Theorem grpidd2 19189

Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 19170. (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 7437	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = ( 0 (+g‘𝐺) 0 ))
3		oveq2 7428	. . . . . 6 ⊢ (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
4		id 23	. . . . . 6 ⊢ (𝑥 = 0 → 𝑥 = 0 )
5	3, 4	eqeq12d 2776	. . . . 5 ⊢ (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
6		grpidd2.i	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
7	6	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥)
8		grpidd2.z	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
9	5, 7, 8	rspcdva 3577	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = 0 )
10	2, 9	eqtr3d 2797	. . 3 ⊢ (𝜑 → ( 0 (+g‘𝐺) 0 ) = 0 )
11		grpidd2.j	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
12		grpidd2.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
13	8, 12	eleqtrd 2862	. . . 4 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
14		eqid 2760	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
15		eqid 2760	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
16		eqid 2760	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
17	14, 15, 16	grpid 19187	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 ))
18	11, 13, 17	syl2anc 596	. . 3 ⊢ (𝜑 → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 ))
19	10, 18	mpbid 235	. 2 ⊢ (𝜑 → (0g‘𝐺) = 0 )
20	19	eqcomd 2766	1 ⊢ (𝜑 → 0 = (0g‘𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ‘cfv 6538  (class class class)co 7420  Basecbs 17388  +_gcplusg 17429  0_gc0g 17611  Grpcgrp 19145

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-riota 7377  df-ov 7423  df-0g 17613  df-mgm 18817  df-sgrp 18909  df-mnd 18925  df-grp 19148

This theorem is used by:  imasgrp2  19266