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| Mirrors > Home > MPE Home > Th. List > dvr1 | Structured version Visualization version GIF version | ||
| Description: A ring element divided by the ring unity is itself. (div1 11901 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 | id 22 | . . 3 β’ (π β π΅ β π β π΅) | |
| 2 | eqid 2724 | . . . 4 β’ (Unitβπ ) = (Unitβπ ) | |
| 3 | dvr1.o | . . . 4 β’ 1 = (1rβπ ) | |
| 4 | 2, 3 | 1unit 20268 | . . 3 β’ (π β Ring β 1 β (Unitβπ )) |
| 5 | dvr1.b | . . . 4 β’ π΅ = (Baseβπ ) | |
| 6 | eqid 2724 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
| 7 | eqid 2724 | . . . 4 β’ (invrβπ ) = (invrβπ ) | |
| 8 | dvr1.d | . . . 4 β’ / = (/rβπ ) | |
| 9 | 5, 6, 2, 7, 8 | dvrval 20297 | . . 3 β’ ((π β π΅ β§ 1 β (Unitβπ )) β (π / 1 ) = (π(.rβπ )((invrβπ )β 1 ))) |
| 10 | 1, 4, 9 | syl2anr 596 | . 2 β’ ((π β Ring β§ π β π΅) β (π / 1 ) = (π(.rβπ )((invrβπ )β 1 ))) |
| 11 | 7, 3 | 1rinv 20289 | . . . 4 β’ (π β Ring β ((invrβπ )β 1 ) = 1 ) |
| 12 | 11 | adantr 480 | . . 3 β’ ((π β Ring β§ π β π΅) β ((invrβπ )β 1 ) = 1 ) |
| 13 | 12 | oveq2d 7418 | . 2 β’ ((π β Ring β§ π β π΅) β (π(.rβπ )((invrβπ )β 1 )) = (π(.rβπ ) 1 )) |
| 14 | 5, 6, 3 | ringridm 20161 | . 2 β’ ((π β Ring β§ π β π΅) β (π(.rβπ ) 1 ) = π) |
| 15 | 10, 13, 14 | 3eqtrd 2768 | 1 β’ ((π β Ring β§ π β π΅) β (π / 1 ) = π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 Basecbs 17145 .rcmulr 17199 1rcur 20078 Ringcrg 20130 Unitcui 20249 invrcinvr 20281 /rcdvr 20294 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-oppr 20228 df-dvdsr 20251 df-unit 20252 df-invr 20282 df-dvr 20295 |
| This theorem is referenced by: qqh0 33456 qqh1 33457 |
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