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Theorem sdc 26450
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 6091 . . . . . . . 8  |-  ( n  =  k  ->  ( M ... n )  =  ( M ... k
) )
1312feq2d 5583 . . . . . . 7  |-  ( n  =  k  ->  (
g : ( M ... n ) --> A  <-> 
g : ( M ... k ) --> A ) )
1413, 4anbi12d 693 . . . . . 6  |-  ( n  =  k  ->  (
( g : ( M ... n ) --> A  /\  ps )  <->  ( g : ( M ... k ) --> A  /\  th ) ) )
1514cbvrexv 2935 . . . . 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 6139 . . 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 1260 . . . . . 6  |-  ( f  =  x  ->  (
( h : ( M ... ( k  +  1 ) ) --> A  /\  f  =  ( h  |`  ( M ... k ) )  /\  si )  <->  ( h : ( M ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k
) )  /\  si ) ) )
2221rexbidv 2728 . . . . 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 6154 . . 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 26449 1  |-  ( ph  ->  E. f ( f : Z --> A  /\  A. n  e.  Z  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   {csn 3816    |` cres 4882   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1c1 8993    + caddc 8995   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-dc 8328  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046
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