Theorem imasplusg 12951

Description: The group operation in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)

Hypotheses

Ref	Expression
imasbas.u	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasbas.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasbas.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasbas.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasplusg.p	⊢ + = (+g‘𝑅)
imasplusg.a	⊢ ✚ = (+g‘𝑈)

Assertion

Ref	Expression
imasplusg	⊢ (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})

Distinct variable groups:   𝐹,𝑝,𝑞   𝑅,𝑝,𝑞   𝑉,𝑝,𝑞   𝜑,𝑝,𝑞

Allowed substitution hints:   𝐵(𝑞,𝑝)   + (𝑞,𝑝)   ✚ (𝑞,𝑝)   𝑈(𝑞,𝑝)   𝑍(𝑞,𝑝)

Proof of Theorem imasplusg

Step	Hyp	Ref	Expression
1		imasplusg.a	. . 3 ⊢ ✚ = (+g‘𝑈)
2		imasbas.u	. . . . . 6 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
3		imasbas.v	. . . . . 6 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
4		eqid 2196	. . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅)
5		eqid 2196	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2196	. . . . . 6 ⊢ ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
7		eqidd 2197	. . . . . 6 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
8		eqidd 2197	. . . . . 6 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩})
9		imasbas.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
10		imasbas.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
11	2, 3, 4, 5, 6, 7, 8, 9, 10	imasival 12949	. . . . 5 ⊢ (𝜑 → 𝑈 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩})
12	11	fveq1d 5560	. . . 4 ⊢ (𝜑 → (𝑈‘(+g‘ndx)) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩}‘(+g‘ndx)))
13		basendxnn 12734	. . . . . . . 8 ⊢ (Base‘ndx) ∈ ℕ
14		basfn 12736	. . . . . . . . . . 11 ⊢ Base Fn V
15	10	elexd 2776	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ V)
16		funfvex 5575	. . . . . . . . . . . 12 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
17	16	funfni 5358	. . . . . . . . . . 11 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
18	14, 15, 17	sylancr 414	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
19	3, 18	eqeltrd 2273	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ V)
20		focdmex 6172	. . . . . . . . 9 ⊢ (𝑉 ∈ V → (𝐹:𝑉–onto→𝐵 → 𝐵 ∈ V))
21	19, 9, 20	sylc 62	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
22		opexg 4261	. . . . . . . 8 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ V) → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
23	13, 21, 22	sylancr 414	. . . . . . 7 ⊢ (𝜑 → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
24		plusgndxnn 12789	. . . . . . . 8 ⊢ (+g‘ndx) ∈ ℕ
25		fof 5480	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
26	9, 25	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
27	26, 19	fexd 5792	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ∈ V)
28		vex 2766	. . . . . . . . . . . . . . . 16 ⊢ 𝑝 ∈ V
29		fvexg 5577	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ V ∧ 𝑝 ∈ V) → (𝐹‘𝑝) ∈ V)
30	27, 28, 29	sylancl 413	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝑝) ∈ V)
31		vex 2766	. . . . . . . . . . . . . . . 16 ⊢ 𝑞 ∈ V
32		fvexg 5577	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ V ∧ 𝑞 ∈ V) → (𝐹‘𝑞) ∈ V)
33	27, 31, 32	sylancl 413	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝑞) ∈ V)
34		opexg 4261	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑝) ∈ V ∧ (𝐹‘𝑞) ∈ V) → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V)
35	30, 33, 34	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V)
36	28	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑝 ∈ V)
37		plusgslid 12790	. . . . . . . . . . . . . . . . . 18 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
38	37	slotex 12705	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑍 → (+g‘𝑅) ∈ V)
39	10, 38	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (+g‘𝑅) ∈ V)
40	31	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑞 ∈ V)
41		ovexg 5956	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ V ∧ (+g‘𝑅) ∈ V ∧ 𝑞 ∈ V) → (𝑝(+g‘𝑅)𝑞) ∈ V)
42	36, 39, 40, 41	syl3anc 1249	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑝(+g‘𝑅)𝑞) ∈ V)
43		fvexg 5577	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ V ∧ (𝑝(+g‘𝑅)𝑞) ∈ V) → (𝐹‘(𝑝(+g‘𝑅)𝑞)) ∈ V)
44	27, 42, 43	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘(𝑝(+g‘𝑅)𝑞)) ∈ V)
45		opexg 4261	. . . . . . . . . . . . . 14 ⊢ ((⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V ∧ (𝐹‘(𝑝(+g‘𝑅)𝑞)) ∈ V) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩ ∈ V)
46	35, 44, 45	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩ ∈ V)
47		snexg 4217	. . . . . . . . . . . . 13 ⊢ (⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩ ∈ V → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
48	46, 47	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
49	48	ralrimivw 2571	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
50		iunexg 6176	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ V ∧ ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V) → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
51	19, 49, 50	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
52	51	ralrimivw 2571	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
