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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 6906 | . . . . . . . 8 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = (+g‘𝑅)) | |
| 4 | oppgval.2 | . . . . . . . 8 ⊢ + = (+g‘𝑅) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = + ) | 
| 6 | 5 | tposeqd 8254 | . . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (+g‘𝑥) = tpos + ) | 
| 7 | 6 | opeq2d 4880 | . . . . 5 ⊢ (𝑥 = 𝑅 → 〈(+g‘ndx), tpos (+g‘𝑥)〉 = 〈(+g‘ndx), tpos + 〉) | 
| 8 | 2, 7 | oveq12d 7449 | . . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet 〈(+g‘ndx), tpos (+g‘𝑥)〉) = (𝑅 sSet 〈(+g‘ndx), tpos + 〉)) | 
| 9 | df-oppg 19364 | . . . 4 ⊢ oppg = (𝑥 ∈ V ↦ (𝑥 sSet 〈(+g‘ndx), tpos (+g‘𝑥)〉)) | |
| 10 | ovex 7464 | . . . 4 ⊢ (𝑅 sSet 〈(+g‘ndx), tpos + 〉) ∈ V | |
| 11 | 8, 9, 10 | fvmpt 7016 | . . 3 ⊢ (𝑅 ∈ V → (oppg‘𝑅) = (𝑅 sSet 〈(+g‘ndx), tpos + 〉)) | 
| 12 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (oppg‘𝑅) = ∅) | |
| 13 | reldmsets 17202 | . . . . 5 ⊢ Rel dom sSet | |
| 14 | 13 | ovprc1 7470 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(+g‘ndx), tpos + 〉) = ∅) | 
| 15 | 12, 14 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝑅 ∈ V → (oppg‘𝑅) = (𝑅 sSet 〈(+g‘ndx), tpos + 〉)) | 
| 16 | 11, 15 | pm2.61i 182 | . 2 ⊢ (oppg‘𝑅) = (𝑅 sSet 〈(+g‘ndx), tpos + 〉) | 
| 17 | 1, 16 | eqtri 2765 | 1 ⊢ 𝑂 = (𝑅 sSet 〈(+g‘ndx), tpos + 〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 ‘cfv 6561 (class class class)co 7431 tpos ctpos 8250 sSet csts 17200 ndxcnx 17230 +gcplusg 17297 oppgcoppg 19363 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-tpos 8251 df-sets 17201 df-oppg 19364 | 
| This theorem is referenced by: oppgplusfval 19366 oppglemOLD 19369 oppgbas 19370 oppgtset 19371 oppgle 32951 | 
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