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Mirrors > Home > MPE Home > Th. List > opprval | Structured version Visualization version GIF version |
Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
opprval.1 | ⊢ 𝐵 = (Base‘𝑅) |
opprval.2 | ⊢ · = (.r‘𝑅) |
opprval.3 | ⊢ 𝑂 = (oppr‘𝑅) |
Ref | Expression |
---|---|
opprval | ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprval.3 | . 2 ⊢ 𝑂 = (oppr‘𝑅) | |
2 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅) | |
3 | fveq2 6839 | . . . . . . . 8 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = (.r‘𝑅)) | |
4 | opprval.2 | . . . . . . . 8 ⊢ · = (.r‘𝑅) | |
5 | 3, 4 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = · ) |
6 | 5 | tposeqd 8152 | . . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (.r‘𝑥) = tpos · ) |
7 | 6 | opeq2d 4835 | . . . . 5 ⊢ (𝑥 = 𝑅 → 〈(.r‘ndx), tpos (.r‘𝑥)〉 = 〈(.r‘ndx), tpos · 〉) |
8 | 2, 7 | oveq12d 7369 | . . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet 〈(.r‘ndx), tpos (.r‘𝑥)〉) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
9 | df-oppr 20001 | . . . 4 ⊢ oppr = (𝑥 ∈ V ↦ (𝑥 sSet 〈(.r‘ndx), tpos (.r‘𝑥)〉)) | |
10 | ovex 7384 | . . . 4 ⊢ (𝑅 sSet 〈(.r‘ndx), tpos · 〉) ∈ V | |
11 | 8, 9, 10 | fvmpt 6945 | . . 3 ⊢ (𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
12 | fvprc 6831 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅) | |
13 | reldmsets 16996 | . . . . 5 ⊢ Rel dom sSet | |
14 | 13 | ovprc1 7390 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(.r‘ndx), tpos · 〉) = ∅) |
15 | 12, 14 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
16 | 11, 15 | pm2.61i 182 | . 2 ⊢ (oppr‘𝑅) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
17 | 1, 16 | eqtri 2765 | 1 ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∅c0 4280 〈cop 4590 ‘cfv 6493 (class class class)co 7351 tpos ctpos 8148 sSet csts 16994 ndxcnx 17024 Basecbs 17042 .rcmulr 17093 opprcoppr 20000 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-res 5643 df-iota 6445 df-fun 6495 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-tpos 8149 df-sets 16995 df-oppr 20001 |
This theorem is referenced by: opprmulfval 20003 opprlem 20006 opprlemOLD 20007 |
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