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Mirrors > Home > MPE Home > Th. List > grpinvid | Structured version Visualization version GIF version |
Description: The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.) |
Ref | Expression |
---|---|
grpinvid.u | ⊢ 0 = (0g‘𝐺) |
grpinvid.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvid | ⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | grpinvid.u | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | 1, 2 | grpidcl 18069 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺)) |
4 | eqid 2818 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | 1, 4, 2 | grplid 18071 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
6 | 3, 5 | mpdan 683 | . 2 ⊢ (𝐺 ∈ Grp → ( 0 (+g‘𝐺) 0 ) = 0 ) |
7 | grpinvid.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
8 | 1, 4, 2, 7 | grpinvid1 18092 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺) ∧ 0 ∈ (Base‘𝐺)) → ((𝑁‘ 0 ) = 0 ↔ ( 0 (+g‘𝐺) 0 ) = 0 )) |
9 | 3, 3, 8 | mpd3an23 1454 | . 2 ⊢ (𝐺 ∈ Grp → ((𝑁‘ 0 ) = 0 ↔ ( 0 (+g‘𝐺) 0 ) = 0 )) |
10 | 6, 9 | mpbird 258 | 1 ⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 Grpcgrp 18041 invgcminusg 18042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-riota 7103 df-ov 7148 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 |
This theorem is referenced by: grpinvnz 18108 grpsubid1 18122 mulgneg 18184 mulginvcom 18190 mulgz 18193 0subg 18242 eqgid 18270 odnncl 18602 gexdvds 18638 gsumzinv 18994 gsumsub 18997 dprdfinv 19070 mplsubglem 20142 dsmmsubg 20815 dchrisum0re 26016 qusker 30845 baerlem3lem1 38723 |
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