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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 16774 | . . . 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 6649 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = (Base‘𝑅)) |
6 | qusval.v | . . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
7 | 6 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑉 = (Base‘𝑅)) |
8 | 5, 7 | eqtr4d 2836 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = 𝑉) |
9 | eceq2 8312 | . . . . . . 7 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ ) | |
10 | 9 | ad2antll 728 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → [𝑥]𝑒 = [𝑥] ∼ ) |
11 | 8, 10 | mpteq12dv 5115 | . . . . 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 2851 | . . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹) |
14 | 13, 4 | oveq12d 7153 | . . 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 3461 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) |
17 | qusval.e | . . . 4 ⊢ (𝜑 → ∼ ∈ 𝑊) | |
18 | 17 | elexd 3461 | . . 3 ⊢ (𝜑 → ∼ ∈ V) |
19 | ovexd 7170 | . . 3 ⊢ (𝜑 → (𝐹 “s 𝑅) ∈ V) | |
20 | 3, 14, 16, 18, 19 | ovmpod 7281 | . 2 ⊢ (𝜑 → (𝑅 /s ∼ ) = (𝐹 “s 𝑅)) |
21 | 1, 20 | eqtrd 2833 | 1 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 [cec 8270 Basecbs 16475 “s cimas 16769 /s cqus 16770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-ec 8274 df-qus 16774 |
This theorem is referenced by: qusin 16809 qusbas 16810 quss 16811 qusaddval 16818 qusaddf 16819 qusmulval 16820 qusmulf 16821 qusgrp2 18209 qusring2 19366 qustps 22327 qustgpopn 22725 qustgplem 22726 qustgphaus 22728 qusscaval 30972 quslmod 30974 quslmhm 30975 |
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