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Theorem grpid 18077
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 2825 . 2 ( 0 = 𝑋𝑋 = 0 )
2 grpinveu.b . . . . . . 7 𝐵 = (Base‘𝐺)
3 grpinveu.o . . . . . . 7 0 = (0g𝐺)
42, 3grpidcl 18069 . . . . . 6 (𝐺 ∈ Grp → 0𝐵)
5 grpinveu.p . . . . . . . 8 + = (+g𝐺)
62, 5grprcan 18075 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵0𝐵𝑋𝐵)) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))
763exp2 1346 . . . . . 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 407 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))
112, 5, 3grplid 18071 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + 𝑋) = 𝑋)
1211eqeq2d 2829 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ (𝑋 + 𝑋) = 𝑋))
1310, 12bitr3d 282 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 = 0 ↔ (𝑋 + 𝑋) = 𝑋))
141, 13syl5rbb 285 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = 𝑋0 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Grpcgrp 18041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-riota 7103  df-ov 7148  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044
This theorem is referenced by:  isgrpid2  18078  grpidd2  18079  subg0  18223  qus0  18276  ghmid  18302  symgid  18459  isdrng2  19441  lmod0vid  19595  psr0  20107  cnfld0  20497  ldual0v  36166  erng0g  38010
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