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Mirrors > Home > MPE Home > Th. List > opprmul | Structured version Visualization version GIF version |
Description: Value of the multiplication operation of an opposite ring. Hypotheses eliminated by a suggestion of Stefan O'Rear, 30-Aug-2015. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
opprval.1 | ⊢ 𝐵 = (Base‘𝑅) |
opprval.2 | ⊢ · = (.r‘𝑅) |
opprval.3 | ⊢ 𝑂 = (oppr‘𝑅) |
opprmulfval.4 | ⊢ ∙ = (.r‘𝑂) |
Ref | Expression |
---|---|
opprmul | ⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprval.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | opprval.2 | . . . 4 ⊢ · = (.r‘𝑅) | |
3 | opprval.3 | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
4 | opprmulfval.4 | . . . 4 ⊢ ∙ = (.r‘𝑂) | |
5 | 1, 2, 3, 4 | opprmulfval 20353 | . . 3 ⊢ ∙ = tpos · |
6 | 5 | oveqi 7444 | . 2 ⊢ (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌) |
7 | ovtpos 8265 | . 2 ⊢ (𝑋tpos · 𝑌) = (𝑌 · 𝑋) | |
8 | 6, 7 | eqtri 2763 | 1 ⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ‘cfv 6563 (class class class)co 7431 tpos ctpos 8249 Basecbs 17245 .rcmulr 17299 opprcoppr 20350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-mulr 17312 df-oppr 20351 |
This theorem is referenced by: crngoppr 20355 opprrng 20362 opprrngb 20363 opprring 20364 opprringb 20365 oppr1 20367 mulgass3 20370 opprunit 20394 unitmulcl 20397 unitgrp 20400 unitpropd 20434 opprirred 20439 irredlmul 20445 rhmopp 20526 opprsubrng 20576 subrguss 20604 subrgunit 20607 opprsubrg 20610 opprdomnb 20734 isdomn4r 20736 isdrng2 20760 isdrngrd 20783 isdrngrdOLD 20785 srngmul 20870 issrngd 20873 rngridlmcl 21245 isridlrng 21247 isridl 21280 2idlcpblrng 21299 psropprmul 22255 invrvald 22698 isunit2 33230 isdrng4 33279 opprlidlabs 33493 opprqusmulr 33499 qsdrngi 33503 ldualsmul 39117 lcdsmul 41585 |
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