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Theorem mgmidcl 18567
Description: The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
ismgmid.b 𝐵 = (Base‘𝐺)
ismgmid.o 0 = (0g𝐺)
ismgmid.p + = (+g𝐺)
mgmidcl.e (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
Assertion
Ref Expression
mgmidcl (𝜑0𝐵)
Distinct variable groups:   𝑥,𝑒, +   0 ,𝑒,𝑥   𝐵,𝑒,𝑥   𝑒,𝐺,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑒)

Proof of Theorem mgmidcl
StepHypRef Expression
1 eqid 2731 . . 3 0 = 0
2 ismgmid.b . . . 4 𝐵 = (Base‘𝐺)
3 ismgmid.o . . . 4 0 = (0g𝐺)
4 ismgmid.p . . . 4 + = (+g𝐺)
5 mgmidcl.e . . . 4 (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
62, 3, 4, 5ismgmid 18566 . . 3 (𝜑 → (( 0𝐵 ∧ ∀𝑥𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) ↔ 0 = 0 ))
71, 6mpbiri 257 . 2 (𝜑 → ( 0𝐵 ∧ ∀𝑥𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
87simpld 495 1 (𝜑0𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3060  wrex 3069  cfv 6532  (class class class)co 7393  Basecbs 17126  +gcplusg 17179  0gc0g 17367
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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fv 6540  df-riota 7349  df-ov 7396  df-0g 17369
This theorem is referenced by:  mndidcl  18617
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