![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > grpinvid | 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 2177 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | grpinvid.u | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | 1, 2 | grpidcl 12836 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺)) |
4 | eqid 2177 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | 1, 4, 2 | grplid 12838 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
6 | 3, 5 | mpdan 421 | . 2 ⊢ (𝐺 ∈ Grp → ( 0 (+g‘𝐺) 0 ) = 0 ) |
7 | grpinvid.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
8 | 1, 4, 2, 7 | grpinvid1 12856 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺) ∧ 0 ∈ (Base‘𝐺)) → ((𝑁‘ 0 ) = 0 ↔ ( 0 (+g‘𝐺) 0 ) = 0 )) |
9 | 3, 3, 8 | mpd3an23 1339 | . 2 ⊢ (𝐺 ∈ Grp → ((𝑁‘ 0 ) = 0 ↔ ( 0 (+g‘𝐺) 0 ) = 0 )) |
10 | 6, 9 | mpbird 167 | 1 ⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ‘cfv 5215 (class class class)co 5872 Basecbs 12454 +gcplusg 12528 0gc0g 12693 Grpcgrp 12809 invgcminusg 12810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-cnex 7899 ax-resscn 7900 ax-1re 7902 ax-addrcl 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-inn 8916 df-2 8974 df-ndx 12457 df-slot 12458 df-base 12460 df-plusg 12541 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 df-minusg 12813 |
This theorem is referenced by: grpinvnz 12873 grpsubid1 12887 mulgneg 12933 mulginvcom 12939 mulgz 12942 0subg 12990 eqgid 13016 |
Copyright terms: Public domain | W3C validator |