Theorem grpidd2 18598

Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 18582. (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 7285	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = ( 0 (+g‘𝐺) 0 ))
3		oveq2 7276	. . . . . 6 ⊢ (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
4		id 22	. . . . . 6 ⊢ (𝑥 = 0 → 𝑥 = 0 )
5	3, 4	eqeq12d 2755	. . . . 5 ⊢ (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
6		grpidd2.i	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
7	6	ralrimiva 3109	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥)
8		grpidd2.z	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
9	5, 7, 8	rspcdva 3562	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = 0 )
10	2, 9	eqtr3d 2781	. . 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 2842	. . . 4 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
14		eqid 2739	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
15		eqid 2739	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
16		eqid 2739	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
17	14, 15, 16	grpid 18596	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 ))
18	11, 13, 17	syl2anc 583	. . 3 ⊢ (𝜑 → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 ))
19	10, 18	mpbid 231	. 2 ⊢ (𝜑 → (0g‘𝐺) = 0 )
20	19	eqcomd 2745	1 ⊢ (𝜑 → 0 = (0g‘𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ‘cfv 6430  (class class class)co 7268  Basecbs 16893  +_gcplusg 16943  0_gc0g 17131  Grpcgrp 18558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-iota 6388  df-fun 6432  df-fv 6438  df-riota 7225  df-ov 7271  df-0g 17133  df-mgm 18307  df-sgrp 18356  df-mnd 18367  df-grp 18561

This theorem is referenced by:  imasgrp2  18671