53		iunexg 6176	. . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V) → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
54	19, 52, 53	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
55		opexg 4261	. . . . . . . 8 ⊢ (((+g‘ndx) ∈ ℕ ∧ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V) → ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩ ∈ V)
56	24, 54, 55	sylancr 414	. . . . . . 7 ⊢ (𝜑 → ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩ ∈ V)
57		mulrslid 12809	. . . . . . . . 9 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
58	57	simpri 113	. . . . . . . 8 ⊢ (.r‘ndx) ∈ ℕ
59	57	slotex 12705	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑍 → (.r‘𝑅) ∈ V)
60	10, 59	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (.r‘𝑅) ∈ V)
61		ovexg 5956	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ V ∧ (.r‘𝑅) ∈ V ∧ 𝑞 ∈ V) → (𝑝(.r‘𝑅)𝑞) ∈ V)
62	36, 60, 40, 61	syl3anc 1249	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑝(.r‘𝑅)𝑞) ∈ V)
63		fvexg 5577	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ V ∧ (𝑝(.r‘𝑅)𝑞) ∈ V) → (𝐹‘(𝑝(.r‘𝑅)𝑞)) ∈ V)
64	27, 62, 63	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘(𝑝(.r‘𝑅)𝑞)) ∈ V)
65		opexg 4261	. . . . . . . . . . . . . 14 ⊢ ((⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V ∧ (𝐹‘(𝑝(.r‘𝑅)𝑞)) ∈ V) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩ ∈ V)
66	35, 64, 65	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩ ∈ V)
67		snexg 4217	. . . . . . . . . . . . 13 ⊢ (⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩ ∈ V → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
68	66, 67	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
69	68	ralrimivw 2571	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
70		iunexg 6176	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ V ∧ ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V) → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
71	19, 69, 70	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
72	71	ralrimivw 2571	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
73		iunexg 6176	. . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V) → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
74	19, 72, 73	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
75		opexg 4261	. . . . . . . 8 ⊢ (((.r‘ndx) ∈ ℕ ∧ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V) → ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩ ∈ V)
76	58, 74, 75	sylancr 414	. . . . . . 7 ⊢ (𝜑 → ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩ ∈ V)
77		tpexg 4479	. . . . . . 7 ⊢ ((⟨(Base‘ndx), 𝐵⟩ ∈ V ∧ ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩ ∈ V ∧ ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩ ∈ V) → {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩} ∈ V)
78	23, 56, 76, 77	syl3anc 1249	. . . . . 6 ⊢ (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩} ∈ V)
79	11, 78	eqeltrd 2273	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ V)
80		plusgid 12788	. . . . 5 ⊢ +g = Slot (+g‘ndx)
81	79, 80, 24	strndxid 12706	. . . 4 ⊢ (𝜑 → (𝑈‘(+g‘ndx)) = (+g‘𝑈))
82	24	a1i 9	. . . . 5 ⊢ (𝜑 → (+g‘ndx) ∈ ℕ)
83		basendxnplusgndx 12802	. . . . . 6 ⊢ (Base‘ndx) ≠ (+g‘ndx)
84	83	a1i 9	. . . . 5 ⊢ (𝜑 → (Base‘ndx) ≠ (+g‘ndx))
85		plusgndxnmulrndx 12810	. . . . . 6 ⊢ (+g‘ndx) ≠ (.r‘ndx)
86	85	a1i 9	. . . . 5 ⊢ (𝜑 → (+g‘ndx) ≠ (.r‘ndx))
87		fvtp2g 5771	. . . . 5 ⊢ ((((+g‘ndx) ∈ ℕ ∧ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V) ∧ ((Base‘ndx) ≠ (+g‘ndx) ∧ (+g‘ndx) ≠ (.r‘ndx))) → ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩}‘(+g‘ndx)) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
88	82, 54, 84, 86, 87	syl22anc 1250	. . . 4 ⊢ (𝜑 → ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩}‘(+g‘ndx)) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
89	12, 81, 88	3eqtr3rd 2238	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} = (+g‘𝑈))
90	1, 89	eqtr4id 2248	. 2 ⊢ (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
91		imasplusg.p	. . . . . . . . . 10 ⊢ + = (+g‘𝑅)
92	91	oveqi 5935	. . . . . . . . 9 ⊢ (𝑝 + 𝑞) = (𝑝(+g‘𝑅)𝑞)
93	92	fveq2i 5561	. . . . . . . 8 ⊢ (𝐹‘(𝑝 + 𝑞)) = (𝐹‘(𝑝(+g‘𝑅)𝑞))
94	93	opeq2i 3812	. . . . . . 7 ⊢ ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩
95	94	sneqi 3634	. . . . . 6 ⊢ {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}
96	95	a1i 9	. . . . 5 ⊢ (𝑞 ∈ 𝑉 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
97	96	iuneq2i 3934	. . . 4 ⊢ ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} = ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}
98	97	a1i 9	. . 3 ⊢ (𝑝 ∈ 𝑉 → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} = ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
99	98	iuneq2i 3934	. 2 ⊢ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}
100	90, 99	eqtr4di 2247	1 ⊢ (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  ∀wral 2475  Vcvv 2763  {csn 3622  {ctp 3624  ⟨cop 3625  ∪ ciun 3916   Fn wfn 5253  ⟶wf 5254  –onto→wfo 5256  ‘cfv 5258  (class class class)co 5922  ℕcn 8990  ndxcnx 12675  Slot cslot 12677  Basecbs 12678  +_gcplusg 12755  ._rcmulr 12756   ·_𝑠 cvsca 12759   “_s cimas 12942

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-mulr 12769  df-iimas 12945

This theorem is referenced by:  imasaddfn  12960  imasaddval  12961  imasaddf  12962  qusaddval  12978  qusaddf  12979