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Theorem erdszelem7 23728
Description: Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n  |-  ( ph  ->  N  e.  NN )
erdsze.f  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
erdszelem.k  |-  K  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.o  |-  O  Or  RR
erdszelem.a  |-  ( ph  ->  A  e.  ( 1 ... N ) )
erdszelem7.r  |-  ( ph  ->  R  e.  NN )
erdszelem7.m  |-  ( ph  ->  -.  ( K `  A )  e.  ( 1 ... ( R  -  1 ) ) )
Assertion
Ref Expression
erdszelem7  |-  ( ph  ->  E. s  e.  ~P  ( 1 ... N
) ( R  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  O  ( s ,  ( F " s
) ) ) )
Distinct variable groups:    x, y,
s, F    K, s    A, s, x, y    O, s, x, y    R, s, x, y    N, s, x, y    ph, s, x, y
Allowed substitution hints:    K( x, y)

Proof of Theorem erdszelem7
StepHypRef Expression
1 hashf 11344 . . . 4  |-  # : _V
--> ( NN0  u.  {  +oo } )
2 ffun 5391 . . . 4  |-  ( # : _V --> ( NN0  u.  { 
+oo } )  ->  Fun  # )
31, 2ax-mp 8 . . 3  |-  Fun  #
4 erdszelem.a . . . 4  |-  ( ph  ->  A  e.  ( 1 ... N ) )
5 erdsze.n . . . . 5  |-  ( ph  ->  N  e.  NN )
6 erdsze.f . . . . 5  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
7 erdszelem.k . . . . 5  |-  K  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
8 erdszelem.o . . . . 5  |-  O  Or  RR
95, 6, 7, 8erdszelem5 23726 . . . 4  |-  ( (
ph  /\  A  e.  ( 1 ... N
) )  ->  ( K `  A )  e.  ( # " {
y  e.  ~P (
1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  A  e.  y ) } ) )
104, 9mpdan 649 . . 3  |-  ( ph  ->  ( K `  A
)  e.  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } ) )
11 fvelima 5574 . . 3  |-  ( ( Fun  #  /\  ( K `  A )  e.  ( # " {
y  e.  ~P (
1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  A  e.  y ) } ) )  ->  E. s  e.  { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) }  ( # `  s )  =  ( K `  A ) )
123, 10, 11sylancr 644 . 2  |-  ( ph  ->  E. s  e.  {
y  e.  ~P (
1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  A  e.  y ) }  ( # `
 s )  =  ( K `  A
) )
13 eqid 2283 . . . . . 6  |-  { y  e.  ~P ( 1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  A  e.  y ) }  =  { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) }
1413erdszelem1 23722 . . . . 5  |-  ( s  e.  { y  e. 
~P ( 1 ... A )  |  ( ( F  |`  y
)  Isom  <  ,  O  ( y ,  ( F " y ) )  /\  A  e.  y ) }  <->  ( s  C_  ( 1 ... A
)  /\  ( F  |`  s )  Isom  <  ,  O  ( s ,  ( F " s
) )  /\  A  e.  s ) )
15 simprl1 1000 . . . . . . . . 9  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  s  C_  ( 1 ... A
) )
16 elfzuz3 10795 . . . . . . . . . . 11  |-  ( A  e.  ( 1 ... N )  ->  N  e.  ( ZZ>= `  A )
)
17 fzss2 10831 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  A
)  ->  ( 1 ... A )  C_  ( 1 ... N
) )
184, 16, 173syl 18 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... A
)  C_  ( 1 ... N ) )
1918adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  (
1 ... A )  C_  ( 1 ... N
) )
2015, 19sstrd 3189 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  s  C_  ( 1 ... N
) )
21 vex 2791 . . . . . . . . 9  |-  s  e. 
_V
2221elpw 3631 . . . . . . . 8  |-  ( s  e.  ~P ( 1 ... N )  <->  s  C_  ( 1 ... N
) )
2320, 22sylibr 203 . . . . . . 7  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  s  e.  ~P ( 1 ... N ) )
24 erdszelem7.m . . . . . . . . . . 11  |-  ( ph  ->  -.  ( K `  A )  e.  ( 1 ... ( R  -  1 ) ) )
255, 6, 7, 8erdszelem6 23727 . . . . . . . . . . . . . . 15  |-  ( ph  ->  K : ( 1 ... N ) --> NN )
26 ffvelrn 5663 . . . . . . . . . . . . . . 15  |-  ( ( K : ( 1 ... N ) --> NN 
/\  A  e.  ( 1 ... N ) )  ->  ( K `  A )  e.  NN )
2725, 4, 26syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( K `  A
)  e.  NN )
28 nnuz 10263 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
2927, 28syl6eleq 2373 . . . . . . . . . . . . 13  |-  ( ph  ->  ( K `  A
)  e.  ( ZZ>= ` 
1 ) )
30 erdszelem7.r . . . . . . . . . . . . . 14  |-  ( ph  ->  R  e.  NN )
31 nnz 10045 . . . . . . . . . . . . . 14  |-  ( R  e.  NN  ->  R  e.  ZZ )
32 peano2zm 10062 . . . . . . . . . . . . . 14  |-  ( R  e.  ZZ  ->  ( R  -  1 )  e.  ZZ )
3330, 31, 323syl 18 . . . . . . . . . . . . 13  |-  ( ph  ->  ( R  -  1 )  e.  ZZ )
34 elfz5 10790 . . . . . . . . . . . . 13  |-  ( ( ( K `  A
)  e.  ( ZZ>= ` 
1 )  /\  ( R  -  1 )  e.  ZZ )  -> 
( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <_  ( R  -  1 ) ) )
3529, 33, 34syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <_  ( R  -  1 ) ) )
36 nnltlem1 10081 . . . . . . . . . . . . 13  |-  ( ( ( K `  A
)  e.  NN  /\  R  e.  NN )  ->  ( ( K `  A )  <  R  <->  ( K `  A )  <_  ( R  - 
1 ) ) )
3727, 30, 36syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( K `  A )  <  R  <->  ( K `  A )  <_  ( R  - 
1 ) ) )
3835, 37bitr4d 247 . . . . . . . . . . 11  |-  ( ph  ->  ( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <  R )
)
3924, 38mtbid 291 . . . . . . . . . 10  |-  ( ph  ->  -.  ( K `  A )  <  R
)
4030nnred 9761 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  RR )
4113erdszelem2 23723 . . . . . . . . . . . . . 14  |-  ( (
# " { y  e.  ~P ( 1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  A  e.  y ) } )  e.  Fin  /\  ( #
" { y  e. 
