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Mirrors > Home > MPE Home > Th. List > grplinv | Structured version Visualization version GIF version |
Description: The left inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
grpinv.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinv.p | ⊢ + = (+g‘𝐺) |
grpinv.u | ⊢ 0 = (0g‘𝐺) |
grpinv.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grplinv | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinv.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpinv.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
3 | grpinv.u | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
4 | grpinv.n | . . . . 5 ⊢ 𝑁 = (invg‘𝐺) | |
5 | 1, 2, 3, 4 | grpinvval 18143 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
6 | 5 | adantl 484 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
7 | 1, 2, 3 | grpinveu 18137 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) |
8 | riotacl2 7129 | . . . 4 ⊢ (∃!𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) ∈ {𝑦 ∈ 𝐵 ∣ (𝑦 + 𝑋) = 0 }) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) ∈ {𝑦 ∈ 𝐵 ∣ (𝑦 + 𝑋) = 0 }) |
10 | 6, 9 | eqeltrd 2913 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ {𝑦 ∈ 𝐵 ∣ (𝑦 + 𝑋) = 0 }) |
11 | oveq1 7162 | . . . . 5 ⊢ (𝑦 = (𝑁‘𝑋) → (𝑦 + 𝑋) = ((𝑁‘𝑋) + 𝑋)) | |
12 | 11 | eqeq1d 2823 | . . . 4 ⊢ (𝑦 = (𝑁‘𝑋) → ((𝑦 + 𝑋) = 0 ↔ ((𝑁‘𝑋) + 𝑋) = 0 )) |
13 | 12 | elrab 3679 | . . 3 ⊢ ((𝑁‘𝑋) ∈ {𝑦 ∈ 𝐵 ∣ (𝑦 + 𝑋) = 0 } ↔ ((𝑁‘𝑋) ∈ 𝐵 ∧ ((𝑁‘𝑋) + 𝑋) = 0 )) |
14 | 13 | simprbi 499 | . 2 ⊢ ((𝑁‘𝑋) ∈ {𝑦 ∈ 𝐵 ∣ (𝑦 + 𝑋) = 0 } → ((𝑁‘𝑋) + 𝑋) = 0 ) |
15 | 10, 14 | syl 17 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃!wreu 3140 {crab 3142 ‘cfv 6354 ℩crio 7112 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 0gc0g 16712 Grpcgrp 18102 invgcminusg 18103 |
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 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-riota 7113 df-ov 7158 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 |
This theorem is referenced by: grprinv 18152 grpinvid1 18153 grpinvid2 18154 isgrpinv 18155 grplrinv 18156 grplcan 18160 grpasscan2 18162 grpinvinv 18165 grpinvssd 18175 grpsubadd 18186 grplactcnv 18201 prdsinvlem 18207 imasgrp 18214 ghmgrp 18222 mulgdirlem 18257 issubg2 18293 isnsg3 18311 nmzsubg 18316 ssnmz 18317 eqger 18329 qusgrp 18334 conjghm 18388 galcan 18433 cntzsubg 18466 lsmmod 18800 lsmdisj2 18807 rngnegr 19344 unitlinv 19426 isdrng2 19511 lmodvneg1 19676 psrlinv 20176 evpmodpmf1o 20739 grpvlinv 21005 tgpconncompeqg 22719 qustgpopn 22727 clmvslinv 23711 ogrpinv0le 30716 ogrpaddltrbid 30721 ogrpinv0lt 30723 ogrpinvlt 30724 lflnegl 36211 dvhgrp 38242 |
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