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Theorem isgrpid2 18078
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 18077 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((𝑍 + 𝑍) = 𝑍0 = 𝑍))
54biimpd 230 . . 3 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((𝑍 + 𝑍) = 𝑍0 = 𝑍))
65expimpd 454 . 2 (𝐺 ∈ Grp → ((𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍) → 0 = 𝑍))
71, 3grpidcl 18069 . . . 4 (𝐺 ∈ Grp → 0𝐵)
81, 2, 3grplid 18071 . . . . 5 ((𝐺 ∈ Grp ∧ 0𝐵) → ( 0 + 0 ) = 0 )
97, 8mpdan 683 . . . 4 (𝐺 ∈ Grp → ( 0 + 0 ) = 0 )
107, 9jca 512 . . 3 (𝐺 ∈ Grp → ( 0𝐵 ∧ ( 0 + 0 ) = 0 ))
11 eleq1 2897 . . . 4 ( 0 = 𝑍 → ( 0𝐵𝑍𝐵))
12 id 22 . . . . . 6 ( 0 = 𝑍0 = 𝑍)
1312, 12oveq12d 7163 . . . . 5 ( 0 = 𝑍 → ( 0 + 0 ) = (𝑍 + 𝑍))
1413, 12eqeq12d 2834 . . . 4 ( 0 = 𝑍 → (( 0 + 0 ) = 0 ↔ (𝑍 + 𝑍) = 𝑍))
1511, 14anbi12d 630 . . 3 ( 0 = 𝑍 → (( 0𝐵 ∧ ( 0 + 0 ) = 0 ) ↔ (𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
1610, 15syl5ibcom 246 . 2 (𝐺 ∈ Grp → ( 0 = 𝑍 → (𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
176, 16impbid 213 1 (𝐺 ∈ Grp → ((𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍))
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:  drngid2  19447  dchr1  25760  erngdvlem4  38007  erngdvlem4-rN  38015
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