045	. . . . . . 7 ⊢ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) → 𝐹 Fn (Base‘𝑅))
111	17, 110	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑅))
112		eqid 2622	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
113	14, 112	mndidcl 17308	. . . . . . 7 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ (Base‘𝑅))
114	3, 113	syl 17	. . . . . 6 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
115		fcoconst 6401	. . . . . 6 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ (0g‘𝑅) ∈ (Base‘𝑅)) → (𝐹 ∘ (𝐴 × {(0g‘𝑅)})) = (𝐴 × {(𝐹‘(0g‘𝑅))}))
116	111, 114, 115	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ (𝐴 × {(0g‘𝑅)})) = (𝐴 × {(𝐹‘(0g‘𝑅))}))
117	5, 112	pws0g 17326	. . . . . . 7 ⊢ ((𝑅 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐴 × {(0g‘𝑅)}) = (0g‘𝑌))
118	3, 4, 117	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝐴 × {(0g‘𝑅)}) = (0g‘𝑌))
119	118	coeq2d 5284	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ (𝐴 × {(0g‘𝑅)})) = (𝐹 ∘ (0g‘𝑌)))
120		eqid 2622	. . . . . . . . 9 ⊢ (0g‘𝑆) = (0g‘𝑆)
121	112, 120	mhm0 17343	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
122	1, 121	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
123	122	sneqd 4189	. . . . . 6 ⊢ (𝜑 → {(𝐹‘(0g‘𝑅))} = {(0g‘𝑆)})
124	123	xpeq2d 5139	. . . . 5 ⊢ (𝜑 → (𝐴 × {(𝐹‘(0g‘𝑅))}) = (𝐴 × {(0g‘𝑆)}))
125	116, 119, 124	3eqtr3d 2664	. . . 4 ⊢ (𝜑 → (𝐹 ∘ (0g‘𝑌)) = (𝐴 × {(0g‘𝑆)}))
126	10, 120	pws0g 17326	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐴 × {(0g‘𝑆)}) = (0g‘𝑍))
127	9, 4, 126	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐴 × {(0g‘𝑆)}) = (0g‘𝑍))
128	109, 125, 127	3eqtrd 2660	. . 3 ⊢ (𝜑 → ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘(0g‘𝑌)) = (0g‘𝑍))
129	32, 101, 128	3jca 1242	. 2 ⊢ (𝜑 → ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔)):𝐵⟶(Base‘𝑍) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘(𝑥(+g‘𝑌)𝑦)) = (((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑥)(+g‘𝑍)((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑦)) ∧ ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘(0g‘𝑌)) = (0g‘𝑍)))
130		eqid 2622	. . 3 ⊢ (0g‘𝑍) = (0g‘𝑍)
131	19, 27, 63, 82, 102, 130	ismhm 17337	. 2 ⊢ ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔)) ∈ (𝑌 MndHom 𝑍) ↔ ((𝑌 ∈ Mnd ∧ 𝑍 ∈ Mnd) ∧ ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔)):𝐵⟶(Base‘𝑍) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘(𝑥(+g‘𝑌)𝑦)) = (((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑥)(+g‘𝑍)((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘𝑦)) ∧ ((𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔))‘(0g‘𝑌)) = (0g‘𝑍))))
132	13, 129, 131	sylanbrc 698	1 ⊢ (𝜑 → (𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔)) ∈ (𝑌 MndHom 𝑍))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  {csn 4177   ↦ cmpt 4729   × cxp 5112   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100   ↑_s cpws 16107  Mndcmnd 17294   MndHom cmhm 17333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335

This theorem is referenced by:  pwsco2rhm  18739