Theorem grpidd 13585

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 2232	. 2 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2232	. 2 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		eqid 2232	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		grpidd.z	. . 3 ⊢ (𝜑 → 0 ∈ 𝐵)
5		grpidd.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
6	4, 5	eleqtrd 2311	. 2 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
7	5	eleq2d 2302	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐺)))
8	7	biimpar 297	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝐵)
9		grpidd.p	. . . . . 6 ⊢ (𝜑 → + = (+g‘𝐺))
10	9	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → + = (+g‘𝐺))
11	10	oveqd 6066	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥))
12		grpidd.i	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
13	11, 12	eqtr3d 2267	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
14	8, 13	syldan 282	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
15	10	oveqd 6066	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 ))
16		grpidd.j	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
17	15, 16	eqtr3d 2267	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺) 0 ) = 𝑥)
18	8, 17	syldan 282	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺) 0 ) = 𝑥)
19	1, 2, 3, 6, 14, 18	ismgmid2 13582	1 ⊢ (𝜑 → 0 = (0g‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  ‘cfv 5351  (class class class)co 6049  Basecbs 13201  +_gcplusg 13279  0_gc0g 13458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-cnex 8214  ax-resscn 8215  ax-1re 8217  ax-addrcl 8220

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-riota 6002  df-ov 6052  df-inn 9234  df-ndx 13204  df-slot 13205  df-base 13207  df-0g 13460

This theorem is referenced by:  ress0g  13645  imasmnd2  13654  mnd1id  13658  isgrpde  13724