Theorem grpidd2 13848

Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 13830. (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 6102	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = ( 0 (+g‘𝐺) 0 ))
3		oveq2 6093	. . . . . 6 ⊢ (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
4		id 19	. . . . . 6 ⊢ (𝑥 = 0 → 𝑥 = 0 )
5	3, 4	eqeq12d 2253	. . . . 5 ⊢ (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
6		grpidd2.i	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
7	6	ralrimiva 2623	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥)
8		grpidd2.z	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
9	5, 7, 8	rspcdva 2934	. . . 4 ⊢ (𝜑 → ( 0 + 0 ) = 0 )
10	2, 9	eqtr3d 2273	. . 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 2317	. . . 4 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
14		eqid 2238	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
15		eqid 2238	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
16		eqid 2238	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
17	14, 15, 16	grpid 13846	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 ))
18	11, 13, 17	syl2anc 415	. . 3 ⊢ (𝜑 → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 ))
19	10, 18	mpbid 147	. 2 ⊢ (𝜑 → (0g‘𝐺) = 0 )
20	19	eqcomd 2244	1 ⊢ (𝜑 → 0 = (0g‘𝐺))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ‘cfv 5377  (class class class)co 6085  Basecbs 13354  +_gcplusg 13433  0_gc0g 13612  Grpcgrp 13807

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9306  df-2 9364  df-ndx 13357  df-slot 13358  df-base 13360  df-plusg 13446  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810

This theorem is used by:  imasgrp2  13915