![]() |
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 17129 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
4 | basendxnmulrndx 17222 | . . 3 ⊢ (Base‘ndx) ≠ (.r‘ndx) | |
5 | 2, 3, 4 | opprlem 20107 | . 2 ⊢ (Base‘𝑅) = (Base‘𝑂) |
6 | 1, 5 | eqtri 2759 | 1 ⊢ 𝐵 = (Base‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ‘cfv 6532 Basecbs 17126 opprcoppr 20101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-2nd 7958 df-tpos 8193 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-mulr 17193 df-oppr 20102 |
This theorem is referenced by: opprring 20113 opprringb 20114 oppr0 20115 oppr1 20116 opprneg 20117 opprsubg 20118 mulgass3 20119 1unit 20140 opprunit 20143 crngunit 20144 unitmulcl 20146 unitgrp 20149 unitnegcl 20163 unitpropd 20181 opprirred 20186 rhmopp 20238 elrhmunit 20239 opprnzr 20249 isdrng2 20278 opprdrng 20296 isdrngrd 20298 isdrngrdOLD 20300 subrguss 20327 subrgunit 20330 opprsubrg 20333 issrngd 20418 2idlcpbl 20807 crngridl 20812 opprdomn 20853 fidomndrng 20860 psropprmul 21691 invrvald 22107 ply1divalg2 25585 isdrng4 32257 crngmxidl 32436 opprabs 32442 oppreqg 32443 opprnsg 32444 opprlidlabs 32445 opprmxidlabs 32447 opprqusbas 32448 opprqusplusg 32449 opprqus0g 32450 opprqusmulr 32451 opprqus1r 32452 opprqusdrng 32453 qsdrngi 32455 qsdrng 32457 ldualsbase 37808 lduallmodlem 37827 lcdsbase 40276 |
Copyright terms: Public domain | W3C validator |