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Mirrors > Home > MPE Home > Th. List > dvrval | Structured version Visualization version GIF version |
Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
dvrval.b | β’ π΅ = (Baseβπ ) |
dvrval.t | β’ Β· = (.rβπ ) |
dvrval.u | β’ π = (Unitβπ ) |
dvrval.i | β’ πΌ = (invrβπ ) |
dvrval.d | β’ / = (/rβπ ) |
Ref | Expression |
---|---|
dvrval | β’ ((π β π΅ β§ π β π) β (π / π) = (π Β· (πΌβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7427 | . 2 β’ (π₯ = π β (π₯ Β· (πΌβπ¦)) = (π Β· (πΌβπ¦))) | |
2 | fveq2 6897 | . . 3 β’ (π¦ = π β (πΌβπ¦) = (πΌβπ)) | |
3 | 2 | oveq2d 7436 | . 2 β’ (π¦ = π β (π Β· (πΌβπ¦)) = (π Β· (πΌβπ))) |
4 | dvrval.b | . . 3 β’ π΅ = (Baseβπ ) | |
5 | dvrval.t | . . 3 β’ Β· = (.rβπ ) | |
6 | dvrval.u | . . 3 β’ π = (Unitβπ ) | |
7 | dvrval.i | . . 3 β’ πΌ = (invrβπ ) | |
8 | dvrval.d | . . 3 β’ / = (/rβπ ) | |
9 | 4, 5, 6, 7, 8 | dvrfval 20341 | . 2 β’ / = (π₯ β π΅, π¦ β π β¦ (π₯ Β· (πΌβπ¦))) |
10 | ovex 7453 | . 2 β’ (π Β· (πΌβπ)) β V | |
11 | 1, 3, 9, 10 | ovmpo 7581 | 1 β’ ((π β π΅ β§ π β π) β (π / π) = (π Β· (πΌβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βcfv 6548 (class class class)co 7420 Basecbs 17180 .rcmulr 17234 Unitcui 20294 invrcinvr 20326 /rcdvr 20339 |
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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 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 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-dvr 20340 |
This theorem is referenced by: dvrcl 20343 unitdvcl 20344 dvrid 20345 dvr1 20346 dvrass 20347 dvrcan1 20348 dvrdir 20351 rdivmuldivd 20352 ringinvdv 20353 subrgdv 20528 abvdiv 20717 cnflddiv 21328 cnflddivOLD 21329 nmdvr 24600 sum2dchr 27220 dvrcan5 32957 |
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