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Theorem ismndo1 33643
Description: The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
ismndo1.1 𝑋 = dom dom 𝐺
Assertion
Ref Expression
ismndo1 (𝐺𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))
Distinct variable groups:   𝑥,𝐺,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem ismndo1
StepHypRef Expression
1 ismndo1.1 . . 3 𝑋 = dom dom 𝐺
21ismndo 33642 . 2 (𝐺𝐴 → (𝐺 ∈ MndOp ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))
31smgrpmgm 33634 . . . . 5 (𝐺 ∈ SemiGrp → 𝐺:(𝑋 × 𝑋)⟶𝑋)
43ad2antrl 763 . . . 4 ((𝐺𝐴 ∧ (𝐺 ∈ SemiGrp ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
51smgrpassOLD 33635 . . . . 5 (𝐺 ∈ SemiGrp → ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
65ad2antrl 763 . . . 4 ((𝐺𝐴 ∧ (𝐺 ∈ SemiGrp ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) → ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
7 simprr 795 . . . 4 ((𝐺𝐴 ∧ (𝐺 ∈ SemiGrp ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) → ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))
84, 6, 73jca 1240 . . 3 ((𝐺𝐴 ∧ (𝐺 ∈ SemiGrp ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
9 3simpa 1056 . . . . . 6 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)) → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
101issmgrpOLD 33633 . . . . . 6 (𝐺𝐴 → (𝐺 ∈ SemiGrp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
119, 10syl5ibr 236 . . . . 5 (𝐺𝐴 → ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)) → 𝐺 ∈ SemiGrp))
1211imp 445 . . . 4 ((𝐺𝐴 ∧ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) → 𝐺 ∈ SemiGrp)
13 simpr3 1067 . . . 4 ((𝐺𝐴 ∧ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) → ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))
1412, 13jca 554 . . 3 ((𝐺𝐴 ∧ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) → (𝐺 ∈ SemiGrp ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
158, 14impbida 876 . 2 (𝐺𝐴 → ((𝐺 ∈ SemiGrp ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))
162, 15bitrd 268 1 (𝐺𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  wrex 2910   × cxp 5102  dom cdm 5104  wf 5872  (class class class)co 6635  SemiGrpcsem 33630  MndOpcmndo 33636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-ass 33613  df-exid 33615  df-mgmOLD 33619  df-sgrOLD 33631  df-mndo 33637
This theorem is referenced by:  ismndo2  33644  rngomndo  33705
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