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Mirrors > Home > ILE Home > Th. List > dvrvald | GIF version |
Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
dvrvald.b | β’ (π β π΅ = (Baseβπ )) |
dvrvald.t | β’ (π β Β· = (.rβπ )) |
dvrvald.u | β’ (π β π = (Unitβπ )) |
dvrvald.i | β’ (π β πΌ = (invrβπ )) |
dvrvald.d | β’ (π β / = (/rβπ )) |
dvrvald.r | β’ (π β π β Ring) |
dvrvald.x | β’ (π β π β π΅) |
dvrvald.y | β’ (π β π β π) |
Ref | Expression |
---|---|
dvrvald | β’ (π β (π / π) = (π Β· (πΌβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvrvald.b | . . 3 β’ (π β π΅ = (Baseβπ )) | |
2 | dvrvald.t | . . 3 β’ (π β Β· = (.rβπ )) | |
3 | dvrvald.u | . . 3 β’ (π β π = (Unitβπ )) | |
4 | dvrvald.i | . . 3 β’ (π β πΌ = (invrβπ )) | |
5 | dvrvald.d | . . 3 β’ (π β / = (/rβπ )) | |
6 | dvrvald.r | . . . 4 β’ (π β π β Ring) | |
7 | ringsrg 13222 | . . . 4 β’ (π β Ring β π β SRing) | |
8 | 6, 7 | syl 14 | . . 3 β’ (π β π β SRing) |
9 | 1, 2, 3, 4, 5, 8 | dvrfvald 13300 | . 2 β’ (π β / = (π₯ β π΅, π¦ β π β¦ (π₯ Β· (πΌβπ¦)))) |
10 | simpl 109 | . . . 4 β’ ((π₯ = π β§ π¦ = π) β π₯ = π) | |
11 | fveq2 5515 | . . . . 5 β’ (π¦ = π β (πΌβπ¦) = (πΌβπ)) | |
12 | 11 | adantl 277 | . . . 4 β’ ((π₯ = π β§ π¦ = π) β (πΌβπ¦) = (πΌβπ)) |
13 | 10, 12 | oveq12d 5892 | . . 3 β’ ((π₯ = π β§ π¦ = π) β (π₯ Β· (πΌβπ¦)) = (π Β· (πΌβπ))) |
14 | 13 | adantl 277 | . 2 β’ ((π β§ (π₯ = π β§ π¦ = π)) β (π₯ Β· (πΌβπ¦)) = (π Β· (πΌβπ))) |
15 | dvrvald.x | . 2 β’ (π β π β π΅) | |
16 | dvrvald.y | . 2 β’ (π β π β π) | |
17 | 2 | oveqd 5891 | . . 3 β’ (π β (π Β· (πΌβπ)) = (π(.rβπ )(πΌβπ))) |
18 | 15, 1 | eleqtrd 2256 | . . . 4 β’ (π β π β (Baseβπ )) |
19 | eqidd 2178 | . . . . 5 β’ (π β (Baseβπ ) = (Baseβπ )) | |
20 | 16, 3 | eleqtrd 2256 | . . . . . . 7 β’ (π β π β (Unitβπ )) |
21 | eqid 2177 | . . . . . . . 8 β’ (Unitβπ ) = (Unitβπ ) | |
22 | eqid 2177 | . . . . . . . 8 β’ (invrβπ ) = (invrβπ ) | |
23 | 21, 22 | unitinvcl 13290 | . . . . . . 7 β’ ((π β Ring β§ π β (Unitβπ )) β ((invrβπ )βπ) β (Unitβπ )) |
24 | 6, 20, 23 | syl2anc 411 | . . . . . 6 β’ (π β ((invrβπ )βπ) β (Unitβπ )) |
25 | 4 | fveq1d 5517 | . . . . . 6 β’ (π β (πΌβπ) = ((invrβπ )βπ)) |
26 | 24, 25, 3 | 3eltr4d 2261 | . . . . 5 β’ (π β (πΌβπ) β π) |
27 | 19, 3, 8, 26 | unitcld 13275 | . . . 4 β’ (π β (πΌβπ) β (Baseβπ )) |
28 | eqid 2177 | . . . . 5 β’ (Baseβπ ) = (Baseβπ ) | |
29 | eqid 2177 | . . . . 5 β’ (.rβπ ) = (.rβπ ) | |
30 | 28, 29 | ringcl 13194 | . . . 4 β’ ((π β Ring β§ π β (Baseβπ ) β§ (πΌβπ) β (Baseβπ )) β (π(.rβπ )(πΌβπ)) β (Baseβπ )) |
31 | 6, 18, 27, 30 | syl3anc 1238 | . . 3 β’ (π β (π(.rβπ )(πΌβπ)) β (Baseβπ )) |
32 | 17, 31 | eqeltrd 2254 | . 2 β’ (π β (π Β· (πΌβπ)) β (Baseβπ )) |
33 | 9, 14, 15, 16, 32 | ovmpod 6001 | 1 β’ (π β (π / π) = (π Β· (πΌβπ))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 βcfv 5216 (class class class)co 5874 Basecbs 12461 .rcmulr 12536 SRingcsrg 13144 Ringcrg 13177 Unitcui 13254 invrcinvr 13287 /rcdvr 13298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-tpos 6245 df-pnf 7993 df-mnf 7994 df-ltxr 7996 df-inn 8919 df-2 8977 df-3 8978 df-ndx 12464 df-slot 12465 df-base 12467 df-sets 12468 df-iress 12469 df-plusg 12548 df-mulr 12549 df-0g 12706 df-mgm 12774 df-sgrp 12807 df-mnd 12817 df-grp 12879 df-minusg 12880 df-cmn 13088 df-abl 13089 df-mgp 13129 df-ur 13141 df-srg 13145 df-ring 13179 df-oppr 13238 df-dvdsr 13256 df-unit 13257 df-invr 13288 df-dvr 13299 |
This theorem is referenced by: dvrcl 13302 unitdvcl 13303 dvrid 13304 dvr1 13305 dvrass 13306 dvrcan1 13307 dvrdir 13310 rdivmuldivd 13311 ringinvdv 13312 subrgdv 13357 |
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