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Theorem imasplusg 12729

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 2177	. . . . . 6 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
5		eqid 2177	. . . . . 6 ⊢ (._r‘𝑅) = (._r‘𝑅)
6		eqid 2177	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑅) = ( ·_𝑠 ‘𝑅)
7		eqidd 2178	. . . . . 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 2178	. . . . . 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 12727	. . . . 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 5518	. . . 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 12518	. . . . . . . 8 ⊢ (Base‘ndx) ∈ ℕ
14		basfn 12520	. . . . . . . . . . 11 ⊢ Base Fn V
15	10	elexd 2751	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ V)
16		funfvex 5533	. . . . . . . . . . . 12 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
17	16	funfni 5317	. . . . . . . . . . 11 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
18	14, 15, 17	sylancr 414	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
19	3, 18	eqeltrd 2254	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ V)
20		focdmex 6116	. . . . . . . . 9 ⊢ (𝑉 ∈ V → (𝐹:𝑉–onto→𝐵 → 𝐵 ∈ V))
21	19, 9, 20	sylc 62	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
22		opexg 4229	. . . . . . . 8 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ V) → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
23	13, 21, 22	sylancr 414	. . . . . . 7 ⊢ (𝜑 → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
24		plusgndxnn 12570	. . . . . . . 8 ⊢ (+_g‘ndx) ∈ ℕ
25		fof 5439	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
26	9, 25	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
27	26, 19	fexd 5747	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ∈ V)
28		vex 2741	. . . . . . . . . . . . . . . 16 ⊢ 𝑝 ∈ V
29		fvexg 5535	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ V ∧ 𝑝 ∈ V) → (𝐹‘𝑝) ∈ V)
30	27, 28, 29	sylancl 413	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝑝) ∈ V)
31		vex 2741	. . . . . . . . . . . . . . . 16 ⊢ 𝑞 ∈ V
32		fvexg 5535	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ V ∧ 𝑞 ∈ V) → (𝐹‘𝑞) ∈ V)
33	27, 31, 32	sylancl 413	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝑞) ∈ V)
34		opexg 4229	. . . . . . . . . . . . . . 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 12571	. . . . . . . . . . . . . . . . . 18 ⊢ (+_g = Slot (+_g‘ndx) ∧ (+_g‘ndx) ∈ ℕ)
38	37	slotex 12489	. . . . . . . . . . . . . . . . 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 5909	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ V ∧ (+_g‘𝑅) ∈ V ∧ 𝑞 ∈ V) → (𝑝(+_g‘𝑅)𝑞) ∈ V)
42	36, 39, 40, 41	syl3anc 1238	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑝(+_g‘𝑅)𝑞) ∈ V)
43		fvexg 5535	. . . . . . . . . . . . . . 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 4229	. . . . . . . . . . . . . 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 4185	. . . . . . . . . . . . 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 2551	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+_g‘𝑅)𝑞))⟩} ∈ V)
50		iunexg 6120	. . . . . . . . . . 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 2551	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+_g‘𝑅)𝑞))⟩} ∈ V)
53		iunexg 6120	. . . . . . . . 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 4229	. . . . . . . 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 12590	. . . . . . . . 9 ⊢ (._r = Slot (._r‘ndx) ∧ (._r‘ndx) ∈ ℕ)
58	57	simpri 113	. . . . . . . 8 ⊢ (._r‘ndx) ∈ ℕ
59	57	slotex 12489	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑍 → (._r‘𝑅) ∈ V)
60	10, 59	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (._r‘𝑅) ∈ V)
61		ovexg 5909	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ V ∧ (._r‘𝑅) ∈ V ∧ 𝑞 ∈ V) → (𝑝(._r‘𝑅)𝑞) ∈ V)
62	36, 60, 40, 61	syl3anc 1238	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑝(._r‘𝑅)𝑞) ∈ V)
63		fvexg 5535	. . . . . . . . . . . . . . 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 4229	. . . . . . . . . . . . . 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 4185	. . . . . . . . . . . . 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 2551	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(._r‘𝑅)𝑞))⟩} ∈ V)
70		iunexg 6120	. . . . . . . . . . 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 2551	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(._r‘𝑅)𝑞))⟩} ∈ V)
73		iunexg 6120	. . . . . . . . 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 4229	. . . . . . . 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 4445	. . . . . . 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 1238	. . . . . 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 2254	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ V)
80		plusgid 12569	. . . . 5 ⊢ +_g = Slot (+_g‘ndx)
81	79, 80, 24	strndxid 12490	. . . 4 ⊢ (𝜑 → (𝑈‘(+_g‘ndx)) = (+_g‘𝑈))
82	24	a1i 9	. . . . 5 ⊢ (𝜑 → (+_g‘ndx) ∈ ℕ)
83		basendxnplusgndx 12583	. . . . . 6 ⊢ (Base‘ndx) ≠ (+_g‘ndx)
84	83	a1i 9	. . . . 5 ⊢ (𝜑 → (Base‘ndx) ≠ (+_g‘ndx))
85		plusgndxnmulrndx 12591	. . . . . 6 ⊢ (+_g‘ndx) ≠ (._r‘ndx)
86	85	a1i 9	. . . . 5 ⊢ (𝜑 → (+_g‘ndx) ≠ (._r‘ndx))
87		fvtp2g 5726	. . . . 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 1239	. . . 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 2219	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+_g‘𝑅)𝑞))⟩} = (+_g‘𝑈))
90	1, 89	eqtr4id 2229	. 2 ⊢ (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+_g‘𝑅)𝑞))⟩})
91		imasplusg.p	. . . . . . . . . 10 ⊢ + = (+_g‘𝑅)
92	91	oveqi 5888	. . . . . . . . 9 ⊢ (𝑝 + 𝑞) = (𝑝(+_g‘𝑅)𝑞)
93	92	fveq2i 5519	. . . . . . . 8 ⊢ (𝐹‘(𝑝 + 𝑞)) = (𝐹‘(𝑝(+_g‘𝑅)𝑞))
94	93	opeq2i 3783	. . . . . . 7 ⊢ ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+_g‘𝑅)𝑞))⟩
95	94	sneqi 3605	. . . . . 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 3905	. . . 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 3905	. 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 2228	1 ⊢ (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 Vcvv 2738 {csn 3593 {ctp 3595 ⟨cop 3596 ∪ ciun 3887 Fn wfn 5212 ⟶wf 5213 –onto→wfo 5215 ‘cfv 5217 (class class class)co 5875 ℕcn 8919 ndxcnx 12459 Slot cslot 12461 Basecbs 12462 +_gcplusg 12536 ._rcmulr 12537 ·_𝑠 cvsca 12540 “_s cimas 12720

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-coll 4119 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-addcom 7911 ax-addass 7913 ax-i2m1 7916 ax-0lt1 7917 ax-0id 7919 ax-rnegex 7920 ax-pre-ltirr 7923 ax-pre-ltadd 7927

This theorem depends on definitions: df-bi 117 df-3or 979 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-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-tp 3601 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-ov 5878 df-oprab 5879 df-mpo 5880 df-pnf 7994 df-mnf 7995 df-ltxr 7997 df-inn 8920 df-2 8978 df-3 8979 df-ndx 12465 df-slot 12466 df-base 12468 df-plusg 12549 df-mulr 12550 df-iimas 12723

This theorem is referenced by: imasaddfn 12738 imasaddval 12739 imasaddf 12740 qusaddval 12754 qusaddf 12755

W3C validator