![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > opprbas | Structured version Visualization version 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 | ⊢ 𝑂 = (oppr‘𝑅) |
opprbas.2 | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
opprbas | ⊢ 𝐵 = (Base‘𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprbas.2 | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
2 | opprbas.1 | . . 3 ⊢ 𝑂 = (oppr‘𝑅) | |
3 | baseid 17176 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
4 | basendxnmulrndx 17269 | . . 3 ⊢ (Base‘ndx) ≠ (.r‘ndx) | |
5 | 2, 3, 4 | opprlem 20271 | . 2 ⊢ (Base‘𝑅) = (Base‘𝑂) |
6 | 1, 5 | eqtri 2756 | 1 ⊢ 𝐵 = (Base‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ‘cfv 6542 Basecbs 17173 opprcoppr 20265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-mulr 17240 df-oppr 20266 |
This theorem is referenced by: opprrng 20277 opprrngb 20278 opprring 20279 opprringb 20280 oppr0 20281 oppr1 20282 opprneg 20283 opprsubg 20284 mulgass3 20285 1unit 20306 opprunit 20309 crngunit 20310 unitmulcl 20312 unitgrp 20315 unitnegcl 20329 unitpropd 20349 opprirred 20354 rhmopp 20441 elrhmunit 20442 opprnzr 20452 opprsubrng 20489 subrguss 20519 subrgunit 20522 opprsubrg 20525 isdrng2 20631 opprdrng 20649 isdrngrd 20651 isdrngrdOLD 20653 issrngd 20734 rngridlmcl 21106 isridlrng 21108 isridl 21139 ridl1 21146 2idlcpblrng 21158 crngridl 21165 opprdomn 21244 fidomndrng 21254 psropprmul 22149 invrvald 22571 ply1divalg2 26067 isdrng4 32956 crngmxidl 33176 opprabs 33187 oppreqg 33188 opprnsg 33189 opprlidlabs 33190 opprmxidlabs 33192 opprqusbas 33193 opprqusplusg 33194 opprqus0g 33195 opprqusmulr 33196 opprqus1r 33197 opprqusdrng 33198 qsdrngi 33200 qsdrng 33202 ldualsbase 38599 lduallmodlem 38618 lcdsbase 41067 |
Copyright terms: Public domain | W3C validator |