Theorem grpidd2 19105

Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 19086. (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 7433	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = ( 0 (+g‘𝐺) 0 ))
3		oveq2 7424	. . . . . 6 ⊢ (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
4		id 23	. . . . . 6 ⊢ (𝑥 = 0 → 𝑥 = 0 )
5	3, 4	eqeq12d 2778	. . . . 5 ⊢ (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
6		grpidd2.i	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
7	6	ralrimiva 3156	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥)
8		grpidd2.z	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
9	5, 7, 8	rspcdva 3580	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = 0 )
10	2, 9	eqtr3d 2799	. . 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 2864	. . . 4 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
14		eqid 2762	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
15		eqid 2762	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
16		eqid 2762	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
17	14, 15, 16	grpid 19103	. . . 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 2768	1 ⊢ (𝜑 → 0 = (0g‘𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  +_gcplusg 17346  0_gc0g 17528  Grpcgrp 19061

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-riota 7373  df-ov 7419  df-0g 17530  df-mgm 18734  df-sgrp 18825  df-mnd 18841  df-grp 19064

This theorem is used by:  imasgrp2  19182