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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 6103 | . . . . . . . 8 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = (.r‘𝑅)) | |
4 | opprval.2 | . . . . . . . 8 ⊢ · = (.r‘𝑅) | |
5 | 3, 4 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = · ) |
6 | 5 | tposeqd 7242 | . . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (.r‘𝑥) = tpos · ) |
7 | 6 | opeq2d 4347 | . . . . 5 ⊢ (𝑥 = 𝑅 → 〈(.r‘ndx), tpos (.r‘𝑥)〉 = 〈(.r‘ndx), tpos · 〉) |
8 | 2, 7 | oveq12d 6567 | . . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet 〈(.r‘ndx), tpos (.r‘𝑥)〉) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
9 | df-oppr 18446 | . . . 4 ⊢ oppr = (𝑥 ∈ V ↦ (𝑥 sSet 〈(.r‘ndx), tpos (.r‘𝑥)〉)) | |
10 | ovex 6577 | . . . 4 ⊢ (𝑅 sSet 〈(.r‘ndx), tpos · 〉) ∈ V | |
11 | 8, 9, 10 | fvmpt 6191 | . . 3 ⊢ (𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
12 | fvprc 6097 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅) | |
13 | reldmsets 15718 | . . . . 5 ⊢ Rel dom sSet | |
14 | 13 | ovprc1 6582 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(.r‘ndx), tpos · 〉) = ∅) |
15 | 12, 14 | eqtr4d 2647 | . . 3 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
16 | 11, 15 | pm2.61i 175 | . 2 ⊢ (oppr‘𝑅) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
17 | 1, 16 | eqtri 2632 | 1 ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 〈cop 4131 ‘cfv 5804 (class class class)co 6549 tpos ctpos 7238 ndxcnx 15692 sSet csts 15693 Basecbs 15695 .rcmulr 15769 opprcoppr 18445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-tpos 7239 df-sets 15701 df-oppr 18446 |
This theorem is referenced by: opprmulfval 18448 opprlem 18451 |
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