|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 2736 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 2 | grpinvid.u | . . . 4 ⊢ 0 = (0g‘𝐺) | |
| 3 | 1, 2 | grpidcl 18984 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺)) | 
| 4 | eqid 2736 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 5 | 1, 4, 2 | grplid 18986 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) | 
| 6 | 3, 5 | mpdan 687 | . 2 ⊢ (𝐺 ∈ Grp → ( 0 (+g‘𝐺) 0 ) = 0 ) | 
| 7 | grpinvid.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
| 8 | 1, 4, 2, 7 | grpinvid1 19010 | . . 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 206 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 0gc0g 17485 Grpcgrp 18952 invgcminusg 18953 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-riota 7389 df-ov 7435 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 | 
| This theorem is referenced by: grpinvnz 19029 grpsubid1 19044 mulgneg 19111 mulginvcom 19118 mulgz 19121 0subg 19170 0subgOLD 19171 eqgid 19199 odnncl 19564 gexdvds 19603 gsumzinv 19964 gsumsub 19967 dprdfinv 20040 dsmmsubg 21764 mplsubglem 22020 mhpinvcl 22157 dchrisum0re 27558 erler 33270 qusker 33378 qsnzr 33484 baerlem3lem1 41710 primrootscoprbij 42104 | 
| Copyright terms: Public domain | W3C validator |