Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sdc Structured version   Visualization version   GIF version

Theorem sdc 33670
Description: Strong dependent choice. Suppose we may choose an element of 𝐴 such that property 𝜓 holds, and suppose that if we have already chosen the first 𝑘 elements (represented here by a function from 1...𝑘 to 𝐴), we may choose another element so that all 𝑘 + 1 elements taken together have property 𝜓. Then there exists an infinite sequence of elements of 𝐴 such that the first 𝑛 terms of this sequence satisfy 𝜓 for all 𝑛. This theorem allows us to construct infinite seqeunces where each term depends on all the previous terms in the sequence. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 3-Jun-2014.)
Hypotheses
Ref Expression
sdc.1 𝑍 = (ℤ𝑀)
sdc.2 (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓𝜒))
sdc.3 (𝑛 = 𝑀 → (𝜓𝜏))
sdc.4 (𝑛 = 𝑘 → (𝜓𝜃))
sdc.5 ((𝑔 = 𝑛 = (𝑘 + 1)) → (𝜓𝜎))
sdc.6 (𝜑𝐴𝑉)
sdc.7 (𝜑𝑀 ∈ ℤ)
sdc.8 (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴𝜏))
sdc.9 ((𝜑𝑘𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴𝜃) → ∃(:(𝑀...(𝑘 + 1))⟶𝐴𝑔 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
Assertion
Ref Expression
sdc (𝜑 → ∃𝑓(𝑓:𝑍𝐴 ∧ ∀𝑛𝑍 𝜒))
Distinct variable groups:   𝑓,𝑔,,𝑘,𝑛,𝐴   𝑓,𝑀,𝑔,,𝑘,𝑛   𝜒,𝑔   𝜓,𝑓,,𝑘   𝜎,𝑓,𝑔,𝑛   𝜑,𝑛   𝜃,𝑛   ,𝑉   𝜏,,𝑘,𝑛   𝑓,𝑍,𝑔,,𝑘,𝑛   𝜑,𝑔,,𝑘
Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑔,𝑛)   𝜒(𝑓,,𝑘,𝑛)   𝜃(𝑓,𝑔,,𝑘)   𝜏(𝑓,𝑔)   𝜎(,𝑘)   𝑉(𝑓,𝑔,𝑘,𝑛)

Proof of Theorem sdc
Dummy variables 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sdc.1 . 2 𝑍 = (ℤ𝑀)
2 sdc.2 . 2 (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓𝜒))
3 sdc.3 . 2 (𝑛 = 𝑀 → (𝜓𝜏))
4 sdc.4 . 2 (𝑛 = 𝑘 → (𝜓𝜃))
5 sdc.5 . 2 ((𝑔 = 𝑛 = (𝑘 + 1)) → (𝜓𝜎))
6 sdc.6 . 2 (𝜑𝐴𝑉)
7 sdc.7 . 2 (𝜑𝑀 ∈ ℤ)
8 sdc.8 . 2 (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴𝜏))
9 sdc.9 . 2 ((𝜑𝑘𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴𝜃) → ∃(:(𝑀...(𝑘 + 1))⟶𝐴𝑔 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
10 eqid 2651 . 2 {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} = {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)}
11 eqid 2651 . . . 4 𝑍 = 𝑍
12 oveq2 6698 . . . . . . . 8 (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘))
1312feq2d 6069 . . . . . . 7 (𝑛 = 𝑘 → (𝑔:(𝑀...𝑛)⟶𝐴𝑔:(𝑀...𝑘)⟶𝐴))
1413, 4anbi12d 747 . . . . . 6 (𝑛 = 𝑘 → ((𝑔:(𝑀...𝑛)⟶𝐴𝜓) ↔ (𝑔:(𝑀...𝑘)⟶𝐴𝜃)))
1514cbvrexv 3202 . . . . 5 (∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓) ↔ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃))
1615abbii 2768 . . . 4 {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} = {𝑔 ∣ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃)}
17 eqid 2651 . . . 4 { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)}
1811, 16, 17mpt2eq123i 6760 . . 3 (𝑗𝑍, 𝑓 ∈ {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)}) = (𝑗𝑍, 𝑓 ∈ {𝑔 ∣ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃)} ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
19 eqidd 2652 . . . 4 (𝑗 = 𝑦 → { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
20 eqeq1 2655 . . . . . . 7 (𝑓 = 𝑥 → (𝑓 = ( ↾ (𝑀...𝑘)) ↔ 𝑥 = ( ↾ (𝑀...𝑘))))
21203anbi2d 1444 . . . . . 6 (𝑓 = 𝑥 → ((:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
2221rexbidv 3081 . . . . 5 (𝑓 = 𝑥 → (∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
2322abbidv 2770 . . . 4 (𝑓 = 𝑥 → { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
2419, 23cbvmpt2v 6777 . . 3 (𝑗𝑍, 𝑓 ∈ {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)}) = (𝑦𝑍, 𝑥 ∈ {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
2518, 24eqtr3i 2675 . 2 (𝑗𝑍, 𝑓 ∈ {𝑔 ∣ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃)} ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)}) = (𝑦𝑍, 𝑥 ∈ {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
261, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25sdclem1 33669 1 (𝜑 → ∃𝑓(𝑓:𝑍𝐴 ∧ ∀𝑛𝑍 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  wrex 2942  {csn 4210  cres 5145  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1c1 9975   + caddc 9977  cz 11415  cuz 11725  ...cfz 12364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-dc 9306  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator