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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 7365 | . 2 β’ (π₯ = π β (π₯ Β· (πΌβπ¦)) = (π Β· (πΌβπ¦))) | |
2 | fveq2 6843 | . . 3 β’ (π¦ = π β (πΌβπ¦) = (πΌβπ)) | |
3 | 2 | oveq2d 7374 | . 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 20114 | . 2 β’ / = (π₯ β π΅, π¦ β π β¦ (π₯ Β· (πΌβπ¦))) |
10 | ovex 7391 | . 2 β’ (π Β· (πΌβπ)) β V | |
11 | 1, 3, 9, 10 | ovmpo 7516 | 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 397 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17084 .rcmulr 17135 Unitcui 20069 invrcinvr 20101 /rcdvr 20112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-dvr 20113 |
This theorem is referenced by: dvrcl 20116 unitdvcl 20117 dvrid 20118 dvr1 20119 dvrass 20120 dvrcan1 20121 ringinvdv 20124 subrgdv 20242 abvdiv 20299 cnflddiv 20830 nmdvr 24037 sum2dchr 26625 dvrdir 32073 rdivmuldivd 32074 dvrcan5 32076 |
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