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Theorem grpid 18403
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 2744 . 2 ( 0 = 𝑋𝑋 = 0 )
2 grpinveu.b . . . . . . 7 𝐵 = (Base‘𝐺)
3 grpinveu.o . . . . . . 7 0 = (0g𝐺)
42, 3grpidcl 18395 . . . . . 6 (𝐺 ∈ Grp → 0𝐵)
5 grpinveu.p . . . . . . . 8 + = (+g𝐺)
62, 5grprcan 18401 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵0𝐵𝑋𝐵)) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))
763exp2 1356 . . . . . 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 410 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))
112, 5, 3grplid 18397 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + 𝑋) = 𝑋)
1211eqeq2d 2748 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ (𝑋 + 𝑋) = 𝑋))
1310, 12bitr3d 284 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 = 0 ↔ (𝑋 + 𝑋) = 𝑋))
141, 13bitr2id 287 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = 𝑋0 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  cfv 6380  (class class class)co 7213  Basecbs 16760  +gcplusg 16802  0gc0g 16944  Grpcgrp 18365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-riota 7170  df-ov 7216  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-grp 18368
This theorem is referenced by:  isgrpid2  18404  grpidd2  18405  subg0  18549  qus0  18602  ghmid  18628  isdrng2  19777  lmod0vid  19931  cnfld0  20387  psr0  20924  ldual0v  36901  erng0g  38745
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