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Theorem grpomndo 36738
Description: A group is a monoid. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
grpomndo (𝐺 ∈ GrpOp β†’ 𝐺 ∈ MndOp)

Proof of Theorem grpomndo
Dummy variables 𝑀 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . 5 ran 𝐺 = ran 𝐺
21isgrpo 29745 . . . 4 (𝐺 ∈ GrpOp β†’ (𝐺 ∈ GrpOp ↔ (𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ βˆƒπ‘€ ∈ ran πΊβˆ€π‘₯ ∈ ran 𝐺((𝑀𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑀))))
32biimpd 228 . . 3 (𝐺 ∈ GrpOp β†’ (𝐺 ∈ GrpOp β†’ (𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ βˆƒπ‘€ ∈ ran πΊβˆ€π‘₯ ∈ ran 𝐺((𝑀𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑀))))
41grpoidinv 29756 . . . . . . . 8 (𝐺 ∈ GrpOp β†’ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦) ∧ βˆƒπ‘€ ∈ ran 𝐺((𝑀𝐺𝑦) = π‘₯ ∧ (𝑦𝐺𝑀) = π‘₯)))
5 simpl 483 . . . . . . . . . . 11 ((((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦) ∧ βˆƒπ‘€ ∈ ran 𝐺((𝑀𝐺𝑦) = π‘₯ ∧ (𝑦𝐺𝑀) = π‘₯)) β†’ ((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦))
65ralimi 3083 . . . . . . . . . 10 (βˆ€π‘¦ ∈ ran 𝐺(((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦) ∧ βˆƒπ‘€ ∈ ran 𝐺((𝑀𝐺𝑦) = π‘₯ ∧ (𝑦𝐺𝑀) = π‘₯)) β†’ βˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦))
76reximi 3084 . . . . . . . . 9 (βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦) ∧ βˆƒπ‘€ ∈ ran 𝐺((𝑀𝐺𝑦) = π‘₯ ∧ (𝑦𝐺𝑀) = π‘₯)) β†’ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦))
81ismndo2 36737 . . . . . . . . . . . . 13 (𝐺 ∈ GrpOp β†’ (𝐺 ∈ MndOp ↔ (𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦))))
98biimprcd 249 . . . . . . . . . . . 12 ((𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦)) β†’ (𝐺 ∈ GrpOp β†’ 𝐺 ∈ MndOp))
1093exp 1119 . . . . . . . . . . 11 (𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) β†’ (βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦) β†’ (𝐺 ∈ GrpOp β†’ 𝐺 ∈ MndOp))))
1110impcom 408 . . . . . . . . . 10 ((βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ 𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺) β†’ (βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦) β†’ (𝐺 ∈ GrpOp β†’ 𝐺 ∈ MndOp)))
1211com3l 89 . . . . . . . . 9 (βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦) β†’ (𝐺 ∈ GrpOp β†’ ((βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ 𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺) β†’ 𝐺 ∈ MndOp)))
137, 12syl 17 . . . . . . . 8 (βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦) ∧ βˆƒπ‘€ ∈ ran 𝐺((𝑀𝐺𝑦) = π‘₯ ∧ (𝑦𝐺𝑀) = π‘₯)) β†’ (𝐺 ∈ GrpOp β†’ ((βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ 𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺) β†’ 𝐺 ∈ MndOp)))
144, 13mpcom 38 . . . . . . 7 (𝐺 ∈ GrpOp β†’ ((βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ 𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺) β†’ 𝐺 ∈ MndOp))
1514expdcom 415 . . . . . 6 (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) β†’ (𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 β†’ (𝐺 ∈ GrpOp β†’ 𝐺 ∈ MndOp)))
1615a1i 11 . . . . 5 (βˆƒπ‘€ ∈ ran πΊβˆ€π‘₯ ∈ ran 𝐺((𝑀𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑀) β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) β†’ (𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 β†’ (𝐺 ∈ GrpOp β†’ 𝐺 ∈ MndOp))))
1716com13 88 . . . 4 (𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) β†’ (βˆƒπ‘€ ∈ ran πΊβˆ€π‘₯ ∈ ran 𝐺((𝑀𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑀) β†’ (𝐺 ∈ GrpOp β†’ 𝐺 ∈ MndOp))))
18173imp 1111 . . 3 ((𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ βˆƒπ‘€ ∈ ran πΊβˆ€π‘₯ ∈ ran 𝐺((𝑀𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑀)) β†’ (𝐺 ∈ GrpOp β†’ 𝐺 ∈ MndOp))
193, 18syli 39 . 2 (𝐺 ∈ GrpOp β†’ (𝐺 ∈ GrpOp β†’ 𝐺 ∈ MndOp))
2019pm2.43i 52 1 (𝐺 ∈ GrpOp β†’ 𝐺 ∈ MndOp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   Γ— cxp 5674  ran crn 5677  βŸΆwf 6539  (class class class)co 7408  GrpOpcgr 29737  MndOpcmndo 36729
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7411  df-grpo 29741  df-ass 36706  df-exid 36708  df-mgmOLD 36712  df-sgrOLD 36724  df-mndo 36730
This theorem is referenced by:  isdrngo2  36821
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