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Theorem isercolllem2 13806
Description: Lemma for isercoll 13808. (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 11854 . . . . . . . 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 5194 . . . . . . . . 9  |-  ( `' G " ( M ... N ) ) 
C_  dom  G
4 isercoll.g . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> Z )
54adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
--> Z )
6 fdm 5745 . . . . . . . . . 10  |-  ( G : NN --> Z  ->  dom  G  =  NN )
75, 6syl 17 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  dom  G  =  NN )
83, 7syl5sseq 3466 . . . . . . . 8  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  NN )
98sseld 3417 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( `' G "
( M ... N
) )  ->  x  e.  NN ) )
10 id 22 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  x  e.  NN )
11 nnuz 11218 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2559 . . . . . . . . . 10  |-  ( x  e.  NN  ->  x  e.  ( ZZ>= `  1 )
)
13 ltso 9732 . . . . . . . . . . . . . 14  |-  <  Or  RR
1413a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  <  Or  RR )
15 fzfid 12224 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( M ... N )  e.  Fin )
16 ffun 5742 . . . . . . . . . . . . . . . . 17  |-  ( G : NN --> Z  ->  Fun  G )
17 funimacnv 5665 . . . . . . . . . . . . . . . . 17  |-  ( Fun 
G  ->  ( G " ( `' G "
( M ... N
) ) )  =  ( ( M ... N )  i^i  ran  G ) )
185, 16, 173syl 18 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  =  ( ( M ... N )  i^i  ran  G ) )
19 inss1 3643 . . . . . . . . . . . . . . . 16  |-  ( ( M ... N )  i^i  ran  G )  C_  ( M ... N
)
2018, 19syl6eqss 3468 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  C_  ( M ... N ) )
21 ssfi 7810 . . . . . . . . . . . . . . 15  |-  ( ( ( M ... N
)  e.  Fin  /\  ( G " ( `' G " ( M ... N ) ) )  C_  ( M ... N ) )  -> 
( G " ( `' G " ( M ... N ) ) )  e.  Fin )
2215, 20, 21syl2anc 673 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  e. 
Fin )
23 ssid 3437 . . . . . . . . . . . . . . . . . . . . 21  |-  NN  C_  NN
24 isercoll.z . . . . . . . . . . . . . . . . . . . . . 22  |-  Z  =  ( ZZ>= `  M )
25 isercoll.m . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  M  e.  ZZ )
26 isercoll.i . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  < 
( G `  (
k  +  1 ) ) )
2724, 25, 4, 26isercolllem1 13805 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  NN  C_  NN )  ->  ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
2823, 27mpan2 685 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  |`  NN ) 
Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
29 ffn 5739 . . . . . . . . . . . . . . . . . . . . 21  |-  ( G : NN --> Z  ->  G  Fn  NN )
30 fnresdm 5695 . . . . . . . . . . . . . . . . . . . . 21  |-  ( G  Fn  NN  ->  ( G  |`  NN )  =  G )
31 isoeq1 6228 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( G  |`  NN )  =  G  ->  ( ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) )  <->  G  Isom  <  ,  <  ( NN , 
( G " NN ) ) ) )
324, 29, 30, 314syl 19 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) )  <->  G  Isom  <  ,  <  ( NN ,  ( G " NN ) ) ) )
3328, 32mpbid 215 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  G  Isom  <  ,  <  ( NN ,  ( G
" NN ) ) )
34 isof1o 6234 . . . . . . . . . . . . . . . . . . 19  |-  ( G 
Isom  <  ,  <  ( NN ,  ( G " NN ) )  ->  G : NN -1-1-onto-> ( G " NN ) )
35 f1ocnv 5840 . . . . . . . . . . . . . . . . . . 19  |-  ( G : NN -1-1-onto-> ( G " NN )  ->  `' G :
( G " NN )
-1-1-onto-> NN )
36 f1ofun 5830 . . . . . . . . . . . . . . . . . . 19  |-  ( `' G : ( G
" NN ) -1-1-onto-> NN  ->  Fun  `' G )
3733, 34, 35, 364syl 19 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Fun  `' G )
38 df-f1 5594 . . . . . . . . . . . . . . . . . 18  |-  ( G : NN -1-1-> Z  <->  ( G : NN --> Z  /\  Fun  `' G ) )
394, 37, 38sylanbrc 677 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G : NN -1-1-> Z
)
4039adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
-1-1-> Z )
41 nnex 10637 . . . . . . . . . . . . . . . . 17  |-  NN  e.  _V
42 ssexg 4542 . . . . . . . . . . . . . . . . 17  |-  ( ( ( `' G "
( M ... N
) )  C_  NN  /\  NN  e.  _V )  ->  ( `' G "
( M ... N
) )  e.  _V )
438, 41, 42sylancl 675 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  e. 
