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Theorem erdszelem7 24844
Description: Lemma for erdsze 24849. (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 11588 . . . 4  |-  # : _V
--> ( NN0  u.  {  +oo } )
2 ffun 5560 . . . 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 24842 . . . 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 650 . . 3  |-  ( ph  ->  ( K `  A
)  e.  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } ) )
11 fvelima 5745 . . 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 645 . 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 2412 . . . . . 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 24838 . . . . 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 1002 . . . . . . . . 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 11020 . . . . . . . . . . 11  |-  ( A  e.  ( 1 ... N )  ->  N  e.  ( ZZ>= `  A )
)
17 fzss2 11056 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  A
)  ->  ( 1 ... A )  C_  ( 1 ... N
) )
184, 16, 173syl 19 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... A
)  C_  ( 1 ... N ) )
1918adantr 452 . . . . . . . . 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 3326 . . . . . . . 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 2927 . . . . . . . . 9  |-  s  e. 
_V
2221elpw 3773 . . . . . . . 8  |-  ( s  e.  ~P ( 1 ... N )  <->  s  C_  ( 1 ... N
) )
2320, 22sylibr 204 . . . . . . 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 24843 . . . . . . . . . . . . . . 15  |-  ( ph  ->  K : ( 1 ... N ) --> NN )
2625, 4ffvelrnd 5838 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( K `  A
)  e.  NN )
27 nnuz 10485 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
2826, 27syl6eleq 2502 . . . . . . . . . . . . 13  |-  ( ph  ->  ( K `  A
)  e.  ( ZZ>= ` 
1 ) )
29 erdszelem7.r . . . . . . . . . . . . . 14  |-  ( ph  ->  R  e.  NN )
30 nnz 10267 . . . . . . . . . . . . . 14  |-  ( R  e.  NN  ->  R  e.  ZZ )
31 peano2zm 10284 . . . . . . . . . . . . . 14  |-  ( R  e.  ZZ  ->  ( R  -  1 )  e.  ZZ )
3229, 30, 313syl 19 . . . . . . . . . . . . 13  |-  ( ph  ->  ( R  -  1 )  e.  ZZ )
33 elfz5 11015 . . . . . . . . . . . . 13  |-  ( ( ( K `  A
)  e.  ( ZZ>= ` 
1 )  /\  ( R  -  1 )  e.  ZZ )  -> 
( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <_  ( R  -  1 ) ) )
3428, 32, 33syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <_  ( R  -  1 ) ) )
35 nnltlem1 10303 . . . . . . . . . . . . 13  |-  ( ( ( K `  A
)  e.  NN  /\  R  e.  NN )  ->  ( ( K `  A )  <  R  <->  ( K `  A )  <_  ( R  - 
1 ) ) )
3626, 29, 35syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( K `  A )  <  R  <->  ( K `  A )  <_  ( R  - 
1 ) ) )
3734, 36bitr4d 248 . . . . . . . . . . 11  |-  ( ph  ->  ( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <  R )
)
3824, 37mtbid 292 . . . . . . . . . 10  |-  ( ph  ->  -.  ( K `  A )  <  R
)
3929nnred 9979 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  RR )
4013erdszelem2 24839 . . . . . . . . . . . . . 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 )
4140simpri 449 . . . . . . . . . . . . 13  |-  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } )  C_  NN
42 nnssre 9968 . . . . . . . . . . . . 13  |-  NN  C_  RR
4341, 42sstri 3325 . . . . . . . . . . . 12  |-  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } )  C_  RR
4443, 10sseldi 3314 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  A
)  e.  RR )
4539, 44lenltd 9183 . . . . . . . . . 10  |-  ( ph  ->  ( R  <_  ( K `  A )  <->  -.  ( K `  A
)  <  R )
)
4638, 45mpbird 224 . . . . . . . . 9  |-  ( ph  ->  R  <_  ( K `  A ) )
4746adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  R  <_  ( K `  A
) )
48 simprr 734 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  ( # `
 s )  =  ( K `  A
) )
4947, 48breqtrrd 4206 . . . . . . 7  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  R  <_  ( # `  s
) )
50 simprl2 1003 . . . . . . 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 ) ) )
5123, 49, 50jca32 522 . . . . . 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 ) ) ) ) )
5251expr 599 . . . . 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
) ) ) ) ) )
5314, 52sylan2b 462 . . . 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
) ) ) ) ) )
5453expimpd 587 . . 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 ) ) ) ) ) )
5554reximdv2 2783 . 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
) ) ) ) )
5612, 55mpd 15 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2675   {crab 2678   _Vcvv 2924    u. cun 3286    C_ wss 3288   ~Pcpw 3767   {csn 3782   class class class wbr 4180    e. cmpt 4234    Or wor 4470    |` cres 4847   "cima 4848   Fun wfun 5415   -->wf 5417   -1-1->wf1 5418   ` cfv 5421    Isom wiso 5422  (class class class)co 6048   Fincfn 7076   supcsup 7411   RRcr 8953   1c1 8955    +oocpnf 9081    < clt 9084    <_ cle 9085    - cmin 9255   NNcn 9964   NN0cn0 10185   ZZcz 10246   ZZ>=cuz 10452   ...cfz 11007   #chash 11581
This theorem is referenced by:  erdszelem11  24848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-hash 11582
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