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Theorem grpidd2 17282
 Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 17267. (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 6566 . . . 4 (𝜑 → ( 0 + 0 ) = ( 0 (+g𝐺) 0 ))
3 grpidd2.z . . . . 5 (𝜑0𝐵)
4 grpidd2.i . . . . . 6 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
54ralrimiva 2949 . . . . 5 (𝜑 → ∀𝑥𝐵 ( 0 + 𝑥) = 𝑥)
6 oveq2 6557 . . . . . . 7 (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
7 id 22 . . . . . . 7 (𝑥 = 0𝑥 = 0 )
86, 7eqeq12d 2625 . . . . . 6 (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
98rspcv 3278 . . . . 5 ( 0𝐵 → (∀𝑥𝐵 ( 0 + 𝑥) = 𝑥 → ( 0 + 0 ) = 0 ))
103, 5, 9sylc 63 . . . 4 (𝜑 → ( 0 + 0 ) = 0 )
112, 10eqtr3d 2646 . . 3 (𝜑 → ( 0 (+g𝐺) 0 ) = 0 )
12 grpidd2.j . . . 4 (𝜑𝐺 ∈ Grp)
13 grpidd2.b . . . . 5 (𝜑𝐵 = (Base‘𝐺))
143, 13eleqtrd 2690 . . . 4 (𝜑0 ∈ (Base‘𝐺))
15 eqid 2610 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
16 eqid 2610 . . . . 5 (+g𝐺) = (+g𝐺)
17 eqid 2610 . . . . 5 (0g𝐺) = (0g𝐺)
1815, 16, 17grpid 17280 . . . 4 ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g𝐺) 0 ) = 0 ↔ (0g𝐺) = 0 ))
1912, 14, 18syl2anc 691 . . 3 (𝜑 → (( 0 (+g𝐺) 0 ) = 0 ↔ (0g𝐺) = 0 ))
2011, 19mpbid 221 . 2 (𝜑 → (0g𝐺) = 0 )
2120eqcomd 2616 1 (𝜑0 = (0g𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248 This theorem is referenced by:  imasgrp2  17353
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