Theorem qusval 17586

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 17553	. . . 4 ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
3	2	a1i 11	. . 3 ⊢ (𝜑 → /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟)))
4		simprl 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑟 = 𝑅)
5	4	fveq2d 6875	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = (Base‘𝑅))
6		qusval.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
7	6	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑉 = (Base‘𝑅))
8	5, 7	eqtr4d 2803	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = 𝑉)
9		eceq2 8724	. . . . . . 7 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ )
10	9	ad2antll 741	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → [𝑥]𝑒 = [𝑥] ∼ )
11	8, 10	mpteq12dv 5192	. . . . 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 2818	. . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹)
14	13, 4	oveq12d 7418	. . 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 3480	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
17		qusval.e	. . . 4 ⊢ (𝜑 → ∼ ∈ 𝑊)
18	17	elexd 3480	. . 3 ⊢ (𝜑 → ∼ ∈ V)
19		ovexd 7435	. . 3 ⊢ (𝜑 → (𝐹 “s 𝑅) ∈ V)
20	3, 14, 16, 18, 19	ovmpod 7552	. 2 ⊢ (𝜑 → (𝑅 /s ∼ ) = (𝐹 “s 𝑅))
21	1, 20	eqtrd 2800	1 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ↦ cmpt 5186  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  [cec 8680  Basecbs 17259   “_s cimas 17548   /_s cqus 17549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-ec 8684  df-qus 17553

This theorem is referenced by:  qusin  17588  qusbas  17589  quss  17590  qusaddval  17597  qusaddf  17598  qusmulval  17599  qusmulf  17600  qusgrp2  19115  qusrng  20249  qusring2  20407  qustps  23840  qustgpopn  24238  qustgplem  24239  qustgphaus  24241  qusvsval  33587  quslmod  33593  quslmhm  33594