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Mirrors > Home > MPE Home > Th. List > 1rinv | Structured version Visualization version GIF version |
Description: The inverse of the ring unity is the ring unity. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
1rinv.1 | ⊢ 𝐼 = (invr‘𝑅) |
1rinv.2 | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
1rinv | ⊢ (𝑅 ∈ Ring → (𝐼‘ 1 ) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
2 | 1rinv.2 | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
3 | 1, 2 | 1unit 20177 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ (Unit‘𝑅)) |
4 | 1rinv.1 | . . . . 5 ⊢ 𝐼 = (invr‘𝑅) | |
5 | eqid 2733 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 1, 4, 5 | ringinvcl 20195 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ (Unit‘𝑅)) → (𝐼‘ 1 ) ∈ (Base‘𝑅)) |
7 | 3, 6 | mpdan 686 | . . 3 ⊢ (𝑅 ∈ Ring → (𝐼‘ 1 ) ∈ (Base‘𝑅)) |
8 | eqid 2733 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
9 | 5, 8, 2 | ringlidm 20076 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘ 1 ) ∈ (Base‘𝑅)) → ( 1 (.r‘𝑅)(𝐼‘ 1 )) = (𝐼‘ 1 )) |
10 | 7, 9 | mpdan 686 | . 2 ⊢ (𝑅 ∈ Ring → ( 1 (.r‘𝑅)(𝐼‘ 1 )) = (𝐼‘ 1 )) |
11 | 1, 4, 8, 2 | unitrinv 20197 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ (Unit‘𝑅)) → ( 1 (.r‘𝑅)(𝐼‘ 1 )) = 1 ) |
12 | 3, 11 | mpdan 686 | . 2 ⊢ (𝑅 ∈ Ring → ( 1 (.r‘𝑅)(𝐼‘ 1 )) = 1 ) |
13 | 10, 12 | eqtr3d 2775 | 1 ⊢ (𝑅 ∈ Ring → (𝐼‘ 1 ) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 .rcmulr 17194 1rcur 19996 Ringcrg 20047 Unitcui 20158 invrcinvr 20190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-2nd 7971 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-mgp 19980 df-ur 19997 df-ring 20049 df-oppr 20139 df-dvdsr 20160 df-unit 20161 df-invr 20191 |
This theorem is referenced by: dvr1 20210 |
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