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Theorem isercolllem2 12451
Description: Lemma for isercoll 12453. (Contributed by Mario Carneiro, 6-Apr-2015.)
Hypotheses
Ref Expression
isercoll.z  |-  Z  =  ( ZZ>= `  M )
isercoll.m  |-  ( ph  ->  M  e.  ZZ )
isercoll.g  |-  ( ph  ->  G : NN --> Z )
isercoll.i  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  < 
( G `  (
k  +  1 ) ) )
Assertion
Ref Expression
isercolllem2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) )  =  ( `' G " ( M ... N ) ) )
Distinct variable groups:    k, N    ph, k    k, G    k, M
Allowed substitution hint:    Z( k)

Proof of Theorem isercolllem2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfznn 11072 . . . . . . . 8  |-  ( x  e.  ( 1 ...
sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  ->  x  e.  NN )
21a1i 11 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( 1 ... sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  ->  x  e.  NN )
)
3 cnvimass 5216 . . . . . . . . 9  |-  ( `' G " ( M ... N ) ) 
C_  dom  G
4 isercoll.g . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> Z )
54adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
--> Z )
6 fdm 5587 . . . . . . . . . 10  |-  ( G : NN --> Z  ->  dom  G  =  NN )
75, 6syl 16 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  dom  G  =  NN )
83, 7syl5sseq 3388 . . . . . . . 8  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  NN )
98sseld 3339 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( `' G "
( M ... N
) )  ->  x  e.  NN ) )
10 id 20 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  x  e.  NN )
11 nnuz 10513 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2525 . . . . . . . . . 10  |-  ( x  e.  NN  ->  x  e.  ( ZZ>= `  1 )
)
13 ltso 9148 . . . . . . . . . . . . . 14  |-  <  Or  RR
1413a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  <  Or  RR )
15 fzfid 11304 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( M ... N )  e.  Fin )
16 ffun 5585 . . . . . . . . . . . . . . . . 17  |-  ( G : NN --> Z  ->  Fun  G )
17 funimacnv 5517 . . . . . . . . . . . . . . . . 17  |-  ( Fun 
G  ->  ( G " ( `' G "
( M ... N
) ) )  =  ( ( M ... N )  i^i  ran  G ) )
185, 16, 173syl 19 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  =  ( ( M ... N )  i^i  ran  G ) )
19 inss1 3553 . . . . . . . . . . . . . . . 16  |-  ( ( M ... N )  i^i  ran  G )  C_  ( M ... N
)
2018, 19syl6eqss 3390 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  C_  ( M ... N ) )
21 ssfi 7321 . . . . . . . . . . . . . . 15  |-  ( ( ( M ... N
)  e.  Fin  /\  ( G " ( `' G " ( M ... N ) ) )  C_  ( M ... N ) )  -> 
( G " ( `' G " ( M ... N ) ) )  e.  Fin )
2215, 20, 21syl2anc 643 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  e. 
Fin )
23 ssid 3359 . . . . . . . . . . . . . . . . . . . . . 22  |-  NN  C_  NN
24 isercoll.z . . . . . . . . . . . . . . . . . . . . . . 23  |-  Z  =  ( ZZ>= `  M )
25 isercoll.m . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  M  e.  ZZ )
26 isercoll.i . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  < 
( G `  (
k  +  1 ) ) )
2724, 25, 4, 26isercolllem1 12450 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  NN  C_  NN )  ->  ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
2823, 27mpan2 653 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( G  |`  NN ) 
Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
29 ffn 5583 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( G : NN --> Z  ->  G  Fn  NN )
304, 29syl 16 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  G  Fn  NN )
31 fnresdm 5546 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( G  Fn  NN  ->  ( G  |`  NN )  =  G )
3230, 31syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( G  |`  NN )  =  G )
33 isoeq1 6031 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( G  |`  NN )  =  G  ->  ( ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) )  <->  G  Isom  <  ,  <  ( NN , 
( G " NN ) ) ) )
3432, 33syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) )  <->  G  Isom  <  ,  <  ( NN ,  ( G " NN ) ) ) )
3528, 34mpbid 202 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  G  Isom  <  ,  <  ( NN ,  ( G
" NN ) ) )
36 isof1o 6037 . . . . . . . . . . . . . . . . . . . 20  |-  ( G 
Isom  <  ,  <  ( NN ,  ( G " NN ) )  ->  G : NN -1-1-onto-> ( G " NN ) )
3735, 36syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  G : NN -1-1-onto-> ( G " NN ) )
38 f1ocnv 5679 . . . . . . . . . . . . . . . . . . 19  |-  ( G : NN -1-1-onto-> ( G " NN )  ->  `' G :
( G " NN )
-1-1-onto-> NN )
39 f1ofun 5668 . . . . . . . . . . . . . . . . . . 19  |-  ( `' G : ( G
" NN ) -1-1-onto-> NN  ->  Fun  `' G )
4037, 38, 393syl 19 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Fun  `' G )
41 df-f1 5451 . . . . . . . . . . . . . . . . . 18  |-  ( G : NN -1-1-> Z  <->  ( G : NN --> Z  /\  Fun  `' G ) )
424, 40, 41sylanbrc 646 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G : NN -1-1-> Z
)
4342adantr 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
-1-1-> Z )
44 nnex 9998 . . . . . . . . . . . . . . . . 17  |-  NN  e.  _V
45 ssexg 4341 . . . . . . . . . . . . . . . . 17  |-  ( ( ( `' G "
( M ... N
) )  C_  NN  /\  NN  e.  _V )  ->  ( `' G "
( M ... N
) )  e.  _V )
468, 44, 45sylancl 644 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  e. 
