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Theorem erdszelem11 28313
Description: Lemma for erdsze 28314. (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 28312 . . 3  |-  ( ph  ->  E. m  e.  ( 1 ... N ) ( -.  ( I `
 m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
101adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  N  e.  NN )
112adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  F : ( 1 ... N ) -1-1-> RR )
12 ltso 9665 . . . . . . 7  |-  <  Or  RR
13 simprl 755 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  m  e.  ( 1 ... N ) )
146adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) ) ) )  ->  R  e.  NN )
15 simprr 756 . . . . . . 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 28309 . . . . . 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 615 . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  N  e.  NN )
192adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  F : ( 1 ... N ) -1-1-> RR )
20 cnvso 5546 . . . . . . . 8  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
2112, 20mpbi 208 . . . . . . 7  |-  `'  <  Or  RR
22 simprl 755 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  m  e.  ( 1 ... N ) )
237adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( 1 ... N
)  /\  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )  ->  S  e.  NN )
24 simprr 756 . . . . . . 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 28309 . . . . . 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 615 . . . . 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 836 . . . 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 2955 . . 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 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
) ) ) ) )
30 r19.43 3017 . 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 212 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 368    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818   ~Pcpw 4010   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    Or wor 4799   `'ccnv 4998    |` cres 5001   "cima 5002   -1-1->wf1 5585   ` cfv 5588    Isom wiso 5589  (class class class)co 6284   supcsup 7900   RRcr 9491   1c1 9493    x. cmul 9497    < clt 9628    <_ cle 9629    - cmin 9805   NNcn 10536   ...cfz 11672   #chash 12373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-hash 12374
This theorem is referenced by:  erdsze  28314
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