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Theorem grpidd2 18904
Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 18885. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
grpidd2.b (𝜑𝐵 = (Base‘𝐺))
grpidd2.p (𝜑+ = (+g𝐺))
grpidd2.z (𝜑0𝐵)
grpidd2.i ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
grpidd2.j (𝜑𝐺 ∈ Grp)
Assertion
Ref Expression
grpidd2 (𝜑0 = (0g𝐺))
Distinct variable groups:   𝑥,𝐵   𝑥, +   𝜑,𝑥   𝑥, 0
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem grpidd2
StepHypRef Expression
1 grpidd2.p . . . . 5 (𝜑+ = (+g𝐺))
21oveqd 7421 . . . 4 (𝜑 → ( 0 + 0 ) = ( 0 (+g𝐺) 0 ))
3 oveq2 7412 . . . . . 6 (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
4 id 22 . . . . . 6 (𝑥 = 0𝑥 = 0 )
53, 4eqeq12d 2742 . . . . 5 (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
6 grpidd2.i . . . . . 6 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
76ralrimiva 3140 . . . . 5 (𝜑 → ∀𝑥𝐵 ( 0 + 𝑥) = 𝑥)
8 grpidd2.z . . . . 5 (𝜑0𝐵)
95, 7, 8rspcdva 3607 . . . 4 (𝜑 → ( 0 + 0 ) = 0 )
102, 9eqtr3d 2768 . . 3 (𝜑 → ( 0 (+g𝐺) 0 ) = 0 )
11 grpidd2.j . . . 4 (𝜑𝐺 ∈ Grp)
12 grpidd2.b . . . . 5 (𝜑𝐵 = (Base‘𝐺))
138, 12eleqtrd 2829 . . . 4 (𝜑0 ∈ (Base‘𝐺))
14 eqid 2726 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
15 eqid 2726 . . . . 5 (+g𝐺) = (+g𝐺)
16 eqid 2726 . . . . 5 (0g𝐺) = (0g𝐺)
1714, 15, 16grpid 18902 . . . 4 ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g𝐺) 0 ) = 0 ↔ (0g𝐺) = 0 ))
1811, 13, 17syl2anc 583 . . 3 (𝜑 → (( 0 (+g𝐺) 0 ) = 0 ↔ (0g𝐺) = 0 ))
1910, 18mpbid 231 . 2 (𝜑 → (0g𝐺) = 0 )
2019eqcomd 2732 1 (𝜑0 = (0g𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  cfv 6536  (class class class)co 7404  Basecbs 17150  +gcplusg 17203  0gc0g 17391  Grpcgrp 18860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-riota 7360  df-ov 7407  df-0g 17393  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-grp 18863
This theorem is referenced by:  imasgrp2  18980
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