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Theorem isgrpid2 18792
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 18791 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((𝑍 + 𝑍) = 𝑍0 = 𝑍))
54biimpd 228 . . 3 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((𝑍 + 𝑍) = 𝑍0 = 𝑍))
65expimpd 455 . 2 (𝐺 ∈ Grp → ((𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍) → 0 = 𝑍))
71, 3grpidcl 18783 . . . 4 (𝐺 ∈ Grp → 0𝐵)
81, 2, 3grplid 18785 . . . . 5 ((𝐺 ∈ Grp ∧ 0𝐵) → ( 0 + 0 ) = 0 )
97, 8mpdan 686 . . . 4 (𝐺 ∈ Grp → ( 0 + 0 ) = 0 )
107, 9jca 513 . . 3 (𝐺 ∈ Grp → ( 0𝐵 ∧ ( 0 + 0 ) = 0 ))
11 eleq1 2822 . . . 4 ( 0 = 𝑍 → ( 0𝐵𝑍𝐵))
12 id 22 . . . . . 6 ( 0 = 𝑍0 = 𝑍)
1312, 12oveq12d 7376 . . . . 5 ( 0 = 𝑍 → ( 0 + 0 ) = (𝑍 + 𝑍))
1413, 12eqeq12d 2749 . . . 4 ( 0 = 𝑍 → (( 0 + 0 ) = 0 ↔ (𝑍 + 𝑍) = 𝑍))
1511, 14anbi12d 632 . . 3 ( 0 = 𝑍 → (( 0𝐵 ∧ ( 0 + 0 ) = 0 ) ↔ (𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
1610, 15syl5ibcom 244 . 2 (𝐺 ∈ Grp → ( 0 = 𝑍 → (𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
176, 16impbid 211 1 (𝐺 ∈ Grp → ((𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  Grpcgrp 18753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-riota 7314  df-ov 7361  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756
This theorem is referenced by:  drngid2  20216  dchr1  26621  erngdvlem4  39500  erngdvlem4-rN  39508
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