MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qusval Structured version   Visualization version   GIF version

Theorem qusval 17588
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 17555 . . . 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 6909 . . . . . . 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 2779 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (Base‘𝑟) = 𝑉)
9 eceq2 8787 . . . . . . 7 (𝑒 = → [𝑥]𝑒 = [𝑥] )
109ad2antll 729 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → [𝑥]𝑒 = [𝑥] )
118, 10mpteq12dv 5232 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥𝑉 ↦ [𝑥] ))
12 qusval.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
1311, 12eqtr4di 2794 . . . 4 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹)
1413, 4oveq12d 7450 . . 3 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = (𝐹s 𝑅))
15 qusval.r . . . 4 (𝜑𝑅𝑍)
1615elexd 3503 . . 3 (𝜑𝑅 ∈ V)
17 qusval.e . . . 4 (𝜑𝑊)
1817elexd 3503 . . 3 (𝜑 ∈ V)
19 ovexd 7467 . . 3 (𝜑 → (𝐹s 𝑅) ∈ V)
203, 14, 16, 18, 19ovmpod 7586 . 2 (𝜑 → (𝑅 /s ) = (𝐹s 𝑅))
211, 20eqtrd 2776 1 (𝜑𝑈 = (𝐹s 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  cmpt 5224  cfv 6560  (class class class)co 7432  cmpo 7434  [cec 8744  Basecbs 17248  s cimas 17550   /s cqus 17551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-ec 8748  df-qus 17555
This theorem is referenced by:  qusin  17590  qusbas  17591  quss  17592  qusaddval  17599  qusaddf  17600  qusmulval  17601  qusmulf  17602  qusgrp2  19077  qusrng  20178  qusring2  20332  qustps  23731  qustgpopn  24129  qustgplem  24130  qustgphaus  24132  qusvsval  33381  quslmod  33387  quslmhm  33388
  Copyright terms: Public domain W3C validator