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Theorem isgrpid2 12918
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 12917 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((𝑍 + 𝑍) = 𝑍0 = 𝑍))
54biimpd 144 . . 3 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((𝑍 + 𝑍) = 𝑍0 = 𝑍))
65expimpd 363 . 2 (𝐺 ∈ Grp → ((𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍) → 0 = 𝑍))
71, 3grpidcl 12909 . . . 4 (𝐺 ∈ Grp → 0𝐵)
81, 2, 3grplid 12911 . . . . 5 ((𝐺 ∈ Grp ∧ 0𝐵) → ( 0 + 0 ) = 0 )
97, 8mpdan 421 . . . 4 (𝐺 ∈ Grp → ( 0 + 0 ) = 0 )
107, 9jca 306 . . 3 (𝐺 ∈ Grp → ( 0𝐵 ∧ ( 0 + 0 ) = 0 ))
11 eleq1 2240 . . . 4 ( 0 = 𝑍 → ( 0𝐵𝑍𝐵))
12 id 19 . . . . . 6 ( 0 = 𝑍0 = 𝑍)
1312, 12oveq12d 5895 . . . . 5 ( 0 = 𝑍 → ( 0 + 0 ) = (𝑍 + 𝑍))
1413, 12eqeq12d 2192 . . . 4 ( 0 = 𝑍 → (( 0 + 0 ) = 0 ↔ (𝑍 + 𝑍) = 𝑍))
1511, 14anbi12d 473 . . 3 ( 0 = 𝑍 → (( 0𝐵 ∧ ( 0 + 0 ) = 0 ) ↔ (𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
1610, 15syl5ibcom 155 . 2 (𝐺 ∈ Grp → ( 0 = 𝑍 → (𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
176, 16impbid 129 1 (𝐺 ∈ Grp → ((𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  cfv 5218  (class class class)co 5877  Basecbs 12464  +gcplusg 12538  0gc0g 12710  Grpcgrp 12882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885
This theorem is referenced by: (None)
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