Theorem imasival 13223

Description: Value of an image structure. The is a lemma for the theorems imasbas 13224, imasplusg 13225, and imasmulr 13226 and should not be needed once they are proved. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Jim Kingdon, 11-Mar-2025.) (New usage is discouraged.)

Hypotheses

Ref	Expression
imasval.u	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasval.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasval.p	⊢ + = (+g‘𝑅)
imasval.m	⊢ × = (.r‘𝑅)
imasval.q	⊢ · = ( ·𝑠 ‘𝑅)
imasval.a	⊢ (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})
imasval.t	⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩})
imasval.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasval.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)

Assertion

Ref	Expression
imasival	⊢ (𝜑 → 𝑈 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), ∙ ⟩})

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)   ∙ (𝑞,𝑝)   · (𝑞,𝑝)   × (𝑞,𝑝)   𝑈(𝑞,𝑝)   𝑍(𝑞,𝑝)

Proof of Theorem imasival

Dummy variables 𝑓 𝑟 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasval.u	. 2 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
2		df-iimas 13219	. . . 4 ⊢ “s = (𝑓 ∈ V, 𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑣⦌{⟨(Base‘ndx), ran 𝑓⟩, ⟨(+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))⟩}⟩})
3	2	a1i 9	. . 3 ⊢ (𝜑 → “s = (𝑓 ∈ V, 𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑣⦌{⟨(Base‘ndx), ran 𝑓⟩, ⟨(+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))⟩}⟩}))
4		basfn 12975	. . . . . 6 ⊢ Base Fn V
5		vex 2776	. . . . . 6 ⊢ 𝑟 ∈ V
6		funfvex 5611	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑟 ∈ dom Base) → (Base‘𝑟) ∈ V)
7	6	funfni 5390	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑟 ∈ V) → (Base‘𝑟) ∈ V)
8	4, 5, 7	mp2an 426	. . . . 5 ⊢ (Base‘𝑟) ∈ V
9	8	a1i 9	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) → (Base‘𝑟) ∈ V)
10		simplrl 535	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 𝑓 = 𝐹)
11	10	rneqd 4921	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ran 𝑓 = ran 𝐹)
12		imasval.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
13		forn 5518	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
14	12, 13	syl 14	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵)
15	14	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ran 𝐹 = 𝐵)
16	11, 15	eqtrd 2239	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ran 𝑓 = 𝐵)
17	16	opeq2d 3835	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ⟨(Base‘ndx), ran 𝑓⟩ = ⟨(Base‘ndx), 𝐵⟩)
18		simplrr 536	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 𝑟 = 𝑅)
19	18	fveq2d 5598	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (Base‘𝑟) = (Base‘𝑅))
20		simpr 110	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 𝑣 = (Base‘𝑟))
21		imasval.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
22	21	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 𝑉 = (Base‘𝑅))
23	19, 20, 22	3eqtr4d 2249	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 𝑣 = 𝑉)
24	10	fveq1d 5596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (𝑓‘𝑝) = (𝐹‘𝑝))
25	10	fveq1d 5596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (𝑓‘𝑞) = (𝐹‘𝑞))
26	24, 25	opeq12d 3836	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩ = ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩)
27	18	fveq2d 5598	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (+g‘𝑟) = (+g‘𝑅))
28		imasval.p	. . . . . . . . . . . . . 14 ⊢ + = (+g‘𝑅)
29	27, 28	eqtr4di 2257	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (+g‘𝑟) = + )
30	29	oveqd 5979	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (𝑝(+g‘𝑟)𝑞) = (𝑝 + 𝑞))
31	10, 30	fveq12d 5601	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (𝑓‘(𝑝(+g‘𝑟)𝑞)) = (𝐹‘(𝑝 + 𝑞)))
32	26, 31	opeq12d 3836	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩)
33	32	sneqd 3651	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})
34	23, 33	iuneq12d 3960	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩} = ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})
35	23, 34	iuneq12d 3960	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})
36		imasval.a	. . . . . . . 8 ⊢ (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})
37	36	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})
38	35, 37	eqtr4d 2242	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩} = ✚ )
39	38	opeq2d 3835	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩}⟩ = ⟨(+g‘ndx), ✚ ⟩)
40	18	fveq2d 5598	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (.r‘𝑟) = (.r‘𝑅))
41		imasval.m	. . . . . . . . . . . . . 14 ⊢ × = (.r‘𝑅)
