Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  erdszelem7 Structured version   Visualization version   Unicode version

Theorem erdszelem7 29932
Description: Lemma for erdsze 29937. (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 12529 . . . 4  |-  # : _V
--> ( NN0  u.  { +oo } )
2 ffun 5736 . . . 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 29930 . . . 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 675 . . 3  |-  ( ph  ->  ( K `  A
)  e.  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } ) )
11 fvelima 5922 . . 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 670 . 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 2453 . . . . . 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 29926 . . . . 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 1054 . . . . . . . . 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 11804 . . . . . . . . . . 11  |-  ( A  e.  ( 1 ... N )  ->  N  e.  ( ZZ>= `  A )
)
17 fzss2 11845 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  A
)  ->  ( 1 ... A )  C_  ( 1 ... N
) )
184, 16, 173syl 18 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... A
)  C_  ( 1 ... N ) )
1918adantr 467 . . . . . . . . 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 3444 . . . . . . . 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 3960 . . . . . . . 8  |-  ( s  e.  ~P ( 1 ... N )  <->  s  C_  ( 1 ... N
) )
2220, 21sylibr 216 . . . . . . 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 29931 . . . . . . . . . . . . . . 15  |-  ( ph  ->  K : ( 1 ... N ) --> NN )
2524, 4ffvelrnd 6028 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( K `  A
)  e.  NN )
26 nnuz 11201 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
2725, 26syl6eleq 2541 . . . . . . . . . . . . 13  |-  ( ph  ->  ( K `  A
)  e.  ( ZZ>= ` 
1 ) )
28 erdszelem7.r . . . . . . . . . . . . . 14  |-  ( ph  ->  R  e.  NN )
29 nnz 10966 . . . . . . . . . . . . . 14  |-  ( R  e.  NN  ->  R  e.  ZZ )
30 peano2zm 10987 . . . . . . . . . . . . . 14  |-  ( R  e.  ZZ  ->  ( R  -  1 )  e.  ZZ )
3128, 29, 303syl 18 . . . . . . . . . . . . 13  |-  ( ph  ->  ( R  -  1 )  e.  ZZ )
32 elfz5 11799 . . . . . . . . . . . . 13  |-  ( ( ( K `  A
)  e.  ( ZZ>= ` 
1 )  /\  ( R  -  1 )  e.  ZZ )  -> 
( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <_  ( R  -  1 ) ) )
3327, 31, 32syl2anc 667 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <_  ( R  -  1 ) ) )
34 nnltlem1 11010 . . . . . . . . . . . . 13  |-  ( ( ( K `  A
)  e.  NN  /\  R  e.  NN )  ->  ( ( K `  A )  <  R  <->  ( K `  A )  <_  ( R  - 
1 ) ) )
3525, 28, 34syl2anc 667 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( K `  A )  <  R  <->  ( K `  A )  <_  ( R  - 
1 ) ) )
3633, 35bitr4d 260 . . . . . . . . . . 11  |-  ( ph  ->  ( ( K `  A )  e.  ( 1 ... ( R  -  1 ) )  <-> 
( K `  A
)  <  R )
)
3723, 36mtbid 302 . . . . . . . . . 10  |-  ( ph  ->  -.  ( K `  A )  <  R
)
3828nnred 10631 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  RR )
3913erdszelem2 29927 . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . 13  |-  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } )  C_  NN
41 nnssre 10620 . . . . . . . . . . . . 13  |-  NN  C_  RR
4240, 41sstri 3443 . . . . . . . . . . . 12  |-  ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } )  C_  RR
4342, 10sseldi 3432 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  A
)  e.  RR )
4438, 43lenltd 9786 . . . . . . . . . 10  |-  ( ph  ->  ( R  <_  ( K `  A )  <->  -.  ( K `  A
)  <  R )
)
4537, 44mpbird 236 . . . . . . . . 9  |-  ( ph  ->  R  <_  ( K `  A ) )
4645adantr 467 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  R  <_  ( K `  A
) )
47 simprr 767 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  ( # `
 s )  =  ( K `  A
) )
4846, 47breqtrrd 4432 . . . . . . 7  |-  ( (
ph  /\  ( (
s  C_  ( 1 ... A )  /\  ( F  |`  s ) 
Isom  <  ,  O  ( s ,  ( F
" s ) )  /\  A  e.  s )  /\  ( # `  s )  =  ( K `  A ) ) )  ->  R  <_  ( # `  s
) )
49 simprl2 1055 . . . . . . 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 538 . . . . . 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 620 . . . . 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 478 . . . 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 608 . . 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 2860 . 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 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   E.wrex 2740   {crab 2743   _Vcvv 3047    u. cun 3404    C_ wss 3406   ~Pcpw 3953   {csn 3970   class class class wbr 4405    |-> cmpt 4464    Or wor 4757    |` cres 4839   "cima 4840   Fun wfun 5579   -->wf 5581   -1-1->wf1 5582   ` cfv 5585    Isom wiso 5586  (class class class)co 6295   Fincfn 7574   supcsup 7959   RRcr 9543   1c1 9545   +oocpnf 9677    < clt 9680    <_ cle 9681    - cmin 9865   NNcn 10616   NN0cn0 10876   ZZcz 10944   ZZ>=cuz 11166   ...cfz 11791   #chash 12522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12523
This theorem is referenced by:  erdszelem11  29936
  Copyright terms: Public domain W3C validator