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

Theorem sdc 26454
Description: Strong dependent choice. Suppose we may choose an element of 
A such that property  ps 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  ps. Then there exists an infinite sequence of elements of  A such that the first  n terms of this sequence satisfy  ps 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  =  ( ZZ>= `  M )
sdc.2  |-  ( g  =  ( f  |`  ( M ... n ) )  ->  ( ps  <->  ch ) )
sdc.3  |-  ( n  =  M  ->  ( ps 
<->  ta ) )
sdc.4  |-  ( n  =  k  ->  ( ps 
<->  th ) )
sdc.5  |-  ( ( g  =  h  /\  n  =  ( k  +  1 ) )  ->  ( ps  <->  si )
)
sdc.6  |-  ( ph  ->  A  e.  V )
sdc.7  |-  ( ph  ->  M  e.  ZZ )
sdc.8  |-  ( ph  ->  E. g ( g : { M } --> A  /\  ta ) )
sdc.9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( g : ( M ... k ) --> A  /\  th )  ->  E. h ( h : ( M ... ( k  +  1 ) ) --> A  /\  g  =  ( h  |`  ( M ... k
) )  /\  si ) ) )
Assertion
Ref Expression
sdc  |-  ( ph  ->  E. f ( f : Z --> A  /\  A. n  e.  Z  ch ) )
Distinct variable groups:    f, g, h, k, n, A    f, M, g, h, k, n    ch, g    ps, f, h, k    si, f, g, n    ph, n    th, n    h, V    ta, h, k, n   
f, Z, g, h, k, n    ph, g, h, k
Allowed substitution hints:    ph( f)    ps( g, n)    ch( f, h, k, n)    th( f,
g, h, k)    ta( f, g)    si( h, 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  =  ( ZZ>= `  M )
2 sdc.2 . 2  |-  ( g  =  ( f  |`  ( M ... n ) )  ->  ( ps  <->  ch ) )
3 sdc.3 . 2  |-  ( n  =  M  ->  ( ps 
<->  ta ) )
4 sdc.4 . 2  |-  ( n  =  k  ->  ( ps 
<->  th ) )
5 sdc.5 . 2  |-  ( ( g  =  h  /\  n  =  ( k  +  1 ) )  ->  ( ps  <->  si )
)
6 sdc.6 . 2  |-  ( ph  ->  A  e.  V )
7 sdc.7 . 2  |-  ( ph  ->  M  e.  ZZ )
8 sdc.8 . 2  |-  ( ph  ->  E. g ( g : { M } --> A  /\  ta ) )
9 sdc.9 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  (
( g : ( M ... k ) --> A  /\  th )  ->  E. h ( h : ( M ... ( k  +  1 ) ) --> A  /\  g  =  ( h  |`  ( M ... k
) )  /\  si ) ) )
10 eqid 2283 . 2  |-  { g  |  E. n  e.  Z  ( g : ( M ... n
) --> A  /\  ps ) }  =  {
g  |  E. n  e.  Z  ( g : ( M ... n ) --> A  /\  ps ) }
11 eqid 2283 . . . 4  |-  Z  =  Z
12 oveq2 5866 . . . . . . . 8  |-  ( n  =  k  ->  ( M ... n )  =  ( M ... k
) )
1312feq2d 5380 . . . . . . 7  |-  ( n  =  k  ->  (
g : ( M ... n ) --> A  <-> 
g : ( M ... k ) --> A ) )
1413, 4anbi12d 691 . . . . . 6  |-  ( n  =  k  ->  (
( g : ( M ... n ) --> A  /\  ps )  <->  ( g : ( M ... k ) --> A  /\  th ) ) )
1514cbvrexv 2765 . . . . 5  |-  ( E. n  e.  Z  ( g : ( M ... n ) --> A  /\  ps )  <->  E. k  e.  Z  ( g : ( M ... k ) --> A  /\  th ) )
1615abbii 2395 . . . 4  |-  { g  |  E. n  e.  Z  ( g : ( M ... n
) --> A  /\  ps ) }  =  {
g  |  E. k  e.  Z  ( g : ( M ... k ) --> A  /\  th ) }
17 eqid 2283 . . . 4  |-  { h  |  E. k  e.  Z  ( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si ) }  =  { h  |  E. k  e.  Z  ( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si ) }
1811, 16, 17mpt2eq123i 5911 . . 3  |-  ( j  e.  Z ,  f  e.  { g  |  E. n  e.  Z  ( g : ( M ... n ) --> A  /\  ps ) }  |->  { h  |  E. k  e.  Z  ( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si ) } )  =  ( j  e.  Z ,  f  e.  { g  |  E. k  e.  Z  ( g : ( M ... k ) --> A  /\  th ) }  |->  { h  |  E. k  e.  Z  ( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si ) } )
19 eqidd 2284 . . . 4  |-  ( j  =  y  ->  { h  |  E. k  e.  Z  ( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si ) }  =  { h  |  E. k  e.  Z  ( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si ) } )
20 eqeq1 2289 . . . . . . 7  |-  ( f  =  x  ->  (
f  =  ( h  |`  ( M ... k
) )  <->  x  =  ( h  |`  ( M ... k ) ) ) )
21203anbi2d 1257 . . . . . 6  |-  ( f  =  x  ->  (
( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si )  <->  ( h : ( M ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k
) )  /\  si ) ) )
2221rexbidv 2564 . . . . 5  |-  ( f  =  x  ->  ( E. k  e.  Z  ( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si )  <->  E. k  e.  Z  ( h : ( M ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k
) )  /\  si ) ) )
2322abbidv 2397 . . . 4  |-  ( f  =  x  ->  { h  |  E. k  e.  Z  ( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si ) }  =  { h  |  E. k  e.  Z  ( h : ( M ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k ) )  /\  si ) } )
2419, 23cbvmpt2v 5926 . . 3  |-  ( j  e.  Z ,  f  e.  { g  |  E. n  e.  Z  ( g : ( M ... n ) --> A  /\  ps ) }  |->  { h  |  E. k  e.  Z  ( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si ) } )  =  ( y  e.  Z ,  x  e.  { g  |  E. n  e.  Z  (
g : ( M ... n ) --> A  /\  ps ) } 
|->  { h  |  E. k  e.  Z  (
h : ( M ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k ) )  /\  si ) } )
2518, 24eqtr3i 2305 . 2  |-  ( j  e.  Z ,  f  e.  { g  |  E. k  e.  Z  ( g : ( M ... k ) --> A  /\  th ) }  |->  { h  |  E. k  e.  Z  ( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si ) } )  =  ( y  e.  Z ,  x  e.  { g  |  E. n  e.  Z  (
g : ( M ... n ) --> A  /\  ps ) } 
|->  { h  |  E. k  e.  Z  (
h : ( M ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k ) )  /\  si ) } )
261, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25sdclem1 26453 1  |-  ( ph  ->  E. f ( f : Z --> A  /\  A. n  e.  Z  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   {csn 3640    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1c1 8738    + caddc 8740   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-dc 8072  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783
  Copyright terms: Public domain W3C validator