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Mirrors > Home > ILE Home > Th. List > dvr1 | GIF version |
Description: A ring element divided by the ring unity is itself. (div1 8662 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
dvr1.b | β’ π΅ = (Baseβπ ) |
dvr1.d | β’ / = (/rβπ ) |
dvr1.o | β’ 1 = (1rβπ ) |
Ref | Expression |
---|---|
dvr1 | β’ ((π β Ring β§ π β π΅) β (π / 1 ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvr1.b | . . . 4 β’ π΅ = (Baseβπ ) | |
2 | 1 | a1i 9 | . . 3 β’ ((π β Ring β§ π β π΅) β π΅ = (Baseβπ )) |
3 | eqidd 2178 | . . 3 β’ ((π β Ring β§ π β π΅) β (.rβπ ) = (.rβπ )) | |
4 | eqidd 2178 | . . 3 β’ ((π β Ring β§ π β π΅) β (Unitβπ ) = (Unitβπ )) | |
5 | eqidd 2178 | . . 3 β’ ((π β Ring β§ π β π΅) β (invrβπ ) = (invrβπ )) | |
6 | dvr1.d | . . . 4 β’ / = (/rβπ ) | |
7 | 6 | a1i 9 | . . 3 β’ ((π β Ring β§ π β π΅) β / = (/rβπ )) |
8 | simpl 109 | . . 3 β’ ((π β Ring β§ π β π΅) β π β Ring) | |
9 | simpr 110 | . . 3 β’ ((π β Ring β§ π β π΅) β π β π΅) | |
10 | eqid 2177 | . . . . 5 β’ (Unitβπ ) = (Unitβπ ) | |
11 | dvr1.o | . . . . 5 β’ 1 = (1rβπ ) | |
12 | 10, 11 | 1unit 13281 | . . . 4 β’ (π β Ring β 1 β (Unitβπ )) |
13 | 12 | adantr 276 | . . 3 β’ ((π β Ring β§ π β π΅) β 1 β (Unitβπ )) |
14 | 2, 3, 4, 5, 7, 8, 9, 13 | dvrvald 13308 | . 2 β’ ((π β Ring β§ π β π΅) β (π / 1 ) = (π(.rβπ )((invrβπ )β 1 ))) |
15 | eqid 2177 | . . . . 5 β’ (invrβπ ) = (invrβπ ) | |
16 | 15, 11 | 1rinv 13302 | . . . 4 β’ (π β Ring β ((invrβπ )β 1 ) = 1 ) |
17 | 16 | adantr 276 | . . 3 β’ ((π β Ring β§ π β π΅) β ((invrβπ )β 1 ) = 1 ) |
18 | 17 | oveq2d 5893 | . 2 β’ ((π β Ring β§ π β π΅) β (π(.rβπ )((invrβπ )β 1 )) = (π(.rβπ ) 1 )) |
19 | eqid 2177 | . . 3 β’ (.rβπ ) = (.rβπ ) | |
20 | 1, 19, 11 | ringridm 13212 | . 2 β’ ((π β Ring β§ π β π΅) β (π(.rβπ ) 1 ) = π) |
21 | 14, 18, 20 | 3eqtrd 2214 | 1 β’ ((π β Ring β§ π β π΅) β (π / 1 ) = π) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 βcfv 5218 (class class class)co 5877 Basecbs 12464 .rcmulr 12539 1rcur 13147 Ringcrg 13184 Unitcui 13261 invrcinvr 13294 /rcdvr 13305 |
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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-pre-ltirr 7925 ax-pre-lttrn 7927 ax-pre-ltadd 7929 |
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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-tpos 6248 df-pnf 7996 df-mnf 7997 df-ltxr 7999 df-inn 8922 df-2 8980 df-3 8981 df-ndx 12467 df-slot 12468 df-base 12470 df-sets 12471 df-iress 12472 df-plusg 12551 df-mulr 12552 df-0g 12712 df-mgm 12780 df-sgrp 12813 df-mnd 12823 df-grp 12885 df-minusg 12886 df-cmn 13095 df-abl 13096 df-mgp 13136 df-ur 13148 df-srg 13152 df-ring 13186 df-oppr 13245 df-dvdsr 13263 df-unit 13264 df-invr 13295 df-dvr 13306 |
This theorem is referenced by: (None) |
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