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Theorem erdszelem7 29992
Description: Lemma for erdsze 29997. (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 12560 . . . 4  |-  # : _V
--> ( NN0  u.  { +oo } )
2 ffun 5742 . . . 4  |-  ( # : _V --> ( NN0  u.  { +oo } )  ->  Fun  # )
31, 2ax-mp 5 . . 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 29990 . . . 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 681 . . 3  |-  ( ph  ->  ( K `  A
)  e.  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } ) )
11 fvelima 5931 . . 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 676 . 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 2471 . . . . . 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 29986 . . . . 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 1075 . . . . . . . . 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 11823 . . . . . . . . . . 11  |-  ( A  e.  ( 1 ... N )  ->  N  e.  ( ZZ>= `  A )
)
17 fzss2 11864 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  A
)  ->  ( 1 ... A )  C_  ( 1 ... N
) )
184, 16, 173syl 18 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... A
)  C_  ( 1 ... N ) )
1918adantr 472 . . . . . . . . 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 3428 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  s  C_  ( 1 ... N
) )
21 selpw 3949 . . . . . . . 8  |-  ( s  e.  ~P ( 1 ... N )  <->  s  C_  ( 1 ... N
) )
2220, 21sylibr 217 . . . . . . 7  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  s  e.  ~P ( 1 ... N ) )
23 erdszelem7.m . . . . . . . . . . 11  |-  ( ph  ->  -.  ( K `  A )  e.  ( 1 ... ( R  -  1 ) ) )
245, 6, 7, 8erdszelem6 29991 . . . . . . . . . . . . . . 15  |-  ( ph  ->  K : ( 1 ... N ) --> NN )
2524, 4ffvelrnd 6038 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( K `  A
)  e.  NN )
26 nnuz 11218 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
2725, 26syl6eleq 2559 . . . . . . . . . . . . 13  |-  ( ph  ->  ( K `  A
)  e.  ( ZZ>= ` 
1 ) )
28 erdszelem7.r . . . . . . . . . . . . . 14  |-  ( ph  ->  R  e.  NN )
29 nnz 10983 . . . . . . . . . . . . . 14  |-  ( R  e.  NN  ->  R  e.  ZZ )
30 peano2zm 11004 . . . . . . . . . . . . . 14  |-  ( R  e.  ZZ  ->  ( R  -  1 )  e.  ZZ )
3128, 29, 303syl 18 . . . . . . . . . . . . 13  |-  ( ph  ->  ( R  -  1 )  e.  ZZ )
32 elfz5 11818 . . . . . . . . . . . . 13  |-  ( ( ( K `  A
)  e.  ( ZZ>= ` 
1 )  /\  ( R  -  1 )  e.  ZZ )  -> 
( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <_  ( R  -  1 ) ) )
3327, 31, 32syl2anc 673 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <_  ( R  -  1 ) ) )
34 nnltlem1 11026 . . . . . . . . . . . . 13  |-  ( ( ( K `  A
)  e.  NN  /\  R  e.  NN )  ->  ( ( K `  A )  <  R  <->  ( K `  A )  <_  ( R  - 
1 ) ) )
3525, 28, 34syl2anc 673 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( K `  A )  <  R  <->  ( K `  A )  <_  ( R  - 
1 ) ) )
3633, 35bitr4d 264 . . . . . . . . . . 11  |-  ( ph  ->  ( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <  R )
)
3723, 36mtbid 307 . . . . . . . . . 10  |-  ( ph  ->  -.  ( K `  A )  <  R
)
3828nnred 10646 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  RR )
3913erdszelem2 29987 . . . . . . . . . . . . . 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 )
4039simpri 469 . . . . . . . . . . . . 13  |-  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } )  C_  NN
41 nnssre 10635 . . . . . . . . . . . . 13  |-  NN  C_  RR
4240, 41sstri 3427 . . . . . . . . . . . 12  |-  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } )  C_  RR
4342, 10sseldi 3416 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  A
)  e.  RR )
4438, 43lenltd 9798 . . . . . . . . . 10  |-  ( ph  ->  ( R  <_  ( K `  A )  <->  -.  ( K `  A
)  <  R )
)
4537, 44mpbird 240 . . . . . . . . 9  |-  ( ph  ->  R  <_  ( K `  A ) )
4645adantr 472 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  R  <_  ( K `  A
) )
47 simprr 774 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  ( # `
 s )  =  ( K `  A
) )
4846, 47breqtrrd 4422 . . . . . . 7  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  R  <_  ( # `  s
) )
49 simprl2 1076 . . . . . . 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 ) ) )
5022, 48, 49jca32 544 . . . . . 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 ) ) ) ) )
5150expr 626 . . . . 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
) ) ) ) ) )
5214, 51sylan2b 483 . . . 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
) ) ) ) ) )
5352expimpd 614 . . 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 ) ) ) ) ) )
5453reximdv2 2855 . 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
) ) ) ) )
5512, 54mpd 15 1  |-  ( ph  ->  E. s  e.  ~P  ( 1 ... N
) ( R  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  O  ( s ,  ( F " s
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   E.wrex 2757   {crab 2760   _Vcvv 3031    u. cun 3388    C_ wss 3390   ~Pcpw 3942   {csn 3959   class class class wbr 4395    |-> cmpt 4454    Or wor 4759    |` cres 4841   "cima 4842   Fun wfun 5583   -->wf 5585   -1-1->wf1 5586   ` cfv 5589    Isom wiso 5590  (class class class)co 6308   Fincfn 7587   supcsup 7972   RRcr 9556   1c1 9558   +oocpnf 9690    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   #chash 12553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554
This theorem is referenced by:  erdszelem11  29996
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