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Mirrors > Home > ILE Home > Th. List > opprbasg | GIF version |
Description: Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.) |
Ref | Expression |
---|---|
opprbas.1 | β’ π = (opprβπ ) |
opprbas.2 | β’ π΅ = (Baseβπ ) |
Ref | Expression |
---|---|
opprbasg | β’ (π β π β π΅ = (Baseβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprbas.2 | . 2 β’ π΅ = (Baseβπ ) | |
2 | opprbas.1 | . . 3 β’ π = (opprβπ ) | |
3 | baseslid 12532 | . . 3 β’ (Base = Slot (Baseβndx) β§ (Baseβndx) β β) | |
4 | basendxnmulrndx 12606 | . . 3 β’ (Baseβndx) β (.rβndx) | |
5 | 2, 3, 4 | opprsllem 13307 | . 2 β’ (π β π β (Baseβπ ) = (Baseβπ)) |
6 | 1, 5 | eqtrid 2232 | 1 β’ (π β π β π΅ = (Baseβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1363 β wcel 2158 βcfv 5228 Basecbs 12475 opprcoppr 13300 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-pre-ltirr 7936 ax-pre-lttrn 7938 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-tpos 6259 df-pnf 8007 df-mnf 8008 df-ltxr 8010 df-inn 8933 df-2 8991 df-3 8992 df-ndx 12478 df-slot 12479 df-base 12481 df-sets 12482 df-mulr 12564 df-oppr 13301 |
This theorem is referenced by: opprrng 13310 opprring 13311 opprringbg 13312 oppr0g 13313 oppr1g 13314 opprnegg 13315 opprsubgg 13316 mulgass3 13317 1unit 13339 opprunitd 13342 crngunit 13343 unitmulcl 13345 unitgrp 13348 unitnegcl 13362 unitpropdg 13380 subrguss 13420 subrgunit 13423 isridlrng 13635 isridl 13652 ridl1 13659 |
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