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Theorem qusval 16404
Description: Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
qusval.u (𝜑𝑈 = (𝑅 /s ))
qusval.v (𝜑𝑉 = (Base‘𝑅))
qusval.f 𝐹 = (𝑥𝑉 ↦ [𝑥] )
qusval.e (𝜑𝑊)
qusval.r (𝜑𝑅𝑍)
Assertion
Ref Expression
qusval (𝜑𝑈 = (𝐹s 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.
StepHypRef Expression
1 qusval.u . 2 (𝜑𝑈 = (𝑅 /s ))
2 df-qus 16371 . . . 4 /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
32a1i 11 . . 3 (𝜑 → /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟)))
4 simprl 811 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → 𝑟 = 𝑅)
54fveq2d 6356 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (Base‘𝑟) = (Base‘𝑅))
6 qusval.v . . . . . . . 8 (𝜑𝑉 = (Base‘𝑅))
76adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → 𝑉 = (Base‘𝑅))
85, 7eqtr4d 2797 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (Base‘𝑟) = 𝑉)
9 eceq2 7951 . . . . . . 7 (𝑒 = → [𝑥]𝑒 = [𝑥] )
109ad2antll 767 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → [𝑥]𝑒 = [𝑥] )
118, 10mpteq12dv 4885 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥𝑉 ↦ [𝑥] ))
12 qusval.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
1311, 12syl6eqr 2812 . . . 4 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹)
1413, 4oveq12d 6831 . . 3 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = (𝐹s 𝑅))
15 qusval.r . . . 4 (𝜑𝑅𝑍)
16 elex 3352 . . . 4 (𝑅𝑍𝑅 ∈ V)
1715, 16syl 17 . . 3 (𝜑𝑅 ∈ V)
18 qusval.e . . . 4 (𝜑𝑊)
19 elex 3352 . . . 4 ( 𝑊 ∈ V)
2018, 19syl 17 . . 3 (𝜑 ∈ V)
21 ovexd 6843 . . 3 (𝜑 → (𝐹s 𝑅) ∈ V)
223, 14, 17, 20, 21ovmpt2d 6953 . 2 (𝜑 → (𝑅 /s ) = (𝐹s 𝑅))
231, 22eqtrd 2794 1 (𝜑𝑈 = (𝐹s 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cmpt 4881  cfv 6049  (class class class)co 6813  cmpt2 6815  [cec 7909  Basecbs 16059  s cimas 16366   /s cqus 16367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-ec 7913  df-qus 16371
This theorem is referenced by:  qusin  16406  qusbas  16407  quss  16408  qusaddval  16415  qusaddf  16416  qusmulval  16417  qusmulf  16418  qusgrp2  17734  qusring2  18820  qustps  21727  qustgpopn  22124  qustgplem  22125  qustgphaus  22127
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