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Theorem grpomndo 37342
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 2728 . . . . 5 ran 𝐺 = ran 𝐺
21isgrpo 30300 . . . 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 30311 . . . . . . . 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 3079 . . . . . . . . . 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 3080 . . . . . . . . 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 37341 . . . . . . . . . . . . 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 1117 . . . . . . . . . . 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 1109 . . 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 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3057  βˆƒwrex 3066   Γ— cxp 5670  ran crn 5673  βŸΆwf 6538  (class class class)co 7414  GrpOpcgr 30292  MndOpcmndo 37333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-ov 7417  df-grpo 30296  df-ass 37310  df-exid 37312  df-mgmOLD 37316  df-sgrOLD 37328  df-mndo 37334
This theorem is referenced by:  isdrngo2  37425
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