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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 2730 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | grpinvid.u | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | 1, 2 | grpidcl 18886 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺)) |
4 | eqid 2730 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | 1, 4, 2 | grplid 18888 | . . 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 18912 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺) ∧ 0 ∈ (Base‘𝐺)) → ((𝑁‘ 0 ) = 0 ↔ ( 0 (+g‘𝐺) 0 ) = 0 )) |
9 | 3, 3, 8 | mpd3an23 1461 | . 2 ⊢ (𝐺 ∈ Grp → ((𝑁‘ 0 ) = 0 ↔ ( 0 (+g‘𝐺) 0 ) = 0 )) |
10 | 6, 9 | mpbird 256 | 1 ⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2104 ‘cfv 6542 (class class class)co 7411 Basecbs 17148 +gcplusg 17201 0gc0g 17389 Grpcgrp 18855 invgcminusg 18856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 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-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-riota 7367 df-ov 7414 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 |
This theorem is referenced by: grpinvnz 18930 grpsubid1 18944 mulgneg 19008 mulginvcom 19015 mulgz 19018 0subg 19067 0subgOLD 19068 eqgid 19096 odnncl 19454 gexdvds 19493 gsumzinv 19854 gsumsub 19857 dprdfinv 19930 dsmmsubg 21517 mplsubglem 21777 mhpinvcl 21914 dchrisum0re 27252 qusker 32734 qsnzr 32848 baerlem3lem1 40881 |
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