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Mirrors > Home > MPE Home > Th. List > oppgval | Structured version Visualization version GIF version |
Description: Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.) |
Ref | Expression |
---|---|
oppgval.2 | ⊢ + = (+g‘𝑅) |
oppgval.3 | ⊢ 𝑂 = (oppg‘𝑅) |
Ref | Expression |
---|---|
oppgval | ⊢ 𝑂 = (𝑅 sSet 〈(+g‘ndx), tpos + 〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppgval.3 | . 2 ⊢ 𝑂 = (oppg‘𝑅) | |
2 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅) | |
3 | fveq2 6645 | . . . . . . . 8 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = (+g‘𝑅)) | |
4 | oppgval.2 | . . . . . . . 8 ⊢ + = (+g‘𝑅) | |
5 | 3, 4 | eqtr4di 2851 | . . . . . . 7 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = + ) |
6 | 5 | tposeqd 7878 | . . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (+g‘𝑥) = tpos + ) |
7 | 6 | opeq2d 4772 | . . . . 5 ⊢ (𝑥 = 𝑅 → 〈(+g‘ndx), tpos (+g‘𝑥)〉 = 〈(+g‘ndx), tpos + 〉) |
8 | 2, 7 | oveq12d 7153 | . . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet 〈(+g‘ndx), tpos (+g‘𝑥)〉) = (𝑅 sSet 〈(+g‘ndx), tpos + 〉)) |
9 | df-oppg 18466 | . . . 4 ⊢ oppg = (𝑥 ∈ V ↦ (𝑥 sSet 〈(+g‘ndx), tpos (+g‘𝑥)〉)) | |
10 | ovex 7168 | . . . 4 ⊢ (𝑅 sSet 〈(+g‘ndx), tpos + 〉) ∈ V | |
11 | 8, 9, 10 | fvmpt 6745 | . . 3 ⊢ (𝑅 ∈ V → (oppg‘𝑅) = (𝑅 sSet 〈(+g‘ndx), tpos + 〉)) |
12 | fvprc 6638 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (oppg‘𝑅) = ∅) | |
13 | reldmsets 16503 | . . . . 5 ⊢ Rel dom sSet | |
14 | 13 | ovprc1 7174 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(+g‘ndx), tpos + 〉) = ∅) |
15 | 12, 14 | eqtr4d 2836 | . . 3 ⊢ (¬ 𝑅 ∈ V → (oppg‘𝑅) = (𝑅 sSet 〈(+g‘ndx), tpos + 〉)) |
16 | 11, 15 | pm2.61i 185 | . 2 ⊢ (oppg‘𝑅) = (𝑅 sSet 〈(+g‘ndx), tpos + 〉) |
17 | 1, 16 | eqtri 2821 | 1 ⊢ 𝑂 = (𝑅 sSet 〈(+g‘ndx), tpos + 〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 〈cop 4531 ‘cfv 6324 (class class class)co 7135 tpos ctpos 7874 ndxcnx 16472 sSet csts 16473 +gcplusg 16557 oppgcoppg 18465 |
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-pow 5231 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-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-tpos 7875 df-sets 16482 df-oppg 18466 |
This theorem is referenced by: oppgplusfval 18468 oppglem 18470 |
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