Theorem grpidd 12807

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 2177	. 2 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2177	. 2 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		eqid 2177	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		grpidd.z	. . 3 ⊢ (𝜑 → 0 ∈ 𝐵)
5		grpidd.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
6	4, 5	eleqtrd 2256	. 2 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
7	5	eleq2d 2247	. . . 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 5894	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥))
12		grpidd.i	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
13	11, 12	eqtr3d 2212	. . 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 5894	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 ))
16		grpidd.j	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
17	15, 16	eqtr3d 2212	. . 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 12804	1 ⊢ (𝜑 → 0 = (0g‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  ‘cfv 5218  (class class class)co 5877  Basecbs 12464  +_gcplusg 12538  0_gc0g 12710

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-ndx 12467  df-slot 12468  df-base 12470  df-0g 12712

This theorem is referenced by:  ress0g  12849  mnd1id  12853  isgrpde  12903