~P ( 1 ... A )  |  ( ( F  |`  y
)  Isom  <  ,  O  ( y ,  ( F " y ) )  /\  A  e.  y ) } ) 
C_  NN )
4241simpri 448 . . . . . . . . . . . . 13  |-  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } )  C_  NN
43 nnssre 9750 . . . . . . . . . . . . 13  |-  NN  C_  RR
4442, 43sstri 3188 . . . . . . . . . . . 12  |-  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } )  C_  RR
4544, 10sseldi 3178 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  A
)  e.  RR )
4640, 45lenltd 8965 . . . . . . . . . 10  |-  ( ph  ->  ( R  <_  ( K `  A )  <->  -.  ( K `  A
)  <  R )
)
4739, 46mpbird 223 . . . . . . . . 9  |-  ( ph  ->  R  <_  ( K `  A ) )
4847adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  R  <_  ( K `  A
) )
49 simprr 733 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  ( # `
 s )  =  ( K `  A
) )
5048, 49breqtrrd 4049 . . . . . . 7  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  R  <_  ( # `  s
) )
51 simprl2 1001 . . . . . . 7  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  ( F  |`  s )  Isom  <  ,  O  ( s ,  ( F "
s ) ) )
5223, 50, 51jca32 521 . . . . . 6  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  (
s  e.  ~P (
1 ... N )  /\  ( R  <_  ( # `  s )  /\  ( F  |`  s )  Isom  <  ,  O  ( s ,  ( F "
s ) ) ) ) )
5352expr 598 . . . . 5  |-  ( (
ph  /\  ( s  C_  ( 1 ... A
)  /\  ( F  |`  s )  Isom  <  ,  O  ( s ,  ( F " s
) )  /\  A  e.  s ) )  -> 
( ( # `  s
)  =  ( K `
 A )  -> 
( s  e.  ~P ( 1 ... N
)  /\  ( R  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  O  ( s ,  ( F " s
) ) ) ) ) )
5414, 53sylan2b 461 . . . 4  |-  ( (
ph  /\  s  e.  { y  e.  ~P (
1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  A  e.  y ) } )  ->  ( ( # `  s )  =  ( K `  A )  ->  ( s  e. 
~P ( 1 ... N )  /\  ( R  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  O  ( s ,  ( F " s
) ) ) ) ) )
5554expimpd 586 . . 3  |-  ( ph  ->  ( ( s  e. 
{ y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) }  /\  ( # `
 s )  =  ( K `  A
) )  ->  (
s  e.  ~P (
1 ... N )  /\  ( R  <_  ( # `  s )  /\  ( F  |`  s )  Isom  <  ,  O  ( s ,  ( F "
s ) ) ) ) ) )
5655reximdv2 2652 . 2  |-  ( ph  ->  ( E. s  e. 
{ y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) }  ( # `  s )  =  ( K `  A )  ->  E. s  e.  ~P  ( 1 ... N
) ( R  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  O  ( s ,  ( F " s
) ) ) ) )
5712, 56mpd 14 1  |-  ( ph  ->  E. s  e.  ~P  ( 1 ... N
) ( R  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  O  ( s ,  ( F " s
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788    u. cun 3150    C_ wss 3152   ~Pcpw 3625   {csn 3640   class class class wbr 4023    e. cmpt 4077    Or wor 4313    |` cres 4691   "cima 4692   Fun wfun 5249   -->wf 5251   -1-1->wf1 5252   ` cfv 5255    Isom wiso 5256  (class class class)co 5858   Fincfn 6863   supcsup 7193   RRcr 8736   1c1 8738    +oocpnf 8864    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   #chash 11337
This theorem is referenced by:  erdszelem11  23732
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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-rmo 2551  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-int 3863  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-isom 5264  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-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  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  df-hash 11338
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