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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 18840 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
6 | 5 | adantl 482 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
7 | 1, 2, 3 | grpinveu 18834 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) |
8 | riotacl2 7366 | . . . 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 2832 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ {𝑦 ∈ 𝐵 ∣ (𝑦 + 𝑋) = 0 }) |
11 | oveq1 7400 | . . . . 5 ⊢ (𝑦 = (𝑁‘𝑋) → (𝑦 + 𝑋) = ((𝑁‘𝑋) + 𝑋)) | |
12 | 11 | eqeq1d 2733 | . . . 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 497 | . 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 396 = wceq 1541 ∈ wcel 2106 ∃!wreu 3373 {crab 3431 ‘cfv 6532 ℩crio 7348 (class class class)co 7393 Basecbs 17126 +gcplusg 17179 0gc0g 17367 Grpcgrp 18794 invgcminusg 18795 |
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 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fv 6540 df-riota 7349 df-ov 7396 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-minusg 18798 |
This theorem is referenced by: grprinv 18850 grpinvid1 18851 grpinvid2 18852 isgrpinv 18853 grplinvd 18854 grplrinv 18855 grplcan 18859 grpasscan2 18861 grpinvinv 18864 grpinvssd 18874 grpsubadd 18885 grplactcnv 18900 prdsinvlem 18906 imasgrp 18913 ghmgrp 18921 mulgdirlem 18957 issubg2 18993 isnsg3 19012 nmzsubg 19017 ssnmz 19018 eqger 19030 qusgrp 19035 conjghm 19089 galcan 19134 cntzsubg 19167 lsmmod 19507 lsmdisj2 19514 ringnegr 20072 unitlinv 20159 isdrng2 20278 lmodvneg1 20464 evpmodpmf1o 21082 psrlinv 21447 grpvlinv 21826 tgpconncompeqg 23545 qustgpopn 23553 clmvslinv 24553 ogrpinv0le 32104 ogrpaddltrbid 32109 ogrpinv0lt 32111 ogrpinvlt 32112 quslsm 32374 lflnegl 37751 dvhgrp 39783 |
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