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Mirrors > Home > MPE Home > Th. List > grpsubid | Structured version Visualization version GIF version |
Description: Subtraction of a group element from itself. (Contributed by NM, 31-Mar-2014.) |
Ref | Expression |
---|---|
grpsubid.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubid.o | ⊢ 0 = (0g‘𝐺) |
grpsubid.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grpsubid | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubid.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2732 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2732 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | grpsubid.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubval 18872 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑋))) |
6 | 5 | anidms 567 | . . 3 ⊢ (𝑋 ∈ 𝐵 → (𝑋 − 𝑋) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑋))) |
7 | 6 | adantl 482 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑋))) |
8 | grpsubid.o | . . 3 ⊢ 0 = (0g‘𝐺) | |
9 | 1, 2, 8, 3 | grprinv 18877 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑋)) = 0 ) |
10 | 7, 9 | eqtrd 2772 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6543 (class class class)co 7411 Basecbs 17146 +gcplusg 17199 0gc0g 17387 Grpcgrp 18821 invgcminusg 18822 -gcsg 18823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-0g 17389 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-minusg 18825 df-sbg 18826 |
This theorem is referenced by: grppncan 18916 grpnpncan0 18921 issubg4 19027 0nsg 19051 gexdvds 19454 abladdsub4 19681 ablsubaddsub 19684 ablpncan2 19685 ablpnpcan 19689 ablnncan 19690 telgsums 19863 dprdfeq0 19894 lmodsubid 20537 dmatsubcl 22007 mdetuni0 22130 chpmat0d 22343 chpdmatlem2 22348 tgpconncomp 23624 tgpt0 23630 tgptsmscls 23661 deg1sublt 25635 lgsqrlem1 26856 archiabllem1a 32378 archiabllem2a 32381 archiabllem2c 32382 ornglmulle 32464 orngrmulle 32465 lfl0 38021 eqlkr 38055 lkrlsp 38058 lclkrlem2m 40476 lcfrlem1 40499 hdmapinvlem3 40877 rngqiprngimfolem 46854 rngqiprngfulem5 46879 |
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