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Theorem isercolllem2 13570
Description: Lemma for isercoll 13572. (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 11717 . . . . . . . 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 5345 . . . . . . . . 9  |-  ( `' G " ( M ... N ) ) 
C_  dom  G
4 isercoll.g . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> Z )
54adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
--> Z )
6 fdm 5717 . . . . . . . . . 10  |-  ( G : NN --> Z  ->  dom  G  =  NN )
75, 6syl 16 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  dom  G  =  NN )
83, 7syl5sseq 3537 . . . . . . . 8  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  NN )
98sseld 3488 . . . . . . 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 11117 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2552 . . . . . . . . . 10  |-  ( x  e.  NN  ->  x  e.  ( ZZ>= `  1 )
)
13 ltso 9654 . . . . . . . . . . . . . 14  |-  <  Or  RR
1413a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  <  Or  RR )
15 fzfid 12065 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( M ... N )  e.  Fin )
16 ffun 5715 . . . . . . . . . . . . . . . . 17  |-  ( G : NN --> Z  ->  Fun  G )
17 funimacnv 5642 . . . . . . . . . . . . . . . . 17  |-  ( Fun 
G  ->  ( G " ( `' G "
( M ... N
) ) )  =  ( ( M ... N )  i^i  ran  G ) )
185, 16, 173syl 20 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  =  ( ( M ... N )  i^i  ran  G ) )
19 inss1 3704 . . . . . . . . . . . . . . . 16  |-  ( ( M ... N )  i^i  ran  G )  C_  ( M ... N
)
2018, 19syl6eqss 3539 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  C_  ( M ... N ) )
21 ssfi 7733 . . . . . . . . . . . . . . 15  |-  ( ( ( M ... N
)  e.  Fin  /\  ( G " ( `' G " ( M ... N ) ) )  C_  ( M ... N ) )  -> 
( G " ( `' G " ( M ... N ) ) )  e.  Fin )
2215, 20, 21syl2anc 659 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  e. 
Fin )
23 ssid 3508 . . . . . . . . . . . . . . . . . . . . 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 13569 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  NN  C_  NN )  ->  ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
2823, 27mpan2 669 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  |`  NN ) 
Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
29 ffn 5713 . . . . . . . . . . . . . . . . . . . . 21  |-  ( G : NN --> Z  ->  G  Fn  NN )
30 fnresdm 5672 . . . . . . . . . . . . . . . . . . . . 21  |-  ( G  Fn  NN  ->  ( G  |`  NN )  =  G )
31 isoeq1 6190 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( G  |`  NN )  =  G  ->  ( ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) )  <->  G  Isom  <  ,  <  ( NN , 
( G " NN ) ) ) )
324, 29, 30, 314syl 21 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) )  <->  G  Isom  <  ,  <  ( NN ,  ( G " NN ) ) ) )
3328, 32mpbid 210 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  G  Isom  <  ,  <  ( NN ,  ( G
" NN ) ) )
34 isof1o 6196 . . . . . . . . . . . . . . . . . . 19  |-  ( G 
Isom  <  ,  <  ( NN ,  ( G " NN ) )  ->  G : NN -1-1-onto-> ( G " NN ) )
35 f1ocnv 5810 . . . . . . . . . . . . . . . . . . 19  |-  ( G : NN -1-1-onto-> ( G " NN )  ->  `' G :
( G " NN )
-1-1-onto-> NN )
36 f1ofun 5800 . . . . . . . . . . . . . . . . . . 19  |-  ( `' G : ( G
" NN ) -1-1-onto-> NN  ->  Fun  `' G )
3733, 34, 35, 364syl 21 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Fun  `' G )
38 df-f1 5575 . . . . . . . . . . . . . . . . . 18  |-  ( G : NN -1-1-> Z  <->  ( G : NN --> Z  /\  Fun  `' G ) )
394, 37, 38sylanbrc 662 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G : NN -1-1-> Z
)
4039adantr 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
-1-1-> Z )
41 nnex 10537 . . . . . . . . . . . . . . . . 17  |-  NN  e.  _V
42 ssexg 4583 . . . . . . . . . . . . . . . . 17  |-  ( ( ( `' G "
( M ... N
) )  C_  NN  /\  NN  e.  _V )  ->  ( `' G "
( M ... N
) )  e.  _V )
438, 41, 42sylancl 660 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  e. 
