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Mirrors > Home > MPE Home > Th. List > 1rinv | Structured version Visualization version GIF version |
Description: The inverse of the identity is the identity. (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 2759 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
2 | 1rinv.2 | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
3 | 1, 2 | 1unit 19472 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ (Unit‘𝑅)) |
4 | 1rinv.1 | . . . . 5 ⊢ 𝐼 = (invr‘𝑅) | |
5 | eqid 2759 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 1, 4, 5 | ringinvcl 19490 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ (Unit‘𝑅)) → (𝐼‘ 1 ) ∈ (Base‘𝑅)) |
7 | 3, 6 | mpdan 687 | . . 3 ⊢ (𝑅 ∈ Ring → (𝐼‘ 1 ) ∈ (Base‘𝑅)) |
8 | eqid 2759 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
9 | 5, 8, 2 | ringlidm 19385 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘ 1 ) ∈ (Base‘𝑅)) → ( 1 (.r‘𝑅)(𝐼‘ 1 )) = (𝐼‘ 1 )) |
10 | 7, 9 | mpdan 687 | . 2 ⊢ (𝑅 ∈ Ring → ( 1 (.r‘𝑅)(𝐼‘ 1 )) = (𝐼‘ 1 )) |
11 | 1, 4, 8, 2 | unitrinv 19492 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ (Unit‘𝑅)) → ( 1 (.r‘𝑅)(𝐼‘ 1 )) = 1 ) |
12 | 3, 11 | mpdan 687 | . 2 ⊢ (𝑅 ∈ Ring → ( 1 (.r‘𝑅)(𝐼‘ 1 )) = 1 ) |
13 | 10, 12 | eqtr3d 2796 | 1 ⊢ (𝑅 ∈ Ring → (𝐼‘ 1 ) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 ‘cfv 6336 (class class class)co 7151 Basecbs 16534 .rcmulr 16617 1rcur 19312 Ringcrg 19358 Unitcui 19453 invrcinvr 19485 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5157 ax-sep 5170 ax-nul 5177 ax-pow 5235 ax-pr 5299 ax-un 7460 ax-cnex 10624 ax-resscn 10625 ax-1cn 10626 ax-icn 10627 ax-addcl 10628 ax-addrcl 10629 ax-mulcl 10630 ax-mulrcl 10631 ax-mulcom 10632 ax-addass 10633 ax-mulass 10634 ax-distr 10635 ax-i2m1 10636 ax-1ne0 10637 ax-1rid 10638 ax-rnegex 10639 ax-rrecex 10640 ax-cnre 10641 ax-pre-lttri 10642 ax-pre-lttrn 10643 ax-pre-ltadd 10644 ax-pre-mulgt0 10645 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rmo 3079 df-rab 3080 df-v 3412 df-sbc 3698 df-csb 3807 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-pss 3878 df-nul 4227 df-if 4422 df-pw 4497 df-sn 4524 df-pr 4526 df-tp 4528 df-op 4530 df-uni 4800 df-iun 4886 df-br 5034 df-opab 5096 df-mpt 5114 df-tr 5140 df-id 5431 df-eprel 5436 df-po 5444 df-so 5445 df-fr 5484 df-we 5486 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6127 df-ord 6173 df-on 6174 df-lim 6175 df-suc 6176 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7109 df-ov 7154 df-oprab 7155 df-mpo 7156 df-om 7581 df-tpos 7903 df-wrecs 7958 df-recs 8019 df-rdg 8057 df-er 8300 df-en 8529 df-dom 8530 df-sdom 8531 df-pnf 10708 df-mnf 10709 df-xr 10710 df-ltxr 10711 df-le 10712 df-sub 10903 df-neg 10904 df-nn 11668 df-2 11730 df-3 11731 df-ndx 16537 df-slot 16538 df-base 16540 df-sets 16541 df-ress 16542 df-plusg 16629 df-mulr 16630 df-0g 16766 df-mgm 17911 df-sgrp 17960 df-mnd 17971 df-grp 18165 df-minusg 18166 df-mgp 19301 df-ur 19313 df-ring 19360 df-oppr 19437 df-dvdsr 19455 df-unit 19456 df-invr 19486 |
This theorem is referenced by: dvr1 19503 |
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