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Theorem grpidd 17869
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 2818 . 2 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2818 . 2 (0g𝐺) = (0g𝐺)
3 eqid 2818 . 2 (+g𝐺) = (+g𝐺)
4 grpidd.z . . 3 (𝜑0𝐵)
5 grpidd.b . . 3 (𝜑𝐵 = (Base‘𝐺))
64, 5eleqtrd 2912 . 2 (𝜑0 ∈ (Base‘𝐺))
75eleq2d 2895 . . . 4 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐺)))
87biimpar 478 . . 3 ((𝜑𝑥 ∈ (Base‘𝐺)) → 𝑥𝐵)
9 grpidd.p . . . . . 6 (𝜑+ = (+g𝐺))
109adantr 481 . . . . 5 ((𝜑𝑥𝐵) → + = (+g𝐺))
1110oveqd 7162 . . . 4 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = ( 0 (+g𝐺)𝑥))
12 grpidd.i . . . 4 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
1311, 12eqtr3d 2855 . . 3 ((𝜑𝑥𝐵) → ( 0 (+g𝐺)𝑥) = 𝑥)
148, 13syldan 591 . 2 ((𝜑𝑥 ∈ (Base‘𝐺)) → ( 0 (+g𝐺)𝑥) = 𝑥)
1510oveqd 7162 . . . 4 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = (𝑥(+g𝐺) 0 ))
16 grpidd.j . . . 4 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
1715, 16eqtr3d 2855 . . 3 ((𝜑𝑥𝐵) → (𝑥(+g𝐺) 0 ) = 𝑥)
188, 17syldan 591 . 2 ((𝜑𝑥 ∈ (Base‘𝐺)) → (𝑥(+g𝐺) 0 ) = 𝑥)
191, 2, 3, 6, 14, 18ismgmid2 17866 1 (𝜑0 = (0g𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-riota 7103  df-ov 7148  df-0g 16703
This theorem is referenced by:  ress0g  17927  imasmnd2  17936  isgrpde  18062  xrs0  30589  smndex1id  44011
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