_V )
44 f1imaeng 7568 . . . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  ~~  ( `' G " ( M ... N ) ) )
4645ensymd 7559 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  ~~  ( G " ( `' G " ( M ... N ) ) ) )
47 enfii 7730 . . . . . . . . . . . . . 14  |-  ( ( ( G " ( `' G " ( M ... N ) ) )  e.  Fin  /\  ( `' G " ( M ... N ) ) 
~~  ( G "
( `' G "
( M ... N
) ) ) )  ->  ( `' G " ( M ... N
) )  e.  Fin )
4822, 46, 47syl2anc 659 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  e. 
Fin )
49 1nn 10542 . . . . . . . . . . . . . . . 16  |-  1  e.  NN
5049a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  NN )
51 ffvelrn 6005 . . . . . . . . . . . . . . . . . . 19  |-  ( ( G : NN --> Z  /\  1  e.  NN )  ->  ( G `  1
)  e.  Z )
524, 49, 51sylancl 660 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G `  1
)  e.  Z )
5352, 24syl6eleq 2552 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( G `  1
)  e.  ( ZZ>= `  M ) )
5453adantr 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( ZZ>= `  M )
)
55 simpr 459 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  N  e.  ( ZZ>= `  ( G `  1 ) ) )
56 elfzuzb 11685 . . . . . . . . . . . . . . . 16  |-  ( ( G `  1 )  e.  ( M ... N )  <->  ( ( G `  1 )  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  ( G ` 
1 ) ) ) )
5754, 55, 56sylanbrc 662 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( M ... N
) )
585, 29syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G  Fn  NN )
59 elpreima 5983 . . . . . . . . . . . . . . . 16  |-  ( G  Fn  NN  ->  (
1  e.  ( `' G " ( M ... N ) )  <-> 
( 1  e.  NN  /\  ( G `  1
)  e.  ( M ... N ) ) ) )
6058, 59syl 16 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1  e.  ( `' G " ( M ... N
) )  <->  ( 1  e.  NN  /\  ( G `  1 )  e.  ( M ... N
) ) ) )
6150, 57, 60mpbir2and 920 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  ( `' G " ( M ... N ) ) )
62 ne0i 3789 . . . . . . . . . . . . . 14  |-  ( 1  e.  ( `' G " ( M ... N
) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
6361, 62syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
64 nnssre 10535 . . . . . . . . . . . . . 14  |-  NN  C_  RR
658, 64syl6ss 3501 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  RR )
66 fisupcl 7919 . . . . . . . . . . . . 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 1228 . . . . . . . . . . . 12  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  ( `' G " ( M ... N ) ) )
688, 67sseldd 3490 . . . . . . . . . . 11  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  NN )
6968nnzd 10964 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  ZZ )
70 elfz5 11683 . . . . . . . . . 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 476 . . . . . . . . 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 5983 . . . . . . . . . . . . . . . . . 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 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 ) ) ) )
7467, 73mpbid 210 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN  /\  ( G `  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  e.  ( M ... N
) ) )
7574simprd 461 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  ( M ... N ) )
76 elfzle2 11693 . . . . . . . . . . . . . . 15  |-  ( ( G `  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  ) )  e.  ( M ... N
)  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
7775, 76syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
7877adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
79 uzssz 11101 . . . . . . . . . . . . . . . . 17  |-  ( ZZ>= `  M )  C_  ZZ
8024, 79eqsstri 3519 . . . . . . . . . . . . . . . 16  |-  Z  C_  ZZ
81 zssre 10867 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
8280, 81sstri 3498 . . . . . . . . . . . . . . 15  |-  Z  C_  RR
835ffvelrnda 6007 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  Z )
8482, 83sseldi 3487 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  RR )
855, 68ffvelrnd 6008 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  Z )
8685adantr 463 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  Z )
8782, 86sseldi 3487 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  RR )
88 eluzelz 11091 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ( ZZ>= `  ( G `  1 )
)  ->  N  e.  ZZ )
8988ad2antlr 724 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  N  e.  ZZ )
9081, 89sseldi 3487 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  N  e.  RR )
91 letr 9667 . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . 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 672 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
( G `  x
)  <_  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  ->  ( G `  x )  <_  N
) )
9433ad2antrr 723 . . . . . . . . . . . . 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 9626 . . . . . . . . . . . . . 14  |-  RR  C_  RR*
9795, 96syl6ss 3501 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  NN  C_ 
RR* )
98 imassrn 5336 . . . . . . . . . . . . . . . 16  |-  ( G
" NN )  C_  ran  G
