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Theorem grpid 18901
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 2731 . 2 ( 0 = 𝑋𝑋 = 0 )
2 grpinveu.b . . . . . . 7 𝐵 = (Base‘𝐺)
3 grpinveu.o . . . . . . 7 0 = (0g𝐺)
42, 3grpidcl 18891 . . . . . 6 (𝐺 ∈ Grp → 0𝐵)
5 grpinveu.p . . . . . . . 8 + = (+g𝐺)
62, 5grprcan 18899 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵0𝐵𝑋𝐵)) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))
763exp2 1351 . . . . . 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 18893 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + 𝑋) = 𝑋)
1211eqeq2d 2735 . . 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 1533  wcel 2098  cfv 6534  (class class class)co 7402  Basecbs 17149  +gcplusg 17202  0gc0g 17390  Grpcgrp 18859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-riota 7358  df-ov 7405  df-0g 17392  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-grp 18862
This theorem is referenced by:  isgrpid2  18902  grpidd2  18903  subg0  19055  qus0  19111  ghmid  19143  isdrng2  20597  lmod0vid  20736  cnfld0  21274  psr0  21850  ldual0v  38523  erng0g  40368
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