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Theorem grpidd 17870
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 2826 . 2 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2826 . 2 (0g𝐺) = (0g𝐺)
3 eqid 2826 . 2 (+g𝐺) = (+g𝐺)
4 grpidd.z . . 3 (𝜑0𝐵)
5 grpidd.b . . 3 (𝜑𝐵 = (Base‘𝐺))
64, 5eleqtrd 2920 . 2 (𝜑0 ∈ (Base‘𝐺))
75eleq2d 2903 . . . 4 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐺)))
87biimpar 478 . . 3 ((𝜑𝑥 ∈ (Base‘𝐺)) → 𝑥𝐵)
9 grpidd.p . . . . . 6 (𝜑+ = (+g𝐺))
109adantr 481 . . . . 5 ((𝜑𝑥𝐵) → + = (+g𝐺))
1110oveqd 7165 . . . 4 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = ( 0 (+g𝐺)𝑥))
12 grpidd.i . . . 4 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
1311, 12eqtr3d 2863 . . 3 ((𝜑𝑥𝐵) → ( 0 (+g𝐺)𝑥) = 𝑥)
148, 13syldan 591 . 2 ((𝜑𝑥 ∈ (Base‘𝐺)) → ( 0 (+g𝐺)𝑥) = 𝑥)
1510oveqd 7165 . . . 4 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = (𝑥(+g𝐺) 0 ))
16 grpidd.j . . . 4 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
1715, 16eqtr3d 2863 . . 3 ((𝜑𝑥𝐵) → (𝑥(+g𝐺) 0 ) = 𝑥)
188, 17syldan 591 . 2 ((𝜑𝑥 ∈ (Base‘𝐺)) → (𝑥(+g𝐺) 0 ) = 𝑥)
191, 2, 3, 6, 14, 18ismgmid2 17867 1 (𝜑0 = (0g𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  cfv 6352  (class class class)co 7148  Basecbs 16473  +gcplusg 16555  0gc0g 16703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6312  df-fun 6354  df-fv 6360  df-riota 7106  df-ov 7151  df-0g 16705
This theorem is referenced by:  ress0g  17927  imasmnd2  17936  isgrpde  18054  xrs0  30576
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