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Theorem erdszelem10 24886
Description: Lemma for erdsze 24888. (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 11311 . . . . . . . 8  |-  ( 1 ... ( R  - 
1 ) )  e. 
Fin
2 fzfi 11311 . . . . . . . 8  |-  ( 1 ... ( S  - 
1 ) )  e. 
Fin
3 xpfi 7378 . . . . . . . 8  |-  ( ( ( 1 ... ( R  -  1 ) )  e.  Fin  /\  ( 1 ... ( S  -  1 ) )  e.  Fin )  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  e.  Fin )
41, 2, 3mp2an 654 . . . . . . 7  |-  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  e. 
Fin
5 ssdomg 7153 . . . . . . 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 7233 . . . . . 6  |-  ( ran 
T  ~<_  ( ( 1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  -.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
86, 7syl 16 . . . . 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 11697 . . . . . . . . . 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 654 . . . . . . . . 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 10261 . . . . . . . . . . 11  |-  ( R  e.  NN  ->  ( R  -  1 )  e.  NN0 )
14 hashfz1 11630 . . . . . . . . . . 11  |-  ( ( R  -  1 )  e.  NN0  ->  ( # `  ( 1 ... ( R  -  1 ) ) )  =  ( R  -  1 ) )
1512, 13, 143syl 19 . . . . . . . . . 10  |-  ( ph  ->  ( # `  (
1 ... ( R  - 
1 ) ) )  =  ( R  - 
1 ) )
16 erdszelem.s . . . . . . . . . . 11  |-  ( ph  ->  S  e.  NN )
17 nnm1nn0 10261 . . . . . . . . . . 11  |-  ( S  e.  NN  ->  ( S  -  1 )  e.  NN0 )
18 hashfz1 11630 . . . . . . . . . . 11  |-  ( ( S  -  1 )  e.  NN0  ->  ( # `  ( 1 ... ( S  -  1 ) ) )  =  ( S  -  1 ) )
1916, 17, 183syl 19 . . . . . . . . . 10  |-  ( ph  ->  ( # `  (
1 ... ( S  - 
1 ) ) )  =  ( S  - 
1 ) )
2015, 19oveq12d 6099 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  (
1 ... ( R  - 
1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) )  =  ( ( R  -  1 )  x.  ( S  - 
1 ) ) )
2111, 20syl5eq 2480 . . . . . . . 8  |-  ( ph  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  =  ( ( R  -  1 )  x.  ( S  - 
1 ) ) )
22 erdsze.n . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN )
2322nnnn0d 10274 . . . . . . . . 9  |-  ( ph  ->  N  e.  NN0 )
24 hashfz1 11630 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
2523, 24syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
269, 21, 253brtr4d 4242 . . . . . . 7  |-  ( ph  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  <  ( # `  ( 1 ... N
) ) )
27 fzfid 11312 . . . . . . . 8  |-  ( ph  ->  ( 1 ... N
)  e.  Fin )
28 hashsdom 11655 . . . . . . . 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 645 . . . . . . 7  |-  ( ph  ->  ( ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  <  ( # `  ( 1 ... N
) )  <->  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ~<  (
1 ... N ) ) )
3026, 29mpbid 202 . . . . . 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 24885 . . . . . . 7  |-  ( ph  ->  T : ( 1 ... N ) -1-1-> ( NN  X.  NN ) )
36 f1f1orn 5685 . . . . . . 7  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  ->  T : ( 1 ... N ) -1-1-onto-> ran  T )
37 ovex 6106 . . . . . . . 8  |-  ( 1 ... N )  e. 
_V
3837f1oen 7128 . . . . . . 7  |-  ( T : ( 1 ... N ) -1-1-onto-> ran  T  ->  (
1 ... N )  ~~  ran  T )
3935, 36, 383syl 19 . . . . . 6  |-  ( ph  ->  ( 1 ... N
)  ~~  ran  T )
40 sdomentr 7241 . . . . . 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 643 . . . . 5  |-  ( ph  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
428, 41nsyl3 113 . . . 4  |-  ( ph  ->  -.  ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
43 nss 3406 . . . . 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 2711 . . . . 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 244 . . . 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 189 . . 3  |-  ( ph  ->  E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )
47 f1fn 5640 . . . 4  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  ->  T  Fn  ( 1 ... N ) )
48 eleq1 2496 . . . . . 6  |-  ( s  =  ( T `  m )  ->  (
s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( T `  m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) ) )
4948notbid 286 . . . . 5  |-  ( s  =  ( T `  m )  ->  ( -.  s  e.  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5049rexrn 5872 . . . 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 19 . . 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 202 . 2  |-  ( ph  ->  E. m  e.  ( 1 ... N )  -.  ( T `  m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )
53 fveq2 5728 . . . . . . . . . 10  |-  ( n  =  m  ->  (
I `  n )  =  ( I `  m ) )
54 fveq2 5728 . . . . . . . . . 10  |-  ( n  =  m  ->  ( J `  n )  =  ( J `  m ) )
5553, 54opeq12d 3992 . . . . . . . . 9  |-  ( n  =  m  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  m ) ,  ( J `  m ) >. )
56 opex 4427 . . . . . . . . 9  |-  <. (
I `  m ) ,  ( J `  m ) >.  e.  _V
5755, 34, 56fvmpt 5806 . . . . . . . 8  |-  ( m  e.  ( 1 ... N )  ->  ( T `  m )  =  <. ( I `  m ) ,  ( J `  m )
>. )
5857adantl 453 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( T `  m )  =  <. ( I `  m ) ,  ( J `  m )
>. )
5958eleq1d 2502 . . . . . 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 4908 . . . . . 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 253 . . . . 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 286 . . . 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 475 . . . 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 253 . . 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 2722 . 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 202 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 177    \/ wo 358    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   E.wrex 2706   {crab 2709    C_ wss 3320   ~Pcpw 3799   <.cop 3817   class class class wbr 4212    e. cmpt 4266    X. cxp 4876   `'ccnv 4877   ran crn 4879    |` cres 4880   "cima 4881    Fn wfn 5449   -1-1->wf1 5451   -1-1-onto->wf1o 5453   ` cfv 5454    Isom wiso 5455  (class class class)co 6081    ~~ cen 7106    ~<_ cdom 7107    ~< csdm 7108   Fincfn 7109   supcsup 7445   RRcr 8989   1c1 8991    x. cmul 8995    < clt 9120    - cmin 9291   NNcn 10000   NN0cn0 10221   ...cfz 11043   #chash 11618
This theorem is referenced by:  erdszelem11  24887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-hash 11619
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