_V )
47 f1imaeng 7159 . . . . . . . . . . . . . . . 16  |-  ( ( G : NN -1-1-> Z  /\  ( `' G "
( M ... N
) )  C_  NN  /\  ( `' G "
( M ... N
) )  e.  _V )  ->  ( G "
( `' G "
( M ... N
) ) )  ~~  ( `' G " ( M ... N ) ) )
4843, 8, 46, 47syl3anc 1184 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  ~~  ( `' G " ( M ... N ) ) )
4948ensymd 7150 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  ~~  ( G " ( `' G " ( M ... N ) ) ) )
50 enfii 7318 . . . . . . . . . . . . . 14  |-  ( ( ( G " ( `' G " ( M ... N ) ) )  e.  Fin  /\  ( `' G " ( M ... N ) ) 
~~  ( G "
( `' G "
( M ... N
) ) ) )  ->  ( `' G " ( M ... N
) )  e.  Fin )
5122, 49, 50syl2anc 643 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  e. 
Fin )
52 1nn 10003 . . . . . . . . . . . . . . . 16  |-  1  e.  NN
5352a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  NN )
54 ffvelrn 5860 . . . . . . . . . . . . . . . . . . 19  |-  ( ( G : NN --> Z  /\  1  e.  NN )  ->  ( G `  1
)  e.  Z )
554, 52, 54sylancl 644 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G `  1
)  e.  Z )
5655, 24syl6eleq 2525 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( G `  1
)  e.  ( ZZ>= `  M ) )
5756adantr 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( ZZ>= `  M )
)
58 simpr 448 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  N  e.  ( ZZ>= `  ( G `  1 ) ) )
59 elfzuzb 11045 . . . . . . . . . . . . . . . 16  |-  ( ( G `  1 )  e.  ( M ... N )  <->  ( ( G `  1 )  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  ( G ` 
1 ) ) ) )
6057, 58, 59sylanbrc 646 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( M ... N
) )
615, 29syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G  Fn  NN )
62 elpreima 5842 . . . . . . . . . . . . . . . 16  |-  ( G  Fn  NN  ->  (
1  e.  ( `' G " ( M ... N ) )  <-> 
( 1  e.  NN  /\  ( G `  1
)  e.  ( M ... N ) ) ) )
6361, 62syl 16 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1  e.  ( `' G " ( M ... N
) )  <->  ( 1  e.  NN  /\  ( G `  1 )  e.  ( M ... N
) ) ) )
6453, 60, 63mpbir2and 889 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  ( `' G " ( M ... N ) ) )
65 ne0i 3626 . . . . . . . . . . . . . 14  |-  ( 1  e.  ( `' G " ( M ... N
) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
6664, 65syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
67 nnssre 9996 . . . . . . . . . . . . . 14  |-  NN  C_  RR
688, 67syl6ss 3352 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  RR )
69 fisupcl 7464 . . . . . . . . . . . . 13  |-  ( (  <  Or  RR  /\  ( ( `' G " ( M ... N
) )  e.  Fin  /\  ( `' G "
( M ... N
) )  =/=  (/)  /\  ( `' G " ( M ... N ) ) 
C_  RR ) )  ->  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  )  e.  ( `' G " ( M ... N
) ) )
7014, 51, 66, 68, 69syl13anc 1186 . . . . . . . . . . . 12  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  ( `' G " ( M ... N ) ) )
718, 70sseldd 3341 . . . . . . . . . . 11  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  NN )
7271nnzd 10366 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  ZZ )
73 elfz5 11043 . . . . . . . . . 10  |-  ( ( x  e.  ( ZZ>= ` 
1 )  /\  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  ZZ )  ->  ( x  e.  ( 1 ... sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  <->  x  <_  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) ) )
7412, 72, 73syl2anr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <->  x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  ) ) )
75 elpreima 5842 . . . . . . . . . . . . . . . . . 18  |-  ( G  Fn  NN  ->  ( sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  ( `' G " ( M ... N ) )  <-> 
( sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  )  e.  NN  /\  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  ( M ... N ) ) ) )
7661, 75syl 16 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  ( `' G " ( M ... N ) )  <-> 
( sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  )  e.  NN  /\  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  ( M ... N ) ) ) )
7770, 76mpbid 202 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN  /\  ( G `  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  e.  ( M ... N
) ) )
7877simprd 450 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  ( M ... N ) )
