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Theorem qusval 16665
 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 16632 . . . 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 758 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → 𝑟 = 𝑅)
54fveq2d 6497 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (Base‘𝑟) = (Base‘𝑅))
6 qusval.v . . . . . . . 8 (𝜑𝑉 = (Base‘𝑅))
76adantr 473 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → 𝑉 = (Base‘𝑅))
85, 7eqtr4d 2811 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (Base‘𝑟) = 𝑉)
9 eceq2 8123 . . . . . . 7 (𝑒 = → [𝑥]𝑒 = [𝑥] )
109ad2antll 716 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → [𝑥]𝑒 = [𝑥] )
118, 10mpteq12dv 5006 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥𝑉 ↦ [𝑥] ))
12 qusval.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
1311, 12syl6eqr 2826 . . . 4 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹)
1413, 4oveq12d 6988 . . 3 ((𝜑 ∧ (𝑟 = 𝑅𝑒 = )) → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = (𝐹s 𝑅))
15 qusval.r . . . 4 (𝜑𝑅𝑍)
1615elexd 3429 . . 3 (𝜑𝑅 ∈ V)
17 qusval.e . . . 4 (𝜑𝑊)
1817elexd 3429 . . 3 (𝜑 ∈ V)
19 ovexd 7004 . . 3 (𝜑 → (𝐹s 𝑅) ∈ V)
203, 14, 16, 18, 19ovmpod 7112 . 2 (𝜑 → (𝑅 /s ) = (𝐹s 𝑅))
211, 20eqtrd 2808 1 (𝜑𝑈 = (𝐹s 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  Vcvv 3409   ↦ cmpt 5002  ‘cfv 6182  (class class class)co 6970   ∈ cmpo 6972  [cec 8081  Basecbs 16333   “s cimas 16627   /s cqus 16628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-ec 8085  df-qus 16632 This theorem is referenced by:  qusin  16667  qusbas  16668  quss  16669  qusaddval  16676  qusaddf  16677  qusmulval  16678  qusmulf  16679  qusgrp2  17998  qusring2  19087  qustps  22028  qustgpopn  22425  qustgplem  22426  qustgphaus  22428  qusscaval  30600  quslmod  30602  quslmhm  30603
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