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Theorem erdszelem10 23731
Description: Lemma for erdsze 23733. (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
erdszelem10  |-  ( ph  ->  E. m  e.  ( 1 ... N ) ( -.  ( I `
 m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
Distinct variable groups:    x, y    m, n, x, y, F   
n, I, x, y   
n, J, x, y    R, m, x, y    m, N, n, x, y    ph, m, n, x, y    S, m, x, y    T, m
Allowed substitution hints:    R( n)    S( n)    T( x, y, n)    I( m)    J( m)

Proof of Theorem erdszelem10
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 fzfi 11034 . . . . . . . 8  |-  ( 1 ... ( R  - 
1 ) )  e. 
Fin
2 fzfi 11034 . . . . . . . 8  |-  ( 1 ... ( S  - 
1 ) )  e. 
Fin
3 xpfi 7128 . . . . . . . 8  |-  ( ( ( 1 ... ( R  -  1 ) )  e.  Fin  /\  ( 1 ... ( S  -  1 ) )  e.  Fin )  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  e.  Fin )
41, 2, 3mp2an 653 . . . . . . 7  |-  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  e. 
Fin
5 ssdomg 6907 . . . . . . 7  |-  ( ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  e.  Fin  ->  ( ran  T  C_  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  ran  T  ~<_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) ) )
64, 5ax-mp 8 . . . . . 6  |-  ( ran 
T  C_  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  ran  T  ~<_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
7 domnsym 6987 . . . . . 6  |-  ( ran 
T  ~<_  ( ( 1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  -.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
86, 7syl 15 . . . . 5  |-  ( ran 
T  C_  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  -.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
9 erdszelem.m . . . . . . . 8  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  N )
10 hashxp 11386 . . . . . . . . . 10  |-  ( ( ( 1 ... ( R  -  1 ) )  e.  Fin  /\  ( 1 ... ( S  -  1 ) )  e.  Fin )  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  =  ( (
# `  ( 1 ... ( R  -  1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) ) )
111, 2, 10mp2an 653 . . . . . . . . 9  |-  ( # `  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )  =  ( (
# `  ( 1 ... ( R  -  1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) )
12 erdszelem.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  NN )
13 nnm1nn0 10005 . . . . . . . . . . 11  |-  ( R  e.  NN  ->  ( R  -  1 )  e.  NN0 )
14 hashfz1 11345 . . . . . . . . . . 11  |-  ( ( R  -  1 )  e.  NN0  ->  ( # `  ( 1 ... ( R  -  1 ) ) )  =  ( R  -  1 ) )
1512, 13, 143syl 18 . . . . . . . . . 10  |-  ( ph  ->  ( # `  (
1 ... ( R  - 
1 ) ) )  =  ( R  - 
1 ) )
16 erdszelem.s . . . . . . . . . . 11  |-  ( ph  ->  S  e.  NN )
17 nnm1nn0 10005 . . . . . . . . . . 11  |-  ( S  e.  NN  ->  ( S  -  1 )  e.  NN0 )
18 hashfz1 11345 . . . . . . . . . . 11  |-  ( ( S  -  1 )  e.  NN0  ->  ( # `  ( 1 ... ( S  -  1 ) ) )  =  ( S  -  1 ) )
1916, 17, 183syl 18 . . . . . . . . . 10  |-  ( ph  ->  ( # `  (
1 ... ( S  - 
1 ) ) )  =  ( S  - 
1 ) )
2015, 19oveq12d 5876 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  (
1 ... ( R  - 
1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) )  =  ( ( R  -  1 )  x.  ( S  - 
1 ) ) )
2111, 20syl5eq 2327 . . . . . . . 8  |-  ( ph  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  =  ( ( R  -  1 )  x.  ( S  - 
1 ) ) )
22 erdsze.n . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN )
2322nnnn0d 10018 . . . . . . . . 9  |-  ( ph  ->  N  e.  NN0 )
24 hashfz1 11345 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
2523, 24syl 15 . . . . . . . 8  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
269, 21, 253brtr4d 4053 . . . . . . 7  |-  ( ph  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  <  ( # `  ( 1 ... N
) ) )
27 fzfid 11035 . . . . . . . 8  |-  ( ph  ->  ( 1 ... N
)  e.  Fin )
28 hashsdom 11363 . . . . . . . 8  |-  ( ( ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  e.  Fin  /\  (
1 ... N )  e. 
Fin )  ->  (
( # `  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) )  <  ( # `  (
1 ... N ) )  <-> 
( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ( 1 ... N
) ) )
294, 27, 28sylancr 644 . . . . . . 7  |-  ( ph  ->  ( ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  <  ( # `  ( 1 ... N
) )  <->  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ~<  (
1 ... N ) ) )
3026, 29mpbid 201 . . . . . 6  |-  ( ph  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ( 1 ... N
) )
31 erdsze.f . . . . . . . 8  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
32 erdszelem.i . . . . . . . 8  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
33 erdszelem.j . . . . . . . 8  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
34 erdszelem.t . . . . . . . 8  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
3522, 31, 32, 33, 34erdszelem9 23730 . . . . . . 7  |-  ( ph  ->  T : ( 1 ... N ) -1-1-> ( NN  X.  NN ) )
36 f1f1orn 5483 . . . . . . 7  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  ->  T : ( 1 ... N ) -1-1-onto-> ran  T )
37 ovex 5883 . . . . . . . 8  |-  ( 1 ... N )  e. 