_V )
44 f1imaeng 7647 . . . . . . . . . . . . . . . 16  |-  ( ( G : NN -1-1-> Z  /\  ( `' G "
( M ... N
) )  C_  NN  /\  ( `' G "
( M ... N
) )  e.  _V )  ->  ( G "
( `' G "
( M ... N
) ) )  ~~  ( `' G " ( M ... N ) ) )
4540, 8, 43, 44syl3anc 1292 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  ~~  ( `' G " ( M ... N ) ) )
4645ensymd 7638 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  ~~  ( G " ( `' G " ( M ... N ) ) ) )
47 enfii 7807 . . . . . . . . . . . . . 14  |-  ( ( ( G " ( `' G " ( M ... N ) ) )  e.  Fin  /\  ( `' G " ( M ... N ) ) 
~~  ( G "
( `' G "
( M ... N
) ) ) )  ->  ( `' G " ( M ... N
) )  e.  Fin )
4822, 46, 47syl2anc 673 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  e. 
Fin )
49 1nn 10642 . . . . . . . . . . . . . . . 16  |-  1  e.  NN
5049a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  NN )
51 ffvelrn 6035 . . . . . . . . . . . . . . . . . . 19  |-  ( ( G : NN --> Z  /\  1  e.  NN )  ->  ( G `  1
)  e.  Z )
524, 49, 51sylancl 675 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G `  1
)  e.  Z )
5352, 24syl6eleq 2559 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( G `  1
)  e.  ( ZZ>= `  M ) )
5453adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( ZZ>= `  M )
)
55 simpr 468 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  N  e.  ( ZZ>= `  ( G `  1 ) ) )
56 elfzuzb 11820 . . . . . . . . . . . . . . . 16  |-  ( ( G `  1 )  e.  ( M ... N )  <->  ( ( G `  1 )  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  ( G ` 
1 ) ) ) )
5754, 55, 56sylanbrc 677 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( M ... N
) )
585, 29syl 17 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G  Fn  NN )
59 elpreima 6017 . . . . . . . . . . . . . . . 16  |-  ( G  Fn  NN  ->  (
1  e.  ( `' G " ( M ... N ) )  <-> 
( 1  e.  NN  /\  ( G `  1
)  e.  ( M ... N ) ) ) )
6058, 59syl 17 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1  e.  ( `' G " ( M ... N
) )  <->  ( 1  e.  NN  /\  ( G `  1 )  e.  ( M ... N
) ) ) )
6150, 57, 60mpbir2and 936 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  ( `' G " ( M ... N ) ) )
62 ne0i 3728 . . . . . . . . . . . . . 14  |-  ( 1  e.  ( `' G " ( M ... N
) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
6361, 62syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
64 nnssre 10635 . . . . . . . . . . . . . 14  |-  NN  C_  RR
658, 64syl6ss 3430 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  RR )
66 fisupcl 8003 . . . . . . . . . . . . 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
) ) )
6714, 48, 63, 65, 66syl13anc 1294 . . . . . . . . . . . 12  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  ( `' G " ( M ... N ) ) )
688, 67sseldd 3419 . . . . . . . . . . 11  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  NN )
6968nnzd 11062 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  ZZ )
70 elfz5 11818 . . . . . . . . . 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 ,  <  ) ) )
7112, 69, 70syl2anr 486 . . . . . . . . 9  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <->  x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  ) ) )
72 elpreima 6017 . . . . . . . . . . . . . . . . . 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 ) ) ) )
7358, 72syl 17 . . . . . . . . . . . . . . . . 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 ) ) ) )
7467, 73mpbid 215 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN  /\  ( G `  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  e.  ( M ... N
) ) )
7574simprd 470 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  ( M ... N ) )
76 elfzle2 11829 . . . . . . . . . . . . . . 15  |-  ( ( G `  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  ) )  e.  ( M ... N
)  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
7775, 76syl 17 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
7877adantr 472 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
79 uzssz 11202 . . . . . . . . . . . . . . . . 17  |-  ( ZZ>= `  M )  C_  ZZ
8024, 79eqsstri 3448 . . . . . . . . . . . . . . . 16  |-  Z  C_  ZZ
81 zssre 10968 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
8280, 81sstri 3427 . . . . . . . . . . . . . . 15  |-  Z  C_  RR
835ffvelrnda 6037 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  Z )
8482, 83sseldi 3416 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  RR )
855, 68ffvelrnd 6038 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  Z )
8685adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  Z )
8782, 86sseldi 3416 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  RR )
88 eluzelz 11192 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ( ZZ>= `  ( G `  1 )
)  ->  N  e.  ZZ )
8988ad2antlr 741 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  N  e.  ZZ )
9081, 89sseldi 3416 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  N  e.  RR )
91 letr 9745 . . . . . . . . . . . . . 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 )
)
9284, 87, 90, 91syl3anc 1292 . . . . . . . . . . . . 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 )
)
9378, 92mpan2d 688 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
( G `  x
)  <_  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  ->  ( G `  x )  <_  N
) )
9433ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  G  Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
9564a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  NN  C_  RR )
96 ressxr 9702 . . . . . . . . . . . . . 14  |-  RR  C_  RR*
9795, 96syl6ss 3430 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  NN  C_ 
RR* )
