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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 19010 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
6 | 5 | adantl 481 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
7 | 1, 2, 3 | grpinveu 19004 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) |
8 | riotacl2 7403 | . . . 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 2838 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ {𝑦 ∈ 𝐵 ∣ (𝑦 + 𝑋) = 0 }) |
11 | oveq1 7437 | . . . . 5 ⊢ (𝑦 = (𝑁‘𝑋) → (𝑦 + 𝑋) = ((𝑁‘𝑋) + 𝑋)) | |
12 | 11 | eqeq1d 2736 | . . . 4 ⊢ (𝑦 = (𝑁‘𝑋) → ((𝑦 + 𝑋) = 0 ↔ ((𝑁‘𝑋) + 𝑋) = 0 )) |
13 | 12 | elrab 3694 | . . 3 ⊢ ((𝑁‘𝑋) ∈ {𝑦 ∈ 𝐵 ∣ (𝑦 + 𝑋) = 0 } ↔ ((𝑁‘𝑋) ∈ 𝐵 ∧ ((𝑁‘𝑋) + 𝑋) = 0 )) |
14 | 13 | simprbi 496 | . 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 395 = wceq 1536 ∈ wcel 2105 ∃!wreu 3375 {crab 3432 ‘cfv 6562 ℩crio 7386 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 0gc0g 17485 Grpcgrp 18963 invgcminusg 18964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-riota 7387 df-ov 7433 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 |
This theorem is referenced by: grprinv 19020 grpinvid1 19021 grpinvid2 19022 isgrpinv 19023 grplinvd 19024 grplrinv 19026 grplcan 19030 grpasscan2 19032 grpinvinv 19035 grpraddf1o 19044 grpinvssd 19047 grpsubadd 19058 grplactcnv 19073 prdsinvlem 19079 imasgrp 19086 ghmgrp 19096 mulgdirlem 19135 issubg2 19171 isnsg3 19190 nmzsubg 19195 ssnmz 19196 eqger 19208 qusgrp 19216 conjghm 19279 galcan 19334 cntzsubg 19369 lsmmod 19707 lsmdisj2 19714 ringnegr 20316 unitlinv 20409 isdrng2 20759 lmodvneg1 20919 evpmodpmf1o 21631 psrlinv 21992 grpvlinv 22417 tgpconncompeqg 24135 qustgpopn 24143 clmvslinv 25154 ogrpinv0le 33074 ogrpaddltrbid 33079 ogrpinv0lt 33081 ogrpinvlt 33082 quslsm 33412 lflnegl 39057 dvhgrp 41089 |
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