994ad2antrr 723 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  G : NN --> Z )
100 frn 5719 . . . . . . . . . . . . . . . . 17  |-  ( G : NN --> Z  ->  ran  G  C_  Z )
10199, 100syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ran  G 
C_  Z )
10298, 101syl5ss 3500 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  Z )
103102, 82syl6ss 3501 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  RR )
104103, 96syl6ss 3501 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  RR* )
105 simpr 459 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  x  e.  NN )
10668adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN )
107 leisorel 12493 . . . . . . . . . . . . 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 1235 . . . . . . . . . . . 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 2552 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  ( ZZ>= `  M )
)
110 elfz5 11683 . . . . . . . . . . . . 13  |-  ( ( ( G `  x
)  e.  ( ZZ>= `  M )  /\  N  e.  ZZ )  ->  (
( G `  x
)  e.  ( M ... N )  <->  ( G `  x )  <_  N
) )
111109, 89, 110syl2anc 659 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
( G `  x
)  e.  ( M ... N )  <->  ( G `  x )  <_  N
) )
11293, 108, 1113imtr4d 268 . . . . . . . . . . 11  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  ->  ( G `  x )  e.  ( M ... N
) ) )
113 elpreima 5983 . . . . . . . . . . . . 13  |-  ( G  Fn  NN  ->  (
x  e.  ( `' G " ( M ... N ) )  <-> 
( x  e.  NN  /\  ( G `  x
)  e.  ( M ... N ) ) ) )
114113baibd 907 . . . . . . . . . . . 12  |-  ( ( G  Fn  NN  /\  x  e.  NN )  ->  ( x  e.  ( `' G " ( M ... N ) )  <-> 
( G `  x
)  e.  ( M ... N ) ) )
11558, 114sylan 469 . . . . . . . . . . 11  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( `' G " ( M ... N ) )  <-> 
( G `  x
)  e.  ( M ... N ) ) )
116112, 115sylibrd 234 . . . . . . . . . 10  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  ->  x  e.  ( `' G "
( M ... N
) ) ) )
117 fimaxre2 10486 . . . . . . . . . . . . 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 659 . . . . . . . . . . . 12  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  E. x  e.  RR  A. y  e.  ( `' G "
( M ... N
) ) y  <_  x )
119 suprub 10499 . . . . . . . . . . . . 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 432 . . . . . . . . . . . 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 1226 . . . . . . . . . . 11  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( `' G "
( M ... N
) )  ->  x  <_  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )
122121adantr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( `' G " ( M ... N ) )  ->  x  <_  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) ) )
123116, 122impbid 191 . . . . . . . . 9  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
12471, 123bitrd 253 . . . . . . . 8  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
125124ex 432 . . . . . . 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 352 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( 1 ... sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
127126eqrdv 2451 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  =  ( `' G " ( M ... N ) ) )
128127fveq2d 5852 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  (
# `  ( `' G " ( M ... N ) ) ) )
12968nnnn0d 10848 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  NN0 )
130 hashfz1 12401 . . . . 5  |-  ( sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN0  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )
131129, 130syl 16 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )
132 hashen 12402 . . . . . 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 659 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( # `
 ( `' G " ( M ... N
) ) )  =  ( # `  ( G " ( `' G " ( M ... N
) ) ) )  <-> 
( `' G "
( M ... N
) )  ~~  ( G " ( `' G " ( M ... N
) ) ) ) )
13446, 133mpbird 232 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  ( `' G " ( M ... N ) ) )  =  ( # `  ( G " ( `' G " ( M ... N ) ) ) ) )
135128, 131, 1343eqtr3d 2503 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  =  (
# `  ( G " ( `' G "
( M ... N
) ) ) ) )
136135oveq2d 6286 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  =  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) ) )
137136, 127eqtr3d 2497 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 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106    i^i cin 3460    C_ wss 3461   (/)c0 3783   class class class wbr 4439    Or wor 4788   `'ccnv 4987   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991   Fun wfun 5564    Fn wfn 5565   -->wf 5566   -1-1->wf1 5567   -1-1-onto->wf1o 5569   ` cfv 5570    Isom wiso 5571  (class class class)co 6270    ~~ cen 7506   Fincfn 7509   supcsup 7892   RRcr 9480   1c1 9482    + caddc 9484   RR*cxr 9616    < clt 9617    <_ cle 9618   NNcn 10531   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12388
This theorem is referenced by:  isercolllem3  13571
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