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Theorem erdszelem7 27015
Description: Lemma for erdsze 27020. (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 12106 . . . 4  |-  # : _V
--> ( NN0  u.  { +oo } )
2 ffun 5558 . . . 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 27013 . . . 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 663 . . 3  |-  ( ph  ->  ( K `  A
)  e.  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } ) )
11 fvelima 5740 . . 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 658 . 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 2441 . . . . . 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 27009 . . . . 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 1028 . . . . . . . . 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 11446 . . . . . . . . . . 11  |-  ( A  e.  ( 1 ... N )  ->  N  e.  ( ZZ>= `  A )
)
17 fzss2 11494 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  A
)  ->  ( 1 ... A )  C_  ( 1 ... N
) )
184, 16, 173syl 20 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... A
)  C_  ( 1 ... N ) )
1918adantr 462 . . . . . . . . 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 3363 . . . . . . . 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 3864 . . . . . . . 8  |-  ( s  e.  ~P ( 1 ... N )  <->  s  C_  ( 1 ... N
) )
2220, 21sylibr 212 . . . . . . 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 27014 . . . . . . . . . . . . . . 15  |-  ( ph  ->  K : ( 1 ... N ) --> NN )
2524, 4ffvelrnd 5841 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( K `  A
)  e.  NN )
26 nnuz 10892 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
2725, 26syl6eleq 2531 . . . . . . . . . . . . 13  |-  ( ph  ->  ( K `  A
)  e.  ( ZZ>= ` 
1 ) )
28 erdszelem7.r . . . . . . . . . . . . . 14  |-  ( ph  ->  R  e.  NN )
29 nnz 10664 . . . . . . . . . . . . . 14  |-  ( R  e.  NN  ->  R  e.  ZZ )
30 peano2zm 10684 . . . . . . . . . . . . . 14  |-  ( R  e.  ZZ  ->  ( R  -  1 )  e.  ZZ )
3128, 29, 303syl 20 . . . . . . . . . . . . 13  |-  ( ph  ->  ( R  -  1 )  e.  ZZ )
32 elfz5 11441 . . . . . . . . . . . . 13  |-  ( ( ( K `  A
)  e.  ( ZZ>= ` 
1 )  /\  ( R  -  1 )  e.  ZZ )  -> 
( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <_  ( R  -  1 ) ) )
3327, 31, 32syl2anc 656 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <_  ( R  -  1 ) ) )
34 nnltlem1 10705 . . . . . . . . . . . . 13  |-  ( ( ( K `  A
)  e.  NN  /\  R  e.  NN )  ->  ( ( K `  A )  <  R  <->  ( K `  A )  <_  ( R  - 
1 ) ) )
3525, 28, 34syl2anc 656 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( K `  A )  <  R  <->  ( K `  A )  <_  ( R  - 
1 ) ) )
3633, 35bitr4d 256 . . . . . . . . . . 11  |-  ( ph  ->  ( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <  R )
)
3723, 36mtbid 300 . . . . . . . . . 10  |-  ( ph  ->  -.  ( K `  A )  <  R
)
3828nnred 10333 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  RR )
3913erdszelem2 27010 . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . 13  |-  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } )  C_  NN
41 nnssre 10322 . . . . . . . . . . . . 13  |-  NN  C_  RR
4240, 41sstri 3362 . . . . . . . . . . . 12  |-  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } )  C_  RR
4342, 10sseldi 3351 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  A
)  e.  RR )
4438, 43lenltd 9516 . . . . . . . . . 10  |-  ( ph  ->  ( R  <_  ( K `  A )  <->  -.  ( K `  A
)  <  R )
)
4537, 44mpbird 232 . . . . . . . . 9  |-  ( ph  ->  R  <_  ( K `  A ) )
4645adantr 462 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  R  <_  ( K `  A
) )
47 simprr 751 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  ( # `
 s )  =  ( K `  A
) )
4846, 47breqtrrd 4315 . . . . . . 7  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  R  <_  ( # `  s
) )
49 simprl2 1029 . . . . . . 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 532 . . . . . 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 612 . . . . 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 472 . . . 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 600 . . 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 2823 . 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 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   E.wrex 2714   {crab 2717   _Vcvv 2970    u. cun 3323    C_ wss 3325   ~Pcpw 3857   {csn 3874   class class class wbr 4289    e. cmpt 4347    Or wor 4636    |` cres 4838   "cima 4839   Fun wfun 5409   -->wf 5411   -1-1->wf1 5412   ` cfv 5415    Isom wiso 5416  (class class class)co 6090   Fincfn 7306   supcsup 7686   RRcr 9277   1c1 9279   +oocpnf 9411    < clt 9414    <_ cle 9415    - cmin 9591   NNcn 10318   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   #chash 12099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-hash 12100
This theorem is referenced by:  erdszelem11  27019
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