![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 12537 | . . 3 β’ (Base = Slot (Baseβndx) β§ (Baseβndx) β β) | |
4 | basendxnmulrndx 12611 | . . 3 β’ (Baseβndx) β (.rβndx) | |
5 | 2, 3, 4 | opprsllem 13385 | . 2 β’ (π β π β (Baseβπ ) = (Baseβπ)) |
6 | 1, 5 | eqtrid 2234 | 1 β’ (π β π β π΅ = (Baseβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1364 β wcel 2160 βcfv 5231 Basecbs 12480 opprcoppr 13378 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-i2m1 7934 ax-0lt1 7935 ax-0id 7937 ax-rnegex 7938 ax-pre-ltirr 7941 ax-pre-lttrn 7943 ax-pre-ltadd 7945 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-tpos 6264 df-pnf 8012 df-mnf 8013 df-ltxr 8015 df-inn 8938 df-2 8996 df-3 8997 df-ndx 12483 df-slot 12484 df-base 12486 df-sets 12487 df-mulr 12569 df-oppr 13379 |
This theorem is referenced by: opprrng 13388 opprrngbg 13389 opprring 13390 opprringbg 13391 oppr0g 13392 oppr1g 13393 opprnegg 13394 opprsubgg 13395 mulgass3 13396 1unit 13418 opprunitd 13421 crngunit 13422 unitmulcl 13424 unitgrp 13427 unitnegcl 13441 unitpropdg 13459 rhmopp 13487 elrhmunit 13488 subrguss 13544 subrgunit 13547 isridlrng 13759 isridl 13780 ridl1 13787 2idlcpblrng 13799 crngridl 13805 |
Copyright terms: Public domain | W3C validator |