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Theorem sdc 28593
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 2438 . 2  |-  { g  |  E. n  e.  Z  ( g : ( M ... n
) --> A  /\  ps ) }  =  {
g  |  E. n  e.  Z  ( g : ( M ... n ) --> A  /\  ps ) }
11 eqid 2438 . . . 4  |-  Z  =  Z
12 oveq2 6094 . . . . . . . 8  |-  ( n  =  k  ->  ( M ... n )  =  ( M ... k
) )
1312feq2d 5542 . . . . . . 7  |-  ( n  =  k  ->  (
g : ( M ... n ) --> A  <-> 
g : ( M ... k ) --> A ) )
1413, 4anbi12d 710 . . . . . 6  |-  ( n  =  k  ->  (
( g : ( M ... n ) --> A  /\  ps )  <->  ( g : ( M ... k ) --> A  /\  th ) ) )
1514cbvrexv 2943 . . . . 5  |-  ( E. n  e.  Z  ( g : ( M ... n ) --> A  /\  ps )  <->  E. k  e.  Z  ( g : ( M ... k ) --> A  /\  th ) )
1615abbii 2550 . . . 4  |-  { g  |  E. n  e.  Z  ( g : ( M ... n
) --> A  /\  ps ) }  =  {
g  |  E. k  e.  Z  ( g : ( M ... k ) --> A  /\  th ) }
17 eqid 2438 . . . 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 6144 . . 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 2439 . . . 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 2444 . . . . . . 7  |-  ( f  =  x  ->  (
f  =  ( h  |`  ( M ... k
) )  <->  x  =  ( h  |`  ( M ... k ) ) ) )
21203anbi2d 1294 . . . . . 6  |-  ( f  =  x  ->  (
( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si )  <->  ( h : ( M ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k
) )  /\  si ) ) )
2221rexbidv 2731 . . . . 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 2552 . . . 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 6161 . . 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 2460 . 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 28592 1  |-  ( ph  ->  E. f ( f : Z --> A  /\  A. n  e.  Z  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2424   A.wral 2710   E.wrex 2711   {csn 3872    |` cres 4837   -->wf 5409   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1c1 9275    + caddc 9277   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-dc 8607  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430
This theorem is referenced by: (None)
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