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| Mirrors > Home > ILE Home > Th. List > dvrcan3 | GIF version | ||
| Description: A cancellation law for division. (divcanap3 8654 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 18-Jun-2015.) |
| Ref | Expression |
|---|---|
| dvrass.b | β’ π΅ = (Baseβπ ) |
| dvrass.o | β’ π = (Unitβπ ) |
| dvrass.d | β’ / = (/rβπ ) |
| dvrass.t | β’ Β· = (.rβπ ) |
| Ref | Expression |
|---|---|
| dvrcan3 | β’ ((π β Ring β§ π β π΅ β§ π β π) β ((π Β· π) / π) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 997 | . . 3 β’ ((π β Ring β§ π β π΅ β§ π β π) β π β Ring) | |
| 2 | simp2 998 | . . 3 β’ ((π β Ring β§ π β π΅ β§ π β π) β π β π΅) | |
| 3 | dvrass.b | . . . . 5 β’ π΅ = (Baseβπ ) | |
| 4 | 3 | a1i 9 | . . . 4 β’ ((π β Ring β§ π β π΅ β§ π β π) β π΅ = (Baseβπ )) |
| 5 | dvrass.o | . . . . 5 β’ π = (Unitβπ ) | |
| 6 | 5 | a1i 9 | . . . 4 β’ ((π β Ring β§ π β π΅ β§ π β π) β π = (Unitβπ )) |
| 7 | ringsrg 13222 | . . . . 5 β’ (π β Ring β π β SRing) | |
| 8 | 1, 7 | syl 14 | . . . 4 β’ ((π β Ring β§ π β π΅ β§ π β π) β π β SRing) |
| 9 | simp3 999 | . . . 4 β’ ((π β Ring β§ π β π΅ β§ π β π) β π β π) | |
| 10 | 4, 6, 8, 9 | unitcld 13275 | . . 3 β’ ((π β Ring β§ π β π΅ β§ π β π) β π β π΅) |
| 11 | dvrass.d | . . . 4 β’ / = (/rβπ ) | |
| 12 | dvrass.t | . . . 4 β’ Β· = (.rβπ ) | |
| 13 | 3, 5, 11, 12 | dvrass 13306 | . . 3 β’ ((π β Ring β§ (π β π΅ β§ π β π΅ β§ π β π)) β ((π Β· π) / π) = (π Β· (π / π))) |
| 14 | 1, 2, 10, 9, 13 | syl13anc 1240 | . 2 β’ ((π β Ring β§ π β π΅ β§ π β π) β ((π Β· π) / π) = (π Β· (π / π))) |
| 15 | eqid 2177 | . . . . 5 β’ (1rβπ ) = (1rβπ ) | |
| 16 | 5, 11, 15 | dvrid 13304 | . . . 4 β’ ((π β Ring β§ π β π) β (π / π) = (1rβπ )) |
| 17 | 16 | 3adant2 1016 | . . 3 β’ ((π β Ring β§ π β π΅ β§ π β π) β (π / π) = (1rβπ )) |
| 18 | 17 | oveq2d 5890 | . 2 β’ ((π β Ring β§ π β π΅ β§ π β π) β (π Β· (π / π)) = (π Β· (1rβπ ))) |
| 19 | 3, 12, 15 | ringridm 13205 | . . 3 β’ ((π β Ring β§ π β π΅) β (π Β· (1rβπ )) = π) |
| 20 | 19 | 3adant3 1017 | . 2 β’ ((π β Ring β§ π β π΅ β§ π β π) β (π Β· (1rβπ )) = π) |
| 21 | 14, 18, 20 | 3eqtrd 2214 | 1 β’ ((π β Ring β§ π β π΅ β§ π β π) β ((π Β· π) / π) = π) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ w3a 978 = wceq 1353 β wcel 2148 βcfv 5216 (class class class)co 5874 Basecbs 12461 .rcmulr 12536 1rcur 13140 SRingcsrg 13144 Ringcrg 13177 Unitcui 13254 /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: (None) |
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