Theorem grpidd 17268

Description: Deduce the identity element of a magma from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
grpidd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
grpidd.p	⊢ (𝜑 → + = (+g‘𝐺))
grpidd.z	⊢ (𝜑 → 0 ∈ 𝐵)
grpidd.i	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
grpidd.j	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)

Assertion

Ref	Expression
grpidd	⊢ (𝜑 → 0 = (0g‘𝐺))

Distinct variable groups:   𝑥,𝐺   𝜑,𝑥   𝑥, 0

Allowed substitution hints:   𝐵(𝑥)   + (𝑥)

Proof of Theorem grpidd

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2622	. 2 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		eqid 2622	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		grpidd.z	. . 3 ⊢ (𝜑 → 0 ∈ 𝐵)
5		grpidd.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
6	4, 5	eleqtrd 2703	. 2 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
7	5	eleq2d 2687	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐺)))
8	7	biimpar 502	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝐵)
9		grpidd.p	. . . . . 6 ⊢ (𝜑 → + = (+g‘𝐺))
10	9	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → + = (+g‘𝐺))
11	10	oveqd 6667	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥))
12		grpidd.i	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
13	11, 12	eqtr3d 2658	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
14	8, 13	syldan 487	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
15	10	oveqd 6667	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 ))
16		grpidd.j	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
17	15, 16	eqtr3d 2658	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺) 0 ) = 𝑥)
18	8, 17	syldan 487	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺) 0 ) = 𝑥)
19	1, 2, 3, 6, 14, 18	ismgmid2 17267	1 ⊢ (𝜑 → 0 = (0g‘𝐺))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102

This theorem is referenced by:  ress0g  17319  imasmnd2  17327  isgrpde  17443  xrs0  29675