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Theorem qusval 17602
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 17569 . . . 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 770 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → 𝑟 = 𝑅)
54fveq2d 6924 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (Base‘𝑟) = (Base‘𝑅))
6 qusval.v . . . . . . . 8 (𝜑𝑉 = (Base‘𝑅))
76adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → 𝑉 = (Base‘𝑅))
85, 7eqtr4d 2783 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (Base‘𝑟) = 𝑉)
9 eceq2 8804 . . . . . . 7 (𝑒 = → [𝑥]𝑒 = [𝑥] )
109ad2antll 728 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → [𝑥]𝑒 = [𝑥] )
118, 10mpteq12dv 5257 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥𝑉 ↦ [𝑥] ))
12 qusval.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
1311, 12eqtr4di 2798 . . . 4 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹)
1413, 4oveq12d 7466 . . 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 7483 . . 3 (𝜑 → (𝐹s 𝑅) ∈ V)
203, 14, 16, 18, 19ovmpod 7602 . 2 (𝜑 → (𝑅 /s ) = (𝐹s 𝑅))
211, 20eqtrd 2780 1 (𝜑𝑈 = (𝐹s 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  Vcvv 3488  cmpt 5249  cfv 6573  (class class class)co 7448  cmpo 7450  [cec 8761  Basecbs 17258  s cimas 17564   /s cqus 17565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-ec 8765  df-qus 17569
This theorem is referenced by:  qusin  17604  qusbas  17605  quss  17606  qusaddval  17613  qusaddf  17614  qusmulval  17615  qusmulf  17616  qusgrp2  19098  qusrng  20207  qusring2  20357  qustps  23751  qustgpopn  24149  qustgplem  24150  qustgphaus  24152  qusvsval  33345  quslmod  33351  quslmhm  33352
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