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Mirrors > Home > ILE Home > Th. List > dvreq1 | GIF version |
Description: Equality in terms of ratio equal to ring unity. (diveqap1 8662 analog.) (Contributed by Mario Carneiro, 28-Apr-2016.) |
Ref | Expression |
---|---|
dvreq1.b | β’ π΅ = (Baseβπ ) |
dvreq1.o | β’ π = (Unitβπ ) |
dvreq1.d | β’ / = (/rβπ ) |
dvreq1.t | β’ 1 = (1rβπ ) |
Ref | Expression |
---|---|
dvreq1 | β’ ((π β Ring β§ π β π΅ β§ π β π) β ((π / π) = 1 β π = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5882 | . . 3 β’ ((π / π) = 1 β ((π / π)(.rβπ )π) = ( 1 (.rβπ )π)) | |
2 | dvreq1.b | . . . . 5 β’ π΅ = (Baseβπ ) | |
3 | dvreq1.o | . . . . 5 β’ π = (Unitβπ ) | |
4 | dvreq1.d | . . . . 5 β’ / = (/rβπ ) | |
5 | eqid 2177 | . . . . 5 β’ (.rβπ ) = (.rβπ ) | |
6 | 2, 3, 4, 5 | dvrcan1 13309 | . . . 4 β’ ((π β Ring β§ π β π΅ β§ π β π) β ((π / π)(.rβπ )π) = π) |
7 | 2 | a1i 9 | . . . . . . 7 β’ ((π β Ring β§ π β π) β π΅ = (Baseβπ )) |
8 | 3 | a1i 9 | . . . . . . 7 β’ ((π β Ring β§ π β π) β π = (Unitβπ )) |
9 | ringsrg 13224 | . . . . . . . 8 β’ (π β Ring β π β SRing) | |
10 | 9 | adantr 276 | . . . . . . 7 β’ ((π β Ring β§ π β π) β π β SRing) |
11 | simpr 110 | . . . . . . 7 β’ ((π β Ring β§ π β π) β π β π) | |
12 | 7, 8, 10, 11 | unitcld 13277 | . . . . . 6 β’ ((π β Ring β§ π β π) β π β π΅) |
13 | dvreq1.t | . . . . . . 7 β’ 1 = (1rβπ ) | |
14 | 2, 5, 13 | ringlidm 13206 | . . . . . 6 β’ ((π β Ring β§ π β π΅) β ( 1 (.rβπ )π) = π) |
15 | 12, 14 | syldan 282 | . . . . 5 β’ ((π β Ring β§ π β π) β ( 1 (.rβπ )π) = π) |
16 | 15 | 3adant2 1016 | . . . 4 β’ ((π β Ring β§ π β π΅ β§ π β π) β ( 1 (.rβπ )π) = π) |
17 | 6, 16 | eqeq12d 2192 | . . 3 β’ ((π β Ring β§ π β π΅ β§ π β π) β (((π / π)(.rβπ )π) = ( 1 (.rβπ )π) β π = π)) |
18 | 1, 17 | imbitrid 154 | . 2 β’ ((π β Ring β§ π β π΅ β§ π β π) β ((π / π) = 1 β π = π)) |
19 | 3, 4, 13 | dvrid 13306 | . . . 4 β’ ((π β Ring β§ π β π) β (π / π) = 1 ) |
20 | 19 | 3adant2 1016 | . . 3 β’ ((π β Ring β§ π β π΅ β§ π β π) β (π / π) = 1 ) |
21 | oveq1 5882 | . . . 4 β’ (π = π β (π / π) = (π / π)) | |
22 | 21 | eqeq1d 2186 | . . 3 β’ (π = π β ((π / π) = 1 β (π / π) = 1 )) |
23 | 20, 22 | syl5ibrcom 157 | . 2 β’ ((π β Ring β§ π β π΅ β§ π β π) β (π = π β (π / π) = 1 )) |
24 | 18, 23 | impbid 129 | 1 β’ ((π β Ring β§ π β π΅ β§ π β π) β ((π / π) = 1 β π = π)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wb 105 β§ w3a 978 = wceq 1353 β wcel 2148 βcfv 5217 (class class class)co 5875 Basecbs 12462 .rcmulr 12537 1rcur 13142 SRingcsrg 13146 Ringcrg 13179 Unitcui 13256 /rcdvr 13300 |
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 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-addcom 7911 ax-addass 7913 ax-i2m1 7916 ax-0lt1 7917 ax-0id 7919 ax-rnegex 7920 ax-pre-ltirr 7923 ax-pre-lttrn 7925 ax-pre-ltadd 7927 |
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 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-tpos 6246 df-pnf 7994 df-mnf 7995 df-ltxr 7997 df-inn 8920 df-2 8978 df-3 8979 df-ndx 12465 df-slot 12466 df-base 12468 df-sets 12469 df-iress 12470 df-plusg 12549 df-mulr 12550 df-0g 12707 df-mgm 12775 df-sgrp 12808 df-mnd 12818 df-grp 12880 df-minusg 12881 df-cmn 13090 df-abl 13091 df-mgp 13131 df-ur 13143 df-srg 13147 df-ring 13181 df-oppr 13240 df-dvdsr 13258 df-unit 13259 df-invr 13290 df-dvr 13301 |
This theorem is referenced by: lringuplu 13337 |
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