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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 17284 | . . . 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 17279 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) |
8 | riotacl2 6524 | . . . 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 2688 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ {𝑦 ∈ 𝐵 ∣ (𝑦 + 𝑋) = 0 }) |
11 | oveq1 6556 | . . . . 5 ⊢ (𝑦 = (𝑁‘𝑋) → (𝑦 + 𝑋) = ((𝑁‘𝑋) + 𝑋)) | |
12 | 11 | eqeq1d 2612 | . . . 4 ⊢ (𝑦 = (𝑁‘𝑋) → ((𝑦 + 𝑋) = 0 ↔ ((𝑁‘𝑋) + 𝑋) = 0 )) |
13 | 12 | elrab 3331 | . . 3 ⊢ ((𝑁‘𝑋) ∈ {𝑦 ∈ 𝐵 ∣ (𝑦 + 𝑋) = 0 } ↔ ((𝑁‘𝑋) ∈ 𝐵 ∧ ((𝑁‘𝑋) + 𝑋) = 0 )) |
14 | 13 | simprbi 479 | . 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 383 = wceq 1475 ∈ wcel 1977 ∃!wreu 2898 {crab 2900 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 invgcminusg 17246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 |
This theorem is referenced by: grprinv 17292 grpinvid1 17293 grpinvid2 17294 isgrpinv 17295 grplrinv 17296 grplcan 17300 grpasscan2 17302 grpinvinv 17305 grpinvssd 17315 grpsubadd 17326 grplactcnv 17341 prdsinvlem 17347 imasgrp 17354 ghmgrp 17362 mulgdirlem 17395 issubg2 17432 isnsg3 17451 nmzsubg 17458 ssnmz 17459 eqger 17467 qusgrp 17472 conjghm 17514 galcan 17560 cntzsubg 17592 lsmmod 17911 lsmdisj2 17918 rngnegr 18418 unitlinv 18500 isdrng2 18580 lmodvneg1 18729 psrlinv 19218 evpmodpmf1o 19761 grpvlinv 20020 tgpconcompeqg 21725 qustgpopn 21733 clmvslinv 22716 ogrpinv0le 29047 ogrpaddltrbid 29052 ogrpinv0lt 29054 ogrpinvlt 29055 lflnegl 33381 dvhgrp 35414 |
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