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Theorem grpomndo 36384
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 2733 . . . . 5 ran 𝐺 = ran 𝐺
21isgrpo 29488 . . . 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 29499 . . . . . . . 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 484 . . . . . . . . . . 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 36383 . . . . . . . . . . . . 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 250 . . . . . . . . . . . 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 1120 . . . . . . . . . . 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 409 . . . . . . . . . 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 416 . . . . . 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 1112 . . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   Γ— cxp 5635  ran crn 5638  βŸΆwf 6496  (class class class)co 7361  GrpOpcgr 29480  MndOpcmndo 36375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-ov 7364  df-grpo 29484  df-ass 36352  df-exid 36354  df-mgmOLD 36358  df-sgrOLD 36370  df-mndo 36376
This theorem is referenced by:  isdrngo2  36467
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