_V
3837f1oen 6882 . . . . . . 7  |-  ( T : ( 1 ... N ) -1-1-onto-> ran  T  ->  (
1 ... N )  ~~  ran  T )
3935, 36, 383syl 18 . . . . . 6  |-  ( ph  ->  ( 1 ... N
)  ~~  ran  T )
40 sdomentr 6995 . . . . . 6  |-  ( ( ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ( 1 ... N
)  /\  ( 1 ... N )  ~~  ran  T )  ->  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
4130, 39, 40syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
428, 41nsyl3 111 . . . 4  |-  ( ph  ->  -.  ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
43 nss 3236 . . . . 5  |-  ( -. 
ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  E. s
( s  e.  ran  T  /\  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) ) )
44 df-rex 2549 . . . . 5  |-  ( E. s  e.  ran  T  -.  s  e.  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  E. s ( s  e. 
ran  T  /\  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) ) )
4543, 44bitr4i 243 . . . 4  |-  ( -. 
ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
4642, 45sylib 188 . . 3  |-  ( ph  ->  E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )
47 f1fn 5438 . . . 4  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  ->  T  Fn  ( 1 ... N ) )
48 eleq1 2343 . . . . . 6  |-  ( s  =  ( T `  m )  ->  (
s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( T `  m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) ) )
4948notbid 285 . . . . 5  |-  ( s  =  ( T `  m )  ->  ( -.  s  e.  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5049rexrn 5667 . . . 4  |-  ( T  Fn  ( 1 ... N )  ->  ( E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  E. m  e.  (
1 ... N )  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5135, 47, 503syl 18 . . 3  |-  ( ph  ->  ( E. s  e. 
ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  <->  E. m  e.  (
1 ... N )  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5246, 51mpbid 201 . 2  |-  ( ph  ->  E. m  e.  ( 1 ... N )  -.  ( T `  m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )
53 fveq2 5525 . . . . . . . . . 10  |-  ( n  =  m  ->  (
I `  n )  =  ( I `  m ) )
54 fveq2 5525 . . . . . . . . . 10  |-  ( n  =  m  ->  ( J `  n )  =  ( J `  m ) )
5553, 54opeq12d 3804 . . . . . . . . 9  |-  ( n  =  m  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  m ) ,  ( J `  m ) >. )
56 opex 4237 . . . . . . . . 9  |-  <. (
I `  m ) ,  ( J `  m ) >.  e.  _V
5755, 34, 56fvmpt 5602 . . . . . . . 8  |-  ( m  e.  ( 1 ... N )  ->  ( T `  m )  =  <. ( I `  m ) ,  ( J `  m )
>. )
5857adantl 452 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( T `  m )  =  <. ( I `  m ) ,  ( J `  m )
>. )
5958eleq1d 2349 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  (
( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  <. ( I `
 m ) ,  ( J `  m
) >.  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
60 opelxp 4719 . . . . . 6  |-  ( <.
( I `  m
) ,  ( J `
 m ) >.  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  <-> 
( ( I `  m )  e.  ( 1 ... ( R  -  1 ) )  /\  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
6159, 60syl6bb 252 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  (
( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( (
I `  m )  e.  ( 1 ... ( R  -  1 ) )  /\  ( J `
 m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
6261notbid 285 . . . 4  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  -.  (
( I `  m
)  e.  ( 1 ... ( R  - 
1 ) )  /\  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
63 ianor 474 . . . 4  |-  ( -.  ( ( I `  m )  e.  ( 1 ... ( R  -  1 ) )  /\  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) )  <->  ( -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
6462, 63syl6bb 252 . . 3  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( -.  ( I `  m
)  e.  ( 1 ... ( R  - 
1 ) )  \/ 
-.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
6564rexbidva 2560 . 2  |-  ( ph  ->  ( E. m  e.  ( 1 ... N
)  -.  ( T `
 m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  <->  E. m  e.  (
1 ... N ) ( -.  ( I `  m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `
 m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
6652, 65mpbid 201 1  |-  ( ph  ->  E. m  e.  ( 1 ... N ) ( -.  ( I `
 m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547    C_ wss 3152   ~Pcpw 3625   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256  (class class class)co 5858    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863   supcsup 7193   RRcr 8736   1c1 8738    x. cmul 8742    < clt 8867    - cmin 9037   NNcn 9746   NN0cn0 9965   ...cfz 10782   #chash 11337
This theorem is referenced by:  erdszelem11  23732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-hash 11338
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