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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 2733 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | grpinvid.u | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | 1, 2 | grpidcl 18850 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺)) |
4 | eqid 2733 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | 1, 4, 2 | grplid 18852 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
6 | 3, 5 | mpdan 686 | . 2 ⊢ (𝐺 ∈ Grp → ( 0 (+g‘𝐺) 0 ) = 0 ) |
7 | grpinvid.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
8 | 1, 4, 2, 7 | grpinvid1 18876 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺) ∧ 0 ∈ (Base‘𝐺)) → ((𝑁‘ 0 ) = 0 ↔ ( 0 (+g‘𝐺) 0 ) = 0 )) |
9 | 3, 3, 8 | mpd3an23 1464 | . 2 ⊢ (𝐺 ∈ Grp → ((𝑁‘ 0 ) = 0 ↔ ( 0 (+g‘𝐺) 0 ) = 0 )) |
10 | 6, 9 | mpbird 257 | 1 ⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ‘cfv 6544 (class class class)co 7409 Basecbs 17144 +gcplusg 17197 0gc0g 17385 Grpcgrp 18819 invgcminusg 18820 |
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-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-riota 7365 df-ov 7412 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 |
This theorem is referenced by: grpinvnz 18894 grpsubid1 18908 mulgneg 18972 mulginvcom 18979 mulgz 18982 0subg 19031 0subgOLD 19032 eqgid 19060 odnncl 19413 gexdvds 19452 gsumzinv 19813 gsumsub 19816 dprdfinv 19889 dsmmsubg 21298 mplsubglem 21558 mhpinvcl 21695 dchrisum0re 27016 qusker 32464 qsnzr 32574 baerlem3lem1 40578 |
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