42	40, 41	eqtr4di 2257	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (.r‘𝑟) = × )
43	42	oveqd 5979	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (𝑝(.r‘𝑟)𝑞) = (𝑝 × 𝑞))
44	10, 43	fveq12d 5601	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (𝑓‘(𝑝(.r‘𝑟)𝑞)) = (𝐹‘(𝑝 × 𝑞)))
45	26, 44	opeq12d 3836	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩)
46	45	sneqd 3651	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩})
47	23, 46	iuneq12d 3960	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩} = ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩})
48	23, 47	iuneq12d 3960	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩})
49		imasval.t	. . . . . . . 8 ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩})
50	49	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩})
51	48, 50	eqtr4d 2242	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩} = ∙ )
52	51	opeq2d 3835	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩}⟩ = ⟨(.r‘ndx), ∙ ⟩)
53	17, 39, 52	tpeq123d 3730	. . . 4 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → {⟨(Base‘ndx), ran 𝑓⟩, ⟨(+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))⟩}⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), ∙ ⟩})
54	9, 53	csbied 3144	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) → ⦋(Base‘𝑟) / 𝑣⦌{⟨(Base‘ndx), ran 𝑓⟩, ⟨(+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))⟩}⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), ∙ ⟩})
55		fof 5515	. . . . 5 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
56	12, 55	syl 14	. . . 4 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
57		imasval.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
58	57	elexd 2787	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
59		funfvex 5611	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
60	59	funfni 5390	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
61	4, 58, 60	sylancr 414	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
62	21, 61	eqeltrd 2283	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
63	56, 62	fexd 5832	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
64		basendxnn 12973	. . . . 5 ⊢ (Base‘ndx) ∈ ℕ
65		focdmex 6218	. . . . . 6 ⊢ (𝑉 ∈ V → (𝐹:𝑉–onto→𝐵 → 𝐵 ∈ V))
66	62, 12, 65	sylc 62	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
67		opexg 4285	. . . . 5 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ V) → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
68	64, 66, 67	sylancr 414	. . . 4 ⊢ (𝜑 → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
69		plusgndxnn 13028	. . . . 5 ⊢ (+g‘ndx) ∈ ℕ
70	63	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → 𝐹 ∈ V)
71		vex 2776	. . . . . . . . . . . . . . 15 ⊢ 𝑝 ∈ V
72	71	a1i 9	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → 𝑝 ∈ V)
73		fvexg 5613	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ V ∧ 𝑝 ∈ V) → (𝐹‘𝑝) ∈ V)
74	70, 72, 73	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → (𝐹‘𝑝) ∈ V)
75		vex 2776	. . . . . . . . . . . . . . 15 ⊢ 𝑞 ∈ V
76	75	a1i 9	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → 𝑞 ∈ V)
77		fvexg 5613	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ V ∧ 𝑞 ∈ V) → (𝐹‘𝑞) ∈ V)
78	70, 76, 77	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → (𝐹‘𝑞) ∈ V)
79		opexg 4285	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑝) ∈ V ∧ (𝐹‘𝑞) ∈ V) → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V)
80	74, 78, 79	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V)
81		plusgslid 13029	. . . . . . . . . . . . . . . . . 18 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
82	81	slotex 12944	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑍 → (+g‘𝑅) ∈ V)
83	57, 82	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (+g‘𝑅) ∈ V)
84	28, 83	eqeltrid 2293	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → + ∈ V)
85	84	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → + ∈ V)
86		ovexg 5996	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ V ∧ + ∈ V ∧ 𝑞 ∈ V) → (𝑝 + 𝑞) ∈ V)
87	72, 85, 76, 86	syl3anc 1250	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → (𝑝 + 𝑞) ∈ V)
88		fvexg 5613	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ (𝑝 + 𝑞) ∈ V) → (𝐹‘(𝑝 + 𝑞)) ∈ V)
89	70, 87, 88	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → (𝐹‘(𝑝 + 𝑞)) ∈ V)
90		opexg 4285	. . . . . . . . . . . 12 ⊢ ((⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V ∧ (𝐹‘(𝑝 + 𝑞)) ∈ V) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩ ∈ V)
91	80, 89, 90	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩ ∈ V)
