Theorem grpidd2 19043

Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 19024. (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 7427	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = ( 0 (+g‘𝐺) 0 ))
3		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
4		id 23	. . . . . 6 ⊢ (𝑥 = 0 → 𝑥 = 0 )
5	3, 4	eqeq12d 2777	. . . . 5 ⊢ (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
6		grpidd2.i	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
7	6	ralrimiva 3155	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥)
8		grpidd2.z	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
9	5, 7, 8	rspcdva 3581	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = 0 )
10	2, 9	eqtr3d 2798	. . 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 2863	. . . 4 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
14		eqid 2761	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
15		eqid 2761	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
16		eqid 2761	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
17	14, 15, 16	grpid 19041	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 ))
18	11, 13, 17	syl2anc 595	. . 3 ⊢ (𝜑 → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 ))
19	10, 18	mpbid 235	. 2 ⊢ (𝜑 → (0g‘𝐺) = 0 )
20	19	eqcomd 2767	1 ⊢ (𝜑 → 0 = (0g‘𝐺))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  +_gcplusg 17309  0_gc0g 17491  Grpcgrp 18999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-riota 7367  df-ov 7413  df-0g 17493  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-grp 19002

This theorem is referenced by:  imasgrp2  19120