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Mirrors > Home > ILE Home > Th. List > opprex | GIF version |
Description: Existence of the opposite ring. If you know that π is a ring, see opprring 13249. (Contributed by Jim Kingdon, 10-Jan-2025.) |
Ref | Expression |
---|---|
opprex.o | β’ π = (opprβπ ) |
Ref | Expression |
---|---|
opprex | β’ (π β π β π β V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
2 | eqid 2177 | . . 3 β’ (.rβπ ) = (.rβπ ) | |
3 | opprex.o | . . 3 β’ π = (opprβπ ) | |
4 | 1, 2, 3 | opprvalg 13241 | . 2 β’ (π β π β π = (π sSet β¨(.rβndx), tpos (.rβπ )β©)) |
5 | mulrslid 12590 | . . . . 5 β’ (.r = Slot (.rβndx) β§ (.rβndx) β β) | |
6 | 5 | simpri 113 | . . . 4 β’ (.rβndx) β β |
7 | 6 | a1i 9 | . . 3 β’ (π β π β (.rβndx) β β) |
8 | 5 | slotex 12489 | . . . 4 β’ (π β π β (.rβπ ) β V) |
9 | tposexg 6259 | . . . 4 β’ ((.rβπ ) β V β tpos (.rβπ ) β V) | |
10 | 8, 9 | syl 14 | . . 3 β’ (π β π β tpos (.rβπ ) β V) |
11 | setsex 12494 | . . 3 β’ ((π β π β§ (.rβndx) β β β§ tpos (.rβπ ) β V) β (π sSet β¨(.rβndx), tpos (.rβπ )β©) β V) | |
12 | 7, 10, 11 | mpd3an23 1339 | . 2 β’ (π β π β (π sSet β¨(.rβndx), tpos (.rβπ )β©) β V) |
13 | 4, 12 | eqeltrd 2254 | 1 β’ (π β π β π β V) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 Vcvv 2738 β¨cop 3596 βcfv 5217 (class class class)co 5875 tpos ctpos 6245 βcn 8919 ndxcnx 12459 sSet csts 12460 Slot cslot 12461 Basecbs 12462 .rcmulr 12537 opprcoppr 13239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1re 7905 ax-addrcl 7908 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-fv 5225 df-ov 5878 df-oprab 5879 df-mpo 5880 df-tpos 6246 df-inn 8920 df-2 8978 df-3 8979 df-ndx 12465 df-slot 12466 df-sets 12469 df-mulr 12550 df-oppr 13240 |
This theorem is referenced by: oppr0g 13251 oppr1g 13252 opprnegg 13253 |
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