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Theorem grpidd 18697
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 2735 . 2 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2735 . 2 (0g𝐺) = (0g𝐺)
3 eqid 2735 . 2 (+g𝐺) = (+g𝐺)
4 grpidd.z . . 3 (𝜑0𝐵)
5 grpidd.b . . 3 (𝜑𝐵 = (Base‘𝐺))
64, 5eleqtrd 2841 . 2 (𝜑0 ∈ (Base‘𝐺))
75eleq2d 2825 . . . 4 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐺)))
87biimpar 477 . . 3 ((𝜑𝑥 ∈ (Base‘𝐺)) → 𝑥𝐵)
9 grpidd.p . . . . . 6 (𝜑+ = (+g𝐺))
109adantr 480 . . . . 5 ((𝜑𝑥𝐵) → + = (+g𝐺))
1110oveqd 7448 . . . 4 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = ( 0 (+g𝐺)𝑥))
12 grpidd.i . . . 4 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
1311, 12eqtr3d 2777 . . 3 ((𝜑𝑥𝐵) → ( 0 (+g𝐺)𝑥) = 𝑥)
148, 13syldan 591 . 2 ((𝜑𝑥 ∈ (Base‘𝐺)) → ( 0 (+g𝐺)𝑥) = 𝑥)
1510oveqd 7448 . . . 4 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = (𝑥(+g𝐺) 0 ))
16 grpidd.j . . . 4 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
1715, 16eqtr3d 2777 . . 3 ((𝜑𝑥𝐵) → (𝑥(+g𝐺) 0 ) = 𝑥)
188, 17syldan 591 . 2 ((𝜑𝑥 ∈ (Base‘𝐺)) → (𝑥(+g𝐺) 0 ) = 𝑥)
191, 2, 3, 6, 14, 18ismgmid2 18694 1 (𝜑0 = (0g𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488
This theorem is referenced by:  ress0g  18788  imasmnd2  18800  smndex1id  18937  isgrpde  18988  xrs0  32991
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