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Mirrors > Home > MPE Home > Th. List > grpsubid1 | Structured version Visualization version GIF version |
Description: Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.) |
Ref | Expression |
---|---|
grpsubid.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubid.o | ⊢ 0 = (0g‘𝐺) |
grpsubid.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grpsubid1 | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵) | |
2 | grpsubid.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpsubid.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
4 | 2, 3 | grpidcl 18125 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
5 | eqid 2821 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | eqid 2821 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
7 | grpsubid.m | . . . 4 ⊢ − = (-g‘𝐺) | |
8 | 2, 5, 6, 7 | grpsubval 18143 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 − 0 ) = (𝑋(+g‘𝐺)((invg‘𝐺)‘ 0 ))) |
9 | 1, 4, 8 | syl2anr 598 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = (𝑋(+g‘𝐺)((invg‘𝐺)‘ 0 ))) |
10 | 3, 6 | grpinvid 18154 | . . . 4 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 ) |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝐺)‘ 0 ) = 0 ) |
12 | 11 | oveq2d 7166 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)((invg‘𝐺)‘ 0 )) = (𝑋(+g‘𝐺) 0 )) |
13 | 2, 5, 3 | grprid 18128 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺) 0 ) = 𝑋) |
14 | 9, 12, 13 | 3eqtrd 2860 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 0gc0g 16707 Grpcgrp 18097 invgcminusg 18098 -gcsg 18099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 |
This theorem is referenced by: odmod 18668 sylow3lem1 18746 telgsums 19107 dprdfeq0 19138 chp0mat 21448 tsmsxplem1 22755 tngnm 23254 ply1divex 24724 ply1remlem 24750 qqhcn 31227 lcfrlem33 38705 |
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