Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  erdszelem11 Unicode version

Theorem erdszelem11 23747
Description: Lemma for erdsze 23748. (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.i  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.j  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.t  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
erdszelem.r  |-  ( ph  ->  R  e.  NN )
erdszelem.s  |-  ( ph  ->  S  e.  NN )
erdszelem.m  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  N )
Assertion
Ref Expression
erdszelem11  |-  ( ph  ->  E. s  e.  ~P  ( 1 ... N
) ( ( R  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) )  \/  ( S  <_  ( # `
 s )  /\  ( F  |`  s ) 
Isom  <  ,  `'  <  ( s ,  ( F
" s ) ) ) ) )
Distinct variable groups:    x, y    n, s, x, y, F   
n, I, s, x, y    n, J, s, x, y    R, s, x, y    n, N, s, x, y    ph, n, s, x, y    S, s, x, y    T, s
Allowed substitution hints:    R( n)    S( n)    T( x, y, n)

Proof of Theorem erdszelem11
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 erdsze.n . . . 4  |-  ( ph  ->  N  e.  NN )
2 erdsze.f . . . 4  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
3 erdszelem.i . . . 4  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
4 erdszelem.j . . . 4  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
5 erdszelem.t . . . 4  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
6 erdszelem.r . . . 4  |-  ( ph  ->  R  e.  NN )
7 erdszelem.s . . . 4  |-  ( ph  ->  S  e.  NN )
8 erdszelem.m . . . 4  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  N )
91, 2, 3, 4, 5, 6, 7, 8erdszelem10 23746 . . 3  |-  ( ph  ->  E. m  e.  ( 1 ... N ) ( -.  ( I `
 m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
101adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  N  e.  NN )
112adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  F : ( 1 ... N ) -1-1-> RR )
12 ltso 8919 . . . . . . 7  |-  <  Or  RR
13 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  m  e.  ( 1 ... N ) )
146adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  R  e.  NN )
15 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  -.  ( I `  m
)  e.  ( 1 ... ( R  - 
1 ) ) )
1610, 11, 3, 12, 13, 14, 15erdszelem7 23743 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  E. s  e.  ~P  ( 1 ... N
) ( R  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) ) )
1716expr 598 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( -.  ( I `  m
)  e.  ( 1 ... ( R  - 
1 ) )  ->  E. s  e.  ~P  ( 1 ... N
) ( R  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) ) ) )
181adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  N  e.  NN )
192adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  F : ( 1 ... N ) -1-1-> RR )
20 cnvso 5230 . . . . . . . 8  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
2112, 20mpbi 199 . . . . . . 7  |-  `'  <  Or  RR
22 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  m  e.  ( 1 ... N ) )
237adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  S  e.  NN )
24 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  -.  ( J `  m
)  e.  ( 1 ... ( S  - 
1 ) ) )
2518, 19, 4, 21, 22, 23, 24erdszelem7 23743 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  E. s  e.  ~P  ( 1 ... N
) ( S  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  `'  <  ( s ,  ( F " s
) ) ) )
2625expr 598 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( -.  ( J `  m
)  e.  ( 1 ... ( S  - 
1 ) )  ->  E. s  e.  ~P  ( 1 ... N
) ( S  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  `'  <  ( s ,  ( F " s
) ) ) ) )
2717, 26orim12d 811 . . . 4  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  (
( -.  ( I `
 m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) )  ->  ( E. s  e.  ~P  ( 1 ... N
) ( R  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) )  \/ 
E. s  e.  ~P  ( 1 ... N
) ( S  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  `'  <  ( s ,  ( F " s
) ) ) ) ) )
2827rexlimdva 2680 . . 3  |-  ( ph  ->  ( E. m  e.  ( 1 ... N
) ( -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) )  ->  ( E. s  e.  ~P  ( 1 ... N
) ( R  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) )  \/ 
E. s  e.  ~P  ( 1 ... N
) ( S  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  `'  <  ( s ,  ( F " s
) ) ) ) ) )
299, 28mpd 14 . 2  |-  ( ph  ->  ( E. s  e. 
~P  ( 1 ... N ) ( R  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) )  \/ 
E. s  e.  ~P  ( 1 ... N
) ( S  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  `'  <  ( s ,  ( F " s
) ) ) ) )
30 r19.43 2708 . 2  |-  ( E. s  e.  ~P  (
1 ... N ) ( ( R  <_  ( # `
 s )  /\  ( F  |`  s ) 
Isom  <  ,  <  (
s ,  ( F
" s ) ) )  \/  ( S  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  `'  <  ( s ,  ( F " s
) ) ) )  <-> 
( E. s  e. 
~P  ( 1 ... N ) ( R  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) )  \/ 
E. s  e.  ~P  ( 1 ... N
) ( S  <_ 
( # `  s )  /\  ( F  |`  s )  Isom  <  ,  `'  <  ( s ,  ( F " s
) ) ) ) )
3129, 30sylibr 203 1  |-  ( ph  ->  E. s  e.  ~P  ( 1 ... N
) ( ( R  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) )  \/  ( S  <_  ( # `
 s )  /\  ( F  |`  s ) 
Isom  <  ,  `'  <  ( s ,  ( F
" s ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560   ~Pcpw 3638   <.cop 3656   class class class wbr 4039    e. cmpt 4093    Or wor 4329   `'ccnv 4704    |` cres 4707   "cima 4708   -1-1->wf1 5268   ` cfv 5271    Isom wiso 5272  (class class class)co 5874   supcsup 7209   RRcr 8752   1c1 8754    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   ...cfz 10798   #chash 11353
This theorem is referenced by:  erdsze  23748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354
  Copyright terms: Public domain W3C validator