Theorem qusval 13025

Description: Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
qusval.u	⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
qusval.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
qusval.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
qusval.e	⊢ (𝜑 → ∼ ∈ 𝑊)
qusval.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)

Assertion

Ref	Expression
qusval	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))

Distinct variable groups:   𝑥, ∼   𝜑,𝑥   𝑥,𝑅   𝑥,𝑉

Allowed substitution hints:   𝑈(𝑥)   𝐹(𝑥)   𝑊(𝑥)   𝑍(𝑥)

Proof of Theorem qusval

Dummy variables 𝑒 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusval.u	. 2 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
2		df-qus 13005	. . . 4 ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
3	2	a1i 9	. . 3 ⊢ (𝜑 → /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟)))
4		simprl 529	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑟 = 𝑅)
5	4	fveq2d 5565	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = (Base‘𝑅))
6		qusval.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
7	6	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑉 = (Base‘𝑅))
8	5, 7	eqtr4d 2232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = 𝑉)
9		eceq2 6638	. . . . . . 7 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ )
10	9	ad2antll 491	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → [𝑥]𝑒 = [𝑥] ∼ )
11	8, 10	mpteq12dv 4116	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ))
12		qusval.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
13	11, 12	eqtr4di 2247	. . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹)
14	13, 4	oveq12d 5943	. . 3 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = (𝐹 “s 𝑅))
15		qusval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
16	15	elexd 2776	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
17		qusval.e	. . . 4 ⊢ (𝜑 → ∼ ∈ 𝑊)
18	17	elexd 2776	. . 3 ⊢ (𝜑 → ∼ ∈ V)
19		basfn 12761	. . . . . . . 8 ⊢ Base Fn V
20		funfvex 5578	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
21	20	funfni 5361	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
22	19, 16, 21	sylancr 414	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
23	6, 22	eqeltrd 2273	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ V)
24	23	mptexd 5792	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) ∈ V)
25	12, 24	eqeltrid 2283	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
26		imasex 13007	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑅 ∈ 𝑍) → (𝐹 “s 𝑅) ∈ V)
27	25, 15, 26	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐹 “s 𝑅) ∈ V)
28	3, 14, 16, 18, 27	ovmpod 6054	. 2 ⊢ (𝜑 → (𝑅 /s ∼ ) = (𝐹 “s 𝑅))
29	1, 28	eqtrd 2229	1 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ↦ cmpt 4095   Fn wfn 5254  ‘cfv 5259  (class class class)co 5925   ∈ cmpo 5927  [cec 6599  Basecbs 12703   “_s cimas 13001   /_s cqus 13002

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 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  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-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-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-ec 6603  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-mulr 12794  df-iimas 13004  df-qus 13005

This theorem is referenced by:  qusin  13028  qusbas  13029  qusaddval  13037  qusaddf  13038  qusmulval  13039  qusmulf  13040  qusgrp2  13319  qusrng  13590  qusring2  13698