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Theorem grpid 18931
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 2735 . 2 ( 0 = 𝑋𝑋 = 0 )
2 grpinveu.b . . . . . . 7 𝐵 = (Base‘𝐺)
3 grpinveu.o . . . . . . 7 0 = (0g𝐺)
42, 3grpidcl 18921 . . . . . 6 (𝐺 ∈ Grp → 0𝐵)
5 grpinveu.p . . . . . . . 8 + = (+g𝐺)
62, 5grprcan 18929 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵0𝐵𝑋𝐵)) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))
763exp2 1352 . . . . . 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 406 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))
112, 5, 3grplid 18923 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + 𝑋) = 𝑋)
1211eqeq2d 2739 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ (𝑋 + 𝑋) = 𝑋))
1310, 12bitr3d 281 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 = 0 ↔ (𝑋 + 𝑋) = 𝑋))
141, 13bitr2id 284 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = 𝑋0 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  cfv 6548  (class class class)co 7420  Basecbs 17179  +gcplusg 17232  0gc0g 17420  Grpcgrp 18889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-riota 7376  df-ov 7423  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18892
This theorem is referenced by:  isgrpid2  18932  grpidd2  18933  subg0  19086  qus0  19143  ghmid  19175  isdrng2  20637  lmod0vid  20776  cnfld0  21319  psr0  21900  psd1  22090  ldual0v  38622  erng0g  40467
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