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Theorem sdc 31532
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 2404 . 2  |-  { g  |  E. n  e.  Z  ( g : ( M ... n
) --> A  /\  ps ) }  =  {
g  |  E. n  e.  Z  ( g : ( M ... n ) --> A  /\  ps ) }
11 eqid 2404 . . . 4  |-  Z  =  Z
12 oveq2 6288 . . . . . . . 8  |-  ( n  =  k  ->  ( M ... n )  =  ( M ... k
) )
1312feq2d 5703 . . . . . . 7  |-  ( n  =  k  ->  (
g : ( M ... n ) --> A  <-> 
g : ( M ... k ) --> A ) )
1413, 4anbi12d 711 . . . . . 6  |-  ( n  =  k  ->  (
( g : ( M ... n ) --> A  /\  ps )  <->  ( g : ( M ... k ) --> A  /\  th ) ) )
1514cbvrexv 3037 . . . . 5  |-  ( E. n  e.  Z  ( g : ( M ... n ) --> A  /\  ps )  <->  E. k  e.  Z  ( g : ( M ... k ) --> A  /\  th ) )
1615abbii 2538 . . . 4  |-  { g  |  E. n  e.  Z  ( g : ( M ... n
) --> A  /\  ps ) }  =  {
g  |  E. k  e.  Z  ( g : ( M ... k ) --> A  /\  th ) }
17 eqid 2404 . . . 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 6343 . . 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 2405 . . . 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 2408 . . . . . . 7  |-  ( f  =  x  ->  (
f  =  ( h  |`  ( M ... k
) )  <->  x  =  ( h  |`  ( M ... k ) ) ) )
21203anbi2d 1308 . . . . . 6  |-  ( f  =  x  ->  (
( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si )  <->  ( h : ( M ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k
) )  /\  si ) ) )
2221rexbidv 2920 . . . . 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 2540 . . . 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 6360 . . 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 2435 . 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 31531 1  |-  ( ph  ->  E. f ( f : Z --> A  /\  A. n  e.  Z  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407   E.wex 1635    e. wcel 1844   {cab 2389   A.wral 2756   E.wrex 2757   {csn 3974    |` cres 4827   -->wf 5567   ` cfv 5571  (class class class)co 6280    |-> cmpt2 6282   1c1 9525    + caddc 9527   ZZcz 10907   ZZ>=cuz 11129   ...cfz 11728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-dc 8860  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729
This theorem is referenced by: (None)
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