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Theorem grpidd 18525
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 2736 . 2 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2736 . 2 (0g𝐺) = (0g𝐺)
3 eqid 2736 . 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 478 . . 3 ((𝜑𝑥 ∈ (Base‘𝐺)) → 𝑥𝐵)
9 grpidd.p . . . . . 6 (𝜑+ = (+g𝐺))
109adantr 481 . . . . 5 ((𝜑𝑥𝐵) → + = (+g𝐺))
1110oveqd 7373 . . . 4 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = ( 0 (+g𝐺)𝑥))
12 grpidd.i . . . 4 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
1311, 12eqtr3d 2778 . . 3 ((𝜑𝑥𝐵) → ( 0 (+g𝐺)𝑥) = 𝑥)
148, 13syldan 591 . 2 ((𝜑𝑥 ∈ (Base‘𝐺)) → ( 0 (+g𝐺)𝑥) = 𝑥)
1510oveqd 7373 . . . 4 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = (𝑥(+g𝐺) 0 ))
16 grpidd.j . . . 4 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
1715, 16eqtr3d 2778 . . 3 ((𝜑𝑥𝐵) → (𝑥(+g𝐺) 0 ) = 𝑥)
188, 17syldan 591 . 2 ((𝜑𝑥 ∈ (Base‘𝐺)) → (𝑥(+g𝐺) 0 ) = 𝑥)
191, 2, 3, 6, 14, 18ismgmid2 18522 1 (𝜑0 = (0g𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cfv 6496  (class class class)co 7356  Basecbs 17082  +gcplusg 17132  0gc0g 17320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-riota 7312  df-ov 7359  df-0g 17322
This theorem is referenced by:  ress0g  18583  imasmnd2  18592  smndex1id  18720  isgrpde  18770  xrs0  31861
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