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Theorem grpidd 18143
Description: Deduce the identity element of a magma from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
grpidd.b (𝜑𝐵 = (Base‘𝐺))
grpidd.p (𝜑+ = (+g𝐺))
grpidd.z (𝜑0𝐵)
grpidd.i ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
grpidd.j ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
Assertion
Ref Expression
grpidd (𝜑0 = (0g𝐺))
Distinct variable groups:   𝑥,𝐺   𝜑,𝑥   𝑥, 0
Allowed substitution hints:   𝐵(𝑥)   + (𝑥)

Proof of Theorem grpidd
StepHypRef Expression
1 eqid 2737 . 2 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2737 . 2 (0g𝐺) = (0g𝐺)
3 eqid 2737 . 2 (+g𝐺) = (+g𝐺)
4 grpidd.z . . 3 (𝜑0𝐵)
5 grpidd.b . . 3 (𝜑𝐵 = (Base‘𝐺))
64, 5eleqtrd 2840 . 2 (𝜑0 ∈ (Base‘𝐺))
75eleq2d 2823 . . . 4 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐺)))
87biimpar 481 . . 3 ((𝜑𝑥 ∈ (Base‘𝐺)) → 𝑥𝐵)
9 grpidd.p . . . . . 6 (𝜑+ = (+g𝐺))
109adantr 484 . . . . 5 ((𝜑𝑥𝐵) → + = (+g𝐺))
1110oveqd 7230 . . . 4 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = ( 0 (+g𝐺)𝑥))
12 grpidd.i . . . 4 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
1311, 12eqtr3d 2779 . . 3 ((𝜑𝑥𝐵) → ( 0 (+g𝐺)𝑥) = 𝑥)
148, 13syldan 594 . 2 ((𝜑𝑥 ∈ (Base‘𝐺)) → ( 0 (+g𝐺)𝑥) = 𝑥)
1510oveqd 7230 . . . 4 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = (𝑥(+g𝐺) 0 ))
16 grpidd.j . . . 4 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
1715, 16eqtr3d 2779 . . 3 ((𝜑𝑥𝐵) → (𝑥(+g𝐺) 0 ) = 𝑥)
188, 17syldan 594 . 2 ((𝜑𝑥 ∈ (Base‘𝐺)) → (𝑥(+g𝐺) 0 ) = 𝑥)
191, 2, 3, 6, 14, 18ismgmid2 18140 1 (𝜑0 = (0g𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  cfv 6380  (class class class)co 7213  Basecbs 16760  +gcplusg 16802  0gc0g 16944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-riota 7170  df-ov 7216  df-0g 16946
This theorem is referenced by:  ress0g  18201  imasmnd2  18210  smndex1id  18338  isgrpde  18388  xrs0  31003
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