98 imassrn 5185 . . . . . . . . . . . . . . . 16  |-  ( G
" NN )  C_  ran  G
994ad2antrr 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  G : NN --> Z )
100 frn 5747 . . . . . . . . . . . . . . . . 17  |-  ( G : NN --> Z  ->  ran  G  C_  Z )
10199, 100syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ran  G 
C_  Z )
10298, 101syl5ss 3429 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  Z )
103102, 82syl6ss 3430 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  RR )
104103, 96syl6ss 3430 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  RR* )
105 simpr 468 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  x  e.  NN )
10668adantr 472 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN )
107 leisorel 12664 . . . . . . . . . . . . 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 ,  <  ) ) ) )
10894, 97, 104, 105, 106, 107syl122anc 1301 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  <->  ( G `  x )  <_  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) ) )
10983, 24syl6eleq 2559 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  ( ZZ>= `  M )
)
110 elfz5 11818 . . . . . . . . . . . . 13  |-  ( ( ( G `  x
)  e.  ( ZZ>= `  M )  /\  N  e.  ZZ )  ->  (
( G `  x
)  e.  ( M ... N )  <->  ( G `  x )  <_  N
) )
111109, 89, 110syl2anc 673 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
( G `  x
)  e.  ( M ... N )  <->  ( G `  x )  <_  N
) )
11293, 108, 1113imtr4d 276 . . . . . . . . . . 11  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  ->  ( G `  x )  e.  ( M ... N
) ) )
113 elpreima 6017 . . . . . . . . . . . . 13  |-  ( G  Fn  NN  ->  (
x  e.  ( `' G " ( M ... N ) )  <-> 
( x  e.  NN  /\  ( G `  x
)  e.  ( M ... N ) ) ) )
114113baibd 923 . . . . . . . . . . . 12  |-  ( ( G  Fn  NN  /\  x  e.  NN )  ->  ( x  e.  ( `' G " ( M ... N ) )  <-> 
( G `  x
)  e.  ( M ... N ) ) )
11558, 114sylan 479 . . . . . . . . . . 11  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( `' G " ( M ... N ) )  <-> 
( G `  x
)  e.  ( M ... N ) ) )
116112, 115sylibrd 242 . . . . . . . . . 10  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  ->  x  e.  ( `' G "
( M ... N
) ) ) )
117 fimaxre2 10574 . . . . . . . . . . . . 13  |-  ( ( ( `' G "
( M ... N
) )  C_  RR  /\  ( `' G "
( M ... N
) )  e.  Fin )  ->  E. x  e.  RR  A. y  e.  ( `' G " ( M ... N ) ) y  <_  x )
11865, 48, 117syl2anc 673 . . . . . . . . . . . 12  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  E. x  e.  RR  A. y  e.  ( `' G "
( M ... N
) ) y  <_  x )
119 suprub 10592 . . . . . . . . . . . . 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 ,  <  ) )
120119ex 441 . . . . . . . . . . . 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 ,  <  ) ) )
12165, 63, 118, 120syl3anc 1292 . . . . . . . . . . 11  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( `' G "
( M ... N
) )  ->  x  <_  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )
122121adantr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( `' G " ( M ... N ) )  ->  x  <_  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) ) )
123116, 122impbid 195 . . . . . . . . 9  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
12471, 123bitrd 261 . . . . . . . 8  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
125124ex 441 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  NN  ->  ( x  e.  ( 1 ... sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  <->  x  e.  ( `' G " ( M ... N ) ) ) ) )
1262, 9, 125pm5.21ndd 361 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( 1 ... sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
127126eqrdv 2469 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  =  ( `' G " ( M ... N ) ) )
128127fveq2d 5883 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  (
# `  ( `' G " ( M ... N ) ) ) )
12968nnnn0d 10949 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  NN0 )
130 hashfz1 12567 . . . . 5  |-  ( sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN0  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )
131129, 130syl 17 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )
132 hashen 12568 . . . . . 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 ) ) ) ) )
13348, 22, 132syl2anc 673 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( # `
 ( `' G " ( M ... N
) ) )  =  ( # `  ( G " ( `' G " ( M ... N
) ) ) )  <-> 
( `' G "
( M ... N
) )  ~~  ( G " ( `' G " ( M ... N
) ) ) ) )
13446, 133mpbird 240 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  ( `' G " ( M ... N ) ) )  =  ( # `  ( G " ( `' G " ( M ... N ) ) ) ) )
135128, 131, 1343eqtr3d 2513 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  =  (
# `  ( G " ( `' G "
( M ... N
) ) ) ) )
136135oveq2d 6324 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  =  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) ) )
137136, 127eqtr3d 2507 1  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) )  =  ( `' G " ( M ... N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031    i^i cin 3389    C_ wss 3390   (/)c0 3722   class class class wbr 4395    Or wor 4759   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Fun wfun 5583    Fn wfn 5584   -->wf 5585   -1-1->wf1 5586   -1-1-onto->wf1o 5588   ` cfv 5589    Isom wiso 5590  (class class class)co 6308    ~~ cen 7584   Fincfn 7587   supcsup 7972   RRcr 9556   1c1 9558    + caddc 9560   RR*cxr 9692    < clt 9693    <_ cle 9694   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   #chash 12553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554
This theorem is referenced by:  isercolllem3  13807
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