79 elfzle2 11053 . . . . . . . . . . . . . . 15  |-  ( ( G `  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  ) )  e.  ( M ... N
)  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
8078, 79syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
8180adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
82 uzssz 10497 . . . . . . . . . . . . . . . . 17  |-  ( ZZ>= `  M )  C_  ZZ
8324, 82eqsstri 3370 . . . . . . . . . . . . . . . 16  |-  Z  C_  ZZ
84 zssre 10281 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
8583, 84sstri 3349 . . . . . . . . . . . . . . 15  |-  Z  C_  RR
865ffvelrnda 5862 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  Z )
8785, 86sseldi 3338 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  RR )
885, 71ffvelrnd 5863 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  Z )
8988adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  Z )
9085, 89sseldi 3338 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  RR )
91 eluzelz 10488 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ( ZZ>= `  ( G `  1 )
)  ->  N  e.  ZZ )
9291ad2antlr 708 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  N  e.  ZZ )
9384, 92sseldi 3338 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  N  e.  RR )
94 letr 9159 . . . . . . . . . . . . . 14  |-  ( ( ( G `  x
)  e.  RR  /\  ( G `  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  ) )  e.  RR  /\  N  e.  RR )  ->  (
( ( G `  x )  <_  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  /\  ( G `
 sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )  ->  ( G `  x
)  <_  N )
)
9587, 90, 93, 94syl3anc 1184 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
( ( G `  x )  <_  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  /\  ( G `
 sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )  ->  ( G `  x
)  <_  N )
)
9681, 95mpan2d 656 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
( G `  x
)  <_  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  ->  ( G `  x )  <_  N
) )
9735ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  G  Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
9867a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  NN  C_  RR )
99 ressxr 9121 . . . . . . . . . . . . . 14  |-  RR  C_  RR*
10098, 99syl6ss 3352 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  NN  C_ 
RR* )
101 imassrn 5208 . . . . . . . . . . . . . . . 16  |-  ( G
" NN )  C_  ran  G
1024ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  G : NN --> Z )
103 frn 5589 . . . . . . . . . . . . . . . . 17  |-  ( G : NN --> Z  ->  ran  G  C_  Z )
104102, 103syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ran  G 
C_  Z )
105101, 104syl5ss 3351 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  Z )
106105, 85syl6ss 3352 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  RR )
107106, 99syl6ss 3352 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  RR* )
108 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  x  e.  NN )
10971adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN )
110 leisorel 11701 . . . . . . . . . . . . 13  |-  ( ( G  Isom  <  ,  <  ( NN ,  ( G
" NN ) )  /\  ( NN  C_  RR* 
/\  ( G " NN )  C_  RR* )  /\  ( x  e.  NN  /\ 
sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  )  e.  NN ) )  ->  ( x  <_  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  <->  ( G `  x )  <_  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) ) )
11197, 100, 107, 108, 109, 110syl122anc 1193 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  <->  ( G `  x )  <_  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) ) )
11286, 24syl6eleq 2525 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  ( ZZ>= `  M )
)
113 elfz5 11043 . . . . . . . . . . . . 13  |-  ( ( ( G `  x
)  e.  ( ZZ>= `  M )  /\  N  e.  ZZ )  ->  (
( G `  x
)  e.  ( M ... N )  <->  ( G `  x )  <_  N
) )
114112, 92, 113syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
( G `  x
)  e.  ( M ... N )  <->  ( G `  x )  <_  N
) )
11596, 111, 1143imtr4d 260 . . . . . . . . . . 11  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  ->  ( G `  x )  e.  ( M ... N
) ) )
116 elpreima 5842 . . . . . . . . . . . . 13  |-  ( G  Fn  NN  ->  (
x  e.  ( `' G " ( M ... N ) )  <-> 
( x  e.  NN  /\  ( G `  x
)  e.  ( M ... N ) ) ) )
117116baibd 876 . . . . . . . . . . . 12  |-  ( ( G  Fn  NN  /\  x  e.  NN )  ->  ( x  e.  ( `' G " ( M ... N ) )  <-> 
