Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmff Structured version   Visualization version   GIF version

Theorem lmff 21015
 Description: If 𝐹 converges, there is some upper integer set on which 𝐹 is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
Hypotheses
Ref Expression
lmff.1 𝑍 = (ℤ𝑀)
lmff.3 (𝜑𝐽 ∈ (TopOn‘𝑋))
lmff.4 (𝜑𝑀 ∈ ℤ)
lmff.5 (𝜑𝐹 ∈ dom (⇝𝑡𝐽))
Assertion
Ref Expression
lmff (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋)
Distinct variable groups:   𝑗,𝐹   𝑗,𝐽   𝑗,𝑀   𝜑,𝑗   𝑗,𝑋   𝑗,𝑍

Proof of Theorem lmff
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmff.5 . . . . . 6 (𝜑𝐹 ∈ dom (⇝𝑡𝐽))
2 eldm2g 5280 . . . . . . 7 (𝐹 ∈ dom (⇝𝑡𝐽) → (𝐹 ∈ dom (⇝𝑡𝐽) ↔ ∃𝑦𝐹, 𝑦⟩ ∈ (⇝𝑡𝐽)))
32ibi 256 . . . . . 6 (𝐹 ∈ dom (⇝𝑡𝐽) → ∃𝑦𝐹, 𝑦⟩ ∈ (⇝𝑡𝐽))
41, 3syl 17 . . . . 5 (𝜑 → ∃𝑦𝐹, 𝑦⟩ ∈ (⇝𝑡𝐽))
5 df-br 4614 . . . . . 6 (𝐹(⇝𝑡𝐽)𝑦 ↔ ⟨𝐹, 𝑦⟩ ∈ (⇝𝑡𝐽))
65exbii 1771 . . . . 5 (∃𝑦 𝐹(⇝𝑡𝐽)𝑦 ↔ ∃𝑦𝐹, 𝑦⟩ ∈ (⇝𝑡𝐽))
74, 6sylibr 224 . . . 4 (𝜑 → ∃𝑦 𝐹(⇝𝑡𝐽)𝑦)
8 lmff.3 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
9 toponmax 20643 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
108, 9syl 17 . . . . . 6 (𝜑𝑋𝐽)
1110adantr 481 . . . . 5 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → 𝑋𝐽)
128lmbr 20972 . . . . . . 7 (𝜑 → (𝐹(⇝𝑡𝐽)𝑦 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑦𝑋 ∧ ∀𝑗𝐽 (𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗))))
1312biimpa 501 . . . . . 6 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑦𝑋 ∧ ∀𝑗𝐽 (𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗)))
1413simp3d 1073 . . . . 5 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → ∀𝑗𝐽 (𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗))
15 lmcl 21011 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑦) → 𝑦𝑋)
168, 15sylan 488 . . . . 5 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → 𝑦𝑋)
17 eleq2 2687 . . . . . . 7 (𝑗 = 𝑋 → (𝑦𝑗𝑦𝑋))
18 feq3 5985 . . . . . . . 8 (𝑗 = 𝑋 → ((𝐹𝑥):𝑥𝑗 ↔ (𝐹𝑥):𝑥𝑋))
1918rexbidv 3045 . . . . . . 7 (𝑗 = 𝑋 → (∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗 ↔ ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋))
2017, 19imbi12d 334 . . . . . 6 (𝑗 = 𝑋 → ((𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗) ↔ (𝑦𝑋 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋)))
2120rspcv 3291 . . . . 5 (𝑋𝐽 → (∀𝑗𝐽 (𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗) → (𝑦𝑋 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋)))
2211, 14, 16, 21syl3c 66 . . . 4 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋)
237, 22exlimddv 1860 . . 3 (𝜑 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋)
24 uzf 11634 . . . 4 :ℤ⟶𝒫 ℤ
25 ffn 6002 . . . 4 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
26 reseq2 5351 . . . . . 6 (𝑥 = (ℤ𝑗) → (𝐹𝑥) = (𝐹 ↾ (ℤ𝑗)))
27 id 22 . . . . . 6 (𝑥 = (ℤ𝑗) → 𝑥 = (ℤ𝑗))
2826, 27feq12d 5990 . . . . 5 (𝑥 = (ℤ𝑗) → ((𝐹𝑥):𝑥𝑋 ↔ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋))
2928rexrn 6317 . . . 4 (ℤ Fn ℤ → (∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋))
3024, 25, 29mp2b 10 . . 3 (∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋)
3123, 30sylib 208 . 2 (𝜑 → ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋)
32 lmff.4 . . . 4 (𝜑𝑀 ∈ ℤ)
33 lmff.1 . . . . 5 𝑍 = (ℤ𝑀)
3433rexuz3 14022 . . . 4 (𝑀 ∈ ℤ → (∃𝑗𝑍𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋) ↔ ∃𝑗 ∈ ℤ ∀𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
3532, 34syl 17 . . 3 (𝜑 → (∃𝑗𝑍𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋) ↔ ∃𝑗 ∈ ℤ ∀𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
3613simp1d 1071 . . . . . . 7 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → 𝐹 ∈ (𝑋pm ℂ))
377, 36exlimddv 1860 . . . . . 6 (𝜑𝐹 ∈ (𝑋pm ℂ))
38 pmfun 7821 . . . . . 6 (𝐹 ∈ (𝑋pm ℂ) → Fun 𝐹)
3937, 38syl 17 . . . . 5 (𝜑 → Fun 𝐹)
40 ffvresb 6349 . . . . 5 (Fun 𝐹 → ((𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋 ↔ ∀𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
4139, 40syl 17 . . . 4 (𝜑 → ((𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋 ↔ ∀𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
4241rexbidv 3045 . . 3 (𝜑 → (∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋 ↔ ∃𝑗𝑍𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
4341rexbidv 3045 . . 3 (𝜑 → (∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋 ↔ ∃𝑗 ∈ ℤ ∀𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
4435, 42, 433bitr4d 300 . 2 (𝜑 → (∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋))
4531, 44mpbird 247 1 (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  𝒫 cpw 4130  ⟨cop 4154   class class class wbr 4613  dom cdm 5074  ran crn 5075   ↾ cres 5076  Fun wfun 5841   Fn wfn 5842  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ↑pm cpm 7803  ℂcc 9878  ℤcz 11321  ℤ≥cuz 11631  TopOnctopon 20618  ⇝𝑡clm 20940 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954  ax-pre-lttrn 9955 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-neg 10213  df-z 11322  df-uz 11632  df-top 20621  df-topon 20623  df-lm 20943 This theorem is referenced by:  lmle  23007
 Copyright terms: Public domain W3C validator