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Theorem isgrpid2 19007
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 19006 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((𝑍 + 𝑍) = 𝑍0 = 𝑍))
54biimpd 229 . . 3 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((𝑍 + 𝑍) = 𝑍0 = 𝑍))
65expimpd 453 . 2 (𝐺 ∈ Grp → ((𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍) → 0 = 𝑍))
71, 3grpidcl 18996 . . . 4 (𝐺 ∈ Grp → 0𝐵)
81, 2, 3grplid 18998 . . . . 5 ((𝐺 ∈ Grp ∧ 0𝐵) → ( 0 + 0 ) = 0 )
97, 8mpdan 687 . . . 4 (𝐺 ∈ Grp → ( 0 + 0 ) = 0 )
107, 9jca 511 . . 3 (𝐺 ∈ Grp → ( 0𝐵 ∧ ( 0 + 0 ) = 0 ))
11 eleq1 2827 . . . 4 ( 0 = 𝑍 → ( 0𝐵𝑍𝐵))
12 id 22 . . . . . 6 ( 0 = 𝑍0 = 𝑍)
1312, 12oveq12d 7449 . . . . 5 ( 0 = 𝑍 → ( 0 + 0 ) = (𝑍 + 𝑍))
1413, 12eqeq12d 2751 . . . 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 245 . 2 (𝐺 ∈ Grp → ( 0 = 𝑍 → (𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
176, 16impbid 212 1 (𝐺 ∈ Grp → ((𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967
This theorem is referenced by:  drngid2  20769  dchr1  27316  rloc0g  33258  erngdvlem4  40974  erngdvlem4-rN  40982
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