92		snexg 4239	. . . . . . . . . . 11 ⊢ (⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩ ∈ V → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} ∈ V)
93	91, 92	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} ∈ V)
94	93	ralrimiva 2580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑉) → ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} ∈ V)
95		iunexg 6222	. . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} ∈ V) → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} ∈ V)
96	62, 94, 95	syl2an2r 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑉) → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} ∈ V)
97	96	ralrimiva 2580	. . . . . . 7 ⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} ∈ V)
98		iunexg 6222	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} ∈ V) → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} ∈ V)
99	62, 97, 98	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩} ∈ V)
100	36, 99	eqeltrd 2283	. . . . 5 ⊢ (𝜑 → ✚ ∈ V)
101		opexg 4285	. . . . 5 ⊢ (((+g‘ndx) ∈ ℕ ∧ ✚ ∈ V) → ⟨(+g‘ndx), ✚ ⟩ ∈ V)
102	69, 100, 101	sylancr 414	. . . 4 ⊢ (𝜑 → ⟨(+g‘ndx), ✚ ⟩ ∈ V)
103		mulrslid 13049	. . . . . 6 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
104	103	simpri 113	. . . . 5 ⊢ (.r‘ndx) ∈ ℕ
105	103	slotex 12944	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑍 → (.r‘𝑅) ∈ V)
106	57, 105	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (.r‘𝑅) ∈ V)
107	41, 106	eqeltrid 2293	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → × ∈ V)
108	107	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → × ∈ V)
109		ovexg 5996	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ V ∧ × ∈ V ∧ 𝑞 ∈ V) → (𝑝 × 𝑞) ∈ V)
110	72, 108, 76, 109	syl3anc 1250	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → (𝑝 × 𝑞) ∈ V)
111		fvexg 5613	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ (𝑝 × 𝑞) ∈ V) → (𝐹‘(𝑝 × 𝑞)) ∈ V)
112	70, 110, 111	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → (𝐹‘(𝑝 × 𝑞)) ∈ V)
113		opexg 4285	. . . . . . . . . . . 12 ⊢ ((⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V ∧ (𝐹‘(𝑝 × 𝑞)) ∈ V) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩ ∈ V)
114	80, 112, 113	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩ ∈ V)
115		snexg 4239	. . . . . . . . . . 11 ⊢ (⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩ ∈ V → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩} ∈ V)
116	114, 115	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩} ∈ V)
117	116	ralrimiva 2580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑉) → ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩} ∈ V)
118		iunexg 6222	. . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩} ∈ V) → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩} ∈ V)
119	62, 117, 118	syl2an2r 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑉) → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩} ∈ V)
120	119	ralrimiva 2580	. . . . . . 7 ⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩} ∈ V)
121		iunexg 6222	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩} ∈ V) → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩} ∈ V)
122	62, 120, 121	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩} ∈ V)
123	49, 122	eqeltrd 2283	. . . . 5 ⊢ (𝜑 → ∙ ∈ V)
124		opexg 4285	. . . . 5 ⊢ (((.r‘ndx) ∈ ℕ ∧ ∙ ∈ V) → ⟨(.r‘ndx), ∙ ⟩ ∈ V)
125	104, 123, 124	sylancr 414	. . . 4 ⊢ (𝜑 → ⟨(.r‘ndx), ∙ ⟩ ∈ V)
126		tpexg 4504	. . . 4 ⊢ ((⟨(Base‘ndx), 𝐵⟩ ∈ V ∧ ⟨(+g‘ndx), ✚ ⟩ ∈ V ∧ ⟨(.r‘ndx), ∙ ⟩ ∈ V) → {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), ∙ ⟩} ∈ V)
127	68, 102, 125, 126	syl3anc 1250	. . 3 ⊢ (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), ∙ ⟩} ∈ V)
128	3, 54, 63, 58, 127	ovmpod 6091	. 2 ⊢ (𝜑 → (𝐹 “s 𝑅) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), ∙ ⟩})
129	1, 128	eqtrd 2239	1 ⊢ (𝜑 → 𝑈 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), ∙ ⟩})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ∀wral 2485  Vcvv 2773  ⦋csb 3097  {csn 3638  {ctp 3640  ⟨cop 3641  ∪ ciun 3936  ran crn 4689   Fn wfn 5280  ⟶wf 5281  –onto→wfo 5283  ‘cfv 5285  (class class class)co 5962   ∈ cmpo 5964  ℕcn 9066  ndxcnx 12914  Slot cslot 12916  Basecbs 12917  +_gcplusg 12994  ._rcmulr 12995   ·_𝑠 cvsca 12998   “_s cimas 13216

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1re 8049  ax-addrcl 8052

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-tp 3646  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-inn 9067  df-2 9125  df-3 9126  df-ndx 12920  df-slot 12921  df-base 12923  df-plusg 13007  df-mulr 13008  df-iimas 13219

This theorem is referenced by:  imasbas  13224  imasplusg  13225  imasmulr  13226