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	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
grpidd2.p	⊢ (𝜑 → + = (+g‘𝐺))
grpidd2.z	⊢ (𝜑 → 0 ∈ 𝐵)
grpidd2.i	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
grpidd2.j	⊢ (𝜑 → 𝐺 ∈ Grp)

Assertion

Ref	Expression
grpidd2	⊢ (𝜑 → 0 = (0g‘𝐺))

Distinct variable groups:   𝑥,𝐵   𝑥, +   𝜑,𝑥   𝑥, 0

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem grpidd2

Step	Hyp	Ref	Expression
1		grpidd2.p	. . . . 5 ⊢ (𝜑 → + = (+g‘𝐺))
2	1	oveqd 6566	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = ( 0 (+g‘𝐺) 0 ))
3		grpidd2.z	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
4		grpidd2.i	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
5	4	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥)
6		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
7		id 22	. . . . . . 7 ⊢ (𝑥 = 0 → 𝑥 = 0 )
8	6, 7	eqeq12d 2625	. . . . . 6 ⊢ (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
9	8	rspcv 3278	. . . . 5 ⊢ ( 0 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥 → ( 0 + 0 ) = 0 ))
10	3, 5, 9	sylc 63	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = 0 )
11	2, 10	eqtr3d 2646	. . 3 ⊢ (𝜑 → ( 0 (+g‘𝐺) 0 ) = 0 )
12		grpidd2.j	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
13		grpidd2.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
14	3, 13	eleqtrd 2690	. . . 4 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
15		eqid 2610	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
16		eqid 2610	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
17		eqid 2610	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
18	15, 16, 17	grpid 17280	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 ))
19	12, 14, 18	syl2anc 691	. . 3 ⊢ (𝜑 → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 ))
20	11, 19	mpbid 221	. 2 ⊢ (𝜑 → (0g‘𝐺) = 0 )
21	20	eqcomd 2616	1 ⊢ (𝜑 → 0 = (0g‘𝐺))

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  0_gc0g 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