( G `  x
)  e.  ( M ... N ) ) )
11861, 117sylan 458 . . . . . . . . . . 11  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( `' G " ( M ... N ) )  <-> 
( G `  x
)  e.  ( M ... N ) ) )
119115, 118sylibrd 226 . . . . . . . . . 10  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  ->  x  e.  ( `' G "
( M ... N
) ) ) )
120 fimaxre2 9948 . . . . . . . . . . . . 13  |-  ( ( ( `' G "
( M ... N
) )  C_  RR  /\  ( `' G "
( M ... N
) )  e.  Fin )  ->  E. x  e.  RR  A. y  e.  ( `' G " ( M ... N ) ) y  <_  x )
12168, 51, 120syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  E. x  e.  RR  A. y  e.  ( `' G "
( M ... N
) ) y  <_  x )
122 suprub 9961 . . . . . . . . . . . . 13  |-  ( ( ( ( `' G " ( M ... N
) )  C_  RR  /\  ( `' G "
( M ... N
) )  =/=  (/)  /\  E. x  e.  RR  A. y  e.  ( `' G "
( M ... N
) ) y  <_  x )  /\  x  e.  ( `' G "
( M ... N
) ) )  ->  x  <_  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )
123122ex 424 . . . . . . . . . . . 12  |-  ( ( ( `' G "
( M ... N
) )  C_  RR  /\  ( `' G "
( M ... N
) )  =/=  (/)  /\  E. x  e.  RR  A. y  e.  ( `' G "
( M ... N
) ) y  <_  x )  ->  (
x  e.  ( `' G " ( M ... N ) )  ->  x  <_  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) ) )
12468, 66, 121, 123syl3anc 1184 . . . . . . . . . . 11  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( `' G "
( M ... N
) )  ->  x  <_  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )
125124adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( `' G " ( M ... N ) )  ->  x  <_  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) ) )
126119, 125impbid 184 . . . . . . . . 9  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
12774, 126bitrd 245 . . . . . . . 8  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
128127ex 424 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  NN  ->  ( x  e.  ( 1 ... sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  <->  x  e.  ( `' G " ( M ... N ) ) ) ) )
1292, 9, 128pm5.21ndd 344 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( 1 ... sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
130129eqrdv 2433 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  =  ( `' G " ( M ... N ) ) )
131130fveq2d 5724 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  (
# `  ( `' G " ( M ... N ) ) ) )
13271nnnn0d 10266 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  NN0 )
133 hashfz1 11622 . . . . 5  |-  ( sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN0  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )
134132, 133syl 16 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )
135 hashen 11623 . . . . . 6  |-  ( ( ( `' G "
( M ... N
) )  e.  Fin  /\  ( G " ( `' G " ( M ... N ) ) )  e.  Fin )  ->  ( ( # `  ( `' G " ( M ... N ) ) )  =  ( # `  ( G " ( `' G " ( M ... N ) ) ) )  <->  ( `' G " ( M ... N ) )  ~~  ( G " ( `' G " ( M ... N ) ) ) ) )
13651, 22, 135syl2anc 643 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( # `
 ( `' G " ( M ... N
) ) )  =  ( # `  ( G " ( `' G " ( M ... N
) ) ) )  <-> 
( `' G "
( M ... N
) )  ~~  ( G " ( `' G " ( M ... N
) ) ) ) )
13749, 136mpbird 224 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  ( `' G " ( M ... N ) ) )  =  ( # `  ( G " ( `' G " ( M ... N ) ) ) ) )
138131, 134, 1373eqtr3d 2475 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  =  (
# `  ( G " ( `' G "
( M ... N
) ) ) ) )
139138oveq2d 6089 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  =  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) ) )
140139, 130eqtr3d 2469 1  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) )  =  ( `' G " ( M ... N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620   class class class wbr 4204    Or wor 4494   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440    Fn wfn 5441   -->wf 5442   -1-1->wf1 5443   -1-1-onto->wf1o 5445   ` cfv 5446    Isom wiso 5447  (class class class)co 6073    ~~ cen 7098   Fincfn 7101   supcsup 7437   RRcr 8981   1c1 8983    + caddc 8985   RR*cxr 9111    < clt 9112    <_ cle 9113   NNcn 9992   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035   #chash 11610
This theorem is referenced by:  isercolllem3  12452
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611
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