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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 18998 | . . . 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 18992 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) |
| 8 | riotacl2 7404 | . . . 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 2841 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ {𝑦 ∈ 𝐵 ∣ (𝑦 + 𝑋) = 0 }) |
| 11 | oveq1 7438 | . . . . 5 ⊢ (𝑦 = (𝑁‘𝑋) → (𝑦 + 𝑋) = ((𝑁‘𝑋) + 𝑋)) | |
| 12 | 11 | eqeq1d 2739 | . . . 4 ⊢ (𝑦 = (𝑁‘𝑋) → ((𝑦 + 𝑋) = 0 ↔ ((𝑁‘𝑋) + 𝑋) = 0 )) |
| 13 | 12 | elrab 3692 | . . 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 1540 ∈ wcel 2108 ∃!wreu 3378 {crab 3436 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Grpcgrp 18951 invgcminusg 18952 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 |
| This theorem is referenced by: grprinv 19008 grpinvid1 19009 grpinvid2 19010 isgrpinv 19011 grplinvd 19012 grplrinv 19014 grplcan 19018 grpasscan2 19020 grpinvinv 19023 grpraddf1o 19032 grpinvssd 19035 grpsubadd 19046 grplactcnv 19061 prdsinvlem 19067 imasgrp 19074 ghmgrp 19084 mulgdirlem 19123 issubg2 19159 isnsg3 19178 nmzsubg 19183 ssnmz 19184 eqger 19196 qusgrp 19204 conjghm 19267 galcan 19322 cntzsubg 19357 lsmmod 19693 lsmdisj2 19700 ringnegr 20300 unitlinv 20393 isdrng2 20743 lmodvneg1 20903 evpmodpmf1o 21614 psrlinv 21975 grpvlinv 22402 tgpconncompeqg 24120 qustgpopn 24128 clmvslinv 25141 ogrpinv0le 33092 ogrpaddltrbid 33097 ogrpinv0lt 33099 ogrpinvlt 33100 quslsm 33433 lflnegl 39077 dvhgrp 41109 |
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