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Theorem isgrpid2 17281
 Description: Properties showing that an element 𝑍 is the identity element of a group. (Contributed by NM, 7-Aug-2013.)
Hypotheses
Ref Expression
grpinveu.b 𝐵 = (Base‘𝐺)
grpinveu.p + = (+g𝐺)
grpinveu.o 0 = (0g𝐺)
Assertion
Ref Expression
isgrpid2 (𝐺 ∈ Grp → ((𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍))

Proof of Theorem isgrpid2
StepHypRef Expression
1 grpinveu.b . . . . 5 𝐵 = (Base‘𝐺)
2 grpinveu.p . . . . 5 + = (+g𝐺)
3 grpinveu.o . . . . 5 0 = (0g𝐺)
41, 2, 3grpid 17280 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((𝑍 + 𝑍) = 𝑍0 = 𝑍))
54biimpd 218 . . 3 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((𝑍 + 𝑍) = 𝑍0 = 𝑍))
65expimpd 627 . 2 (𝐺 ∈ Grp → ((𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍) → 0 = 𝑍))
71, 3grpidcl 17273 . . . 4 (𝐺 ∈ Grp → 0𝐵)
81, 2, 3grplid 17275 . . . . 5 ((𝐺 ∈ Grp ∧ 0𝐵) → ( 0 + 0 ) = 0 )
97, 8mpdan 699 . . . 4 (𝐺 ∈ Grp → ( 0 + 0 ) = 0 )
107, 9jca 553 . . 3 (𝐺 ∈ Grp → ( 0𝐵 ∧ ( 0 + 0 ) = 0 ))
11 eleq1 2676 . . . 4 ( 0 = 𝑍 → ( 0𝐵𝑍𝐵))
12 id 22 . . . . . 6 ( 0 = 𝑍0 = 𝑍)
1312, 12oveq12d 6567 . . . . 5 ( 0 = 𝑍 → ( 0 + 0 ) = (𝑍 + 𝑍))
1413, 12eqeq12d 2625 . . . 4 ( 0 = 𝑍 → (( 0 + 0 ) = 0 ↔ (𝑍 + 𝑍) = 𝑍))
1511, 14anbi12d 743 . . 3 ( 0 = 𝑍 → (( 0𝐵 ∧ ( 0 + 0 ) = 0 ) ↔ (𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
1610, 15syl5ibcom 234 . 2 (𝐺 ∈ Grp → ( 0 = 𝑍 → (𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
176, 16impbid 201 1 (𝐺 ∈ Grp → ((𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248 This theorem is referenced by:  drngid2  18586  dchr1  24782  erngdvlem4  35297  erngdvlem4-rN  35305
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