Theorem grpidd2 17459

Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 17444. (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 6667	. . . 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 2966	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥)
6		oveq2 6658	. . . . . . 7 ⊢ (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
7		id 22	. . . . . . 7 ⊢ (𝑥 = 0 → 𝑥 = 0 )
8	6, 7	eqeq12d 2637	. . . . . 6 ⊢ (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
9	8	rspcv 3305	. . . . 5 ⊢ ( 0 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥 → ( 0 + 0 ) = 0 ))
10	3, 5, 9	sylc 65	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = 0 )
11	2, 10	eqtr3d 2658	. . 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 2703	. . . 4 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
15		eqid 2622	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
16		eqid 2622	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
17		eqid 2622	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
18	15, 16, 17	grpid 17457	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 ))
19	12, 14, 18	syl2anc 693	. . 3 ⊢ (𝜑 → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 ))
20	11, 19	mpbid 222	. 2 ⊢ (𝜑 → (0g‘𝐺) = 0 )
21	20	eqcomd 2628	1 ⊢ (𝜑 → 0 = (0g‘𝐺))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  Grpcgrp 17422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425

This theorem is referenced by:  imasgrp2  17530