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Theorem grpidd2 17660
Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 17645. (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 6830 . . . 4 (𝜑 → ( 0 + 0 ) = ( 0 (+g𝐺) 0 ))
3 grpidd2.z . . . . 5 (𝜑0𝐵)
4 grpidd2.i . . . . . 6 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
54ralrimiva 3104 . . . . 5 (𝜑 → ∀𝑥𝐵 ( 0 + 𝑥) = 𝑥)
6 oveq2 6821 . . . . . . 7 (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
7 id 22 . . . . . . 7 (𝑥 = 0𝑥 = 0 )
86, 7eqeq12d 2775 . . . . . 6 (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
98rspcv 3445 . . . . 5 ( 0𝐵 → (∀𝑥𝐵 ( 0 + 𝑥) = 𝑥 → ( 0 + 0 ) = 0 ))
103, 5, 9sylc 65 . . . 4 (𝜑 → ( 0 + 0 ) = 0 )
112, 10eqtr3d 2796 . . 3 (𝜑 → ( 0 (+g𝐺) 0 ) = 0 )
12 grpidd2.j . . . 4 (𝜑𝐺 ∈ Grp)
13 grpidd2.b . . . . 5 (𝜑𝐵 = (Base‘𝐺))
143, 13eleqtrd 2841 . . . 4 (𝜑0 ∈ (Base‘𝐺))
15 eqid 2760 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
16 eqid 2760 . . . . 5 (+g𝐺) = (+g𝐺)
17 eqid 2760 . . . . 5 (0g𝐺) = (0g𝐺)
1815, 16, 17grpid 17658 . . . 4 ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g𝐺) 0 ) = 0 ↔ (0g𝐺) = 0 ))
1912, 14, 18syl2anc 696 . . 3 (𝜑 → (( 0 (+g𝐺) 0 ) = 0 ↔ (0g𝐺) = 0 ))
2011, 19mpbid 222 . 2 (𝜑 → (0g𝐺) = 0 )
2120eqcomd 2766 1 (𝜑0 = (0g𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  0gc0g 16302  Grpcgrp 17623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-riota 6774  df-ov 6816  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626
This theorem is referenced by:  imasgrp2  17731
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