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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 11928 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 2728 | . . . 4 β’ (Unitβπ ) = (Unitβπ ) | |
| 3 | dvr1.o | . . . 4 β’ 1 = (1rβπ ) | |
| 4 | 2, 3 | 1unit 20307 | . . 3 β’ (π β Ring β 1 β (Unitβπ )) |
| 5 | dvr1.b | . . . 4 β’ π΅ = (Baseβπ ) | |
| 6 | eqid 2728 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
| 7 | eqid 2728 | . . . 4 β’ (invrβπ ) = (invrβπ ) | |
| 8 | dvr1.d | . . . 4 β’ / = (/rβπ ) | |
| 9 | 5, 6, 2, 7, 8 | dvrval 20336 | . . 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 20328 | . . . 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 7431 | . 2 β’ ((π β Ring β§ π β π΅) β (π(.rβπ )((invrβπ )β 1 )) = (π(.rβπ ) 1 )) |
| 14 | 5, 6, 3 | ringridm 20200 | . 2 β’ ((π β Ring β§ π β π΅) β (π(.rβπ ) 1 ) = π) |
| 15 | 10, 13, 14 | 3eqtrd 2772 | 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 1534 β wcel 2099 βcfv 6543 (class class class)co 7415 Basecbs 17174 .rcmulr 17228 1rcur 20115 Ringcrg 20167 Unitcui 20288 invrcinvr 20320 /rcdvr 20333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-tpos 8226 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-minusg 18888 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-oppr 20267 df-dvdsr 20290 df-unit 20291 df-invr 20321 df-dvr 20334 |
| This theorem is referenced by: qqh0 33580 qqh1 33581 |
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