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Theorem grpomndo 37237
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 2724 . . . . 5 ran 𝐺 = ran 𝐺
21isgrpo 30222 . . . 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 30233 . . . . . . . 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 482 . . . . . . . . . . 11 ((((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦) ∧ βˆƒπ‘€ ∈ ran 𝐺((𝑀𝐺𝑦) = π‘₯ ∧ (𝑦𝐺𝑀) = π‘₯)) β†’ ((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦))
65ralimi 3075 . . . . . . . . . 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 3076 . . . . . . . . 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 37236 . . . . . . . . . . . . 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 1116 . . . . . . . . . . 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 407 . . . . . . . . . 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 414 . . . . . 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 1108 . . 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062   Γ— cxp 5665  ran crn 5668  βŸΆwf 6530  (class class class)co 7402  GrpOpcgr 30214  MndOpcmndo 37228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fo 6540  df-fv 6542  df-ov 7405  df-grpo 30218  df-ass 37205  df-exid 37207  df-mgmOLD 37211  df-sgrOLD 37223  df-mndo 37229
This theorem is referenced by:  isdrngo2  37320
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