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Theorem qusval 16803
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 16770 . . . 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 767 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → 𝑟 = 𝑅)
54fveq2d 6667 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (Base‘𝑟) = (Base‘𝑅))
6 qusval.v . . . . . . . 8 (𝜑𝑉 = (Base‘𝑅))
76adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → 𝑉 = (Base‘𝑅))
85, 7eqtr4d 2856 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (Base‘𝑟) = 𝑉)
9 eceq2 8318 . . . . . . 7 (𝑒 = → [𝑥]𝑒 = [𝑥] )
109ad2antll 725 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → [𝑥]𝑒 = [𝑥] )
118, 10mpteq12dv 5142 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥𝑉 ↦ [𝑥] ))
12 qusval.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
1311, 12syl6eqr 2871 . . . 4 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹)
1413, 4oveq12d 7163 . . 3 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = (𝐹s 𝑅))
15 qusval.r . . . 4 (𝜑𝑅𝑍)
1615elexd 3512 . . 3 (𝜑𝑅 ∈ V)
17 qusval.e . . . 4 (𝜑𝑊)
1817elexd 3512 . . 3 (𝜑 ∈ V)
19 ovexd 7180 . . 3 (𝜑 → (𝐹s 𝑅) ∈ V)
203, 14, 16, 18, 19ovmpod 7291 . 2 (𝜑 → (𝑅 /s ) = (𝐹s 𝑅))
211, 20eqtrd 2853 1 (𝜑𝑈 = (𝐹s 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cmpt 5137  cfv 6348  (class class class)co 7145  cmpo 7147  [cec 8276  Basecbs 16471  s cimas 16765   /s cqus 16766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-ec 8280  df-qus 16770
This theorem is referenced by:  qusin  16805  qusbas  16806  quss  16807  qusaddval  16814  qusaddf  16815  qusmulval  16816  qusmulf  16817  qusgrp2  18155  qusring2  19299  qustps  22258  qustgpopn  22655  qustgplem  22656  qustgphaus  22658  qusscaval  30848  quslmod  30850  quslmhm  30851
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