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Mirrors > Home > MPE Home > Th. List > oppgplus | Structured version Visualization version GIF version |
Description: Value of the addition operation of an opposite ring. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.) |
Ref | Expression |
---|---|
oppgval.2 | ⊢ + = (+g‘𝑅) |
oppgval.3 | ⊢ 𝑂 = (oppg‘𝑅) |
oppgplusfval.4 | ⊢ ✚ = (+g‘𝑂) |
Ref | Expression |
---|---|
oppgplus | ⊢ (𝑋 ✚ 𝑌) = (𝑌 + 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppgval.2 | . . . 4 ⊢ + = (+g‘𝑅) | |
2 | oppgval.3 | . . . 4 ⊢ 𝑂 = (oppg‘𝑅) | |
3 | oppgplusfval.4 | . . . 4 ⊢ ✚ = (+g‘𝑂) | |
4 | 1, 2, 3 | oppgplusfval 18468 | . . 3 ⊢ ✚ = tpos + |
5 | 4 | oveqi 7148 | . 2 ⊢ (𝑋 ✚ 𝑌) = (𝑋tpos + 𝑌) |
6 | ovtpos 7890 | . 2 ⊢ (𝑋tpos + 𝑌) = (𝑌 + 𝑋) | |
7 | 5, 6 | eqtri 2821 | 1 ⊢ (𝑋 ✚ 𝑌) = (𝑌 + 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ‘cfv 6324 (class class class)co 7135 tpos ctpos 7874 +gcplusg 16557 oppgcoppg 18465 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-sets 16482 df-plusg 16570 df-oppg 18466 |
This theorem is referenced by: oppgmnd 18474 oppgmndb 18475 oppgid 18476 oppggrp 18477 oppggrpb 18478 oppginv 18479 invoppggim 18480 oppgsubm 18482 oppgcntz 18484 gsumwrev 18486 oppglsm 18759 gsumzoppg 19057 oppgtmd 22702 tgpconncomp 22718 qustgpopn 22725 omndaddr 30758 ogrpaddltrd 30770 ogrpaddltrbid 30771 |
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