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Mirrors > Home > MPE Home > Th. List > qusval | Structured version Visualization version GIF version |
Description: Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
qusval.u | ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) |
qusval.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
qusval.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) |
qusval.e | ⊢ (𝜑 → ∼ ∈ 𝑊) |
qusval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
Ref | Expression |
---|---|
qusval | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusval.u | . 2 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) | |
2 | df-qus 17569 | . . . 4 ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟)) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))) |
4 | simprl 770 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑟 = 𝑅) | |
5 | 4 | fveq2d 6924 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = (Base‘𝑅)) |
6 | qusval.v | . . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
7 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑉 = (Base‘𝑅)) |
8 | 5, 7 | eqtr4d 2783 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = 𝑉) |
9 | eceq2 8804 | . . . . . . 7 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ ) | |
10 | 9 | ad2antll 728 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → [𝑥]𝑒 = [𝑥] ∼ ) |
11 | 8, 10 | mpteq12dv 5257 | . . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )) |
12 | qusval.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
13 | 11, 12 | eqtr4di 2798 | . . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹) |
14 | 13, 4 | oveq12d 7466 | . . 3 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = (𝐹 “s 𝑅)) |
15 | qusval.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
16 | 15 | elexd 3512 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) |
17 | qusval.e | . . . 4 ⊢ (𝜑 → ∼ ∈ 𝑊) | |
18 | 17 | elexd 3512 | . . 3 ⊢ (𝜑 → ∼ ∈ V) |
19 | ovexd 7483 | . . 3 ⊢ (𝜑 → (𝐹 “s 𝑅) ∈ V) | |
20 | 3, 14, 16, 18, 19 | ovmpod 7602 | . 2 ⊢ (𝜑 → (𝑅 /s ∼ ) = (𝐹 “s 𝑅)) |
21 | 1, 20 | eqtrd 2780 | 1 ⊢ (𝜑 → 𝑈 = (𝐹 “s 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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