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Theorem grpid 17397
Description: Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.)
Hypotheses
Ref Expression
grpinveu.b 𝐵 = (Base‘𝐺)
grpinveu.p + = (+g𝐺)
grpinveu.o 0 = (0g𝐺)
Assertion
Ref Expression
grpid ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = 𝑋0 = 𝑋))

Proof of Theorem grpid
StepHypRef Expression
1 eqcom 2628 . 2 ( 0 = 𝑋𝑋 = 0 )
2 grpinveu.b . . . . . . 7 𝐵 = (Base‘𝐺)
3 grpinveu.o . . . . . . 7 0 = (0g𝐺)
42, 3grpidcl 17390 . . . . . 6 (𝐺 ∈ Grp → 0𝐵)
5 grpinveu.p . . . . . . . 8 + = (+g𝐺)
62, 5grprcan 17395 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵0𝐵𝑋𝐵)) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))
763exp2 1282 . . . . . 6 (𝐺 ∈ Grp → (𝑋𝐵 → ( 0𝐵 → (𝑋𝐵 → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 )))))
84, 7mpid 44 . . . . 5 (𝐺 ∈ Grp → (𝑋𝐵 → (𝑋𝐵 → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))))
98pm2.43d 53 . . . 4 (𝐺 ∈ Grp → (𝑋𝐵 → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 )))
109imp 445 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))
112, 5, 3grplid 17392 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + 𝑋) = 𝑋)
1211eqeq2d 2631 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ (𝑋 + 𝑋) = 𝑋))
1310, 12bitr3d 270 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 = 0 ↔ (𝑋 + 𝑋) = 𝑋))
141, 13syl5rbb 273 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = 𝑋0 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  0gc0g 16040  Grpcgrp 17362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-riota 6576  df-ov 6618  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365
This theorem is referenced by:  isgrpid2  17398  grpidd2  17399  subg0  17540  qus0  17592  ghmid  17606  symgid  17761  isdrng2  18697  lmod0vid  18835  psr0  19339  cnfld0  19710  ldual0v  33956  erng0g  35801
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