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Theorem erdszelem11 29712
Description: Lemma for erdsze 29713. (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 29711 . . 3  |-  ( ph  ->  E. m  e.  ( 1 ... N ) ( -.  ( I `
 m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
101adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  N  e.  NN )
112adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  F : ( 1 ... N ) -1-1-> RR )
12 ltso 9713 . . . . . . 7  |-  <  Or  RR
13 simprl 762 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  m  e.  ( 1 ... N ) )
146adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  R  e.  NN )
15 simprr 764 . . . . . . 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 29708 . . . . . 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 618 . . . . 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 466 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  N  e.  NN )
192adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  F : ( 1 ... N ) -1-1-> RR )
20 gtso 9714 . . . . . . 7  |-  `'  <  Or  RR
21 simprl 762 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  m  e.  ( 1 ... N ) )
227adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  S  e.  NN )
23 simprr 764 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  -.  ( J `  m
)  e.  ( 1 ... ( S  - 
1 ) ) )
2418, 19, 4, 20, 21, 22, 23erdszelem7 29708 . . . . . 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
) ) ) )
2524expr 618 . . . . 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
) ) ) ) )
2617, 25orim12d 846 . . . 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
) ) ) ) ) )
2726rexlimdva 2924 . . 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
) ) ) ) ) )
289, 27mpd 15 . 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
) ) ) ) )
29 r19.43 2991 . 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
) ) ) ) )
3028, 29sylibr 215 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783   {crab 2786   ~Pcpw 3985   <.cop 4008   class class class wbr 4426    |-> cmpt 4484   `'ccnv 4853    |` cres 4856   "cima 4857   -1-1->wf1 5598   ` cfv 5601    Isom wiso 5602  (class class class)co 6305   supcsup 7960   RRcr 9537   1c1 9539    x. cmul 9543    < clt 9674    <_ cle 9675    - cmin 9859   NNcn 10609   ...cfz 11782   #chash 12512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-hash 12513
This theorem is referenced by:  erdsze  29713
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