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Theorem isercolllem2 13135
Description: Lemma for isercoll 13137. (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 11470 . . . . . . . 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 5184 . . . . . . . . 9  |-  ( `' G " ( M ... N ) ) 
C_  dom  G
4 isercoll.g . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> Z )
54adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
--> Z )
6 fdm 5558 . . . . . . . . . 10  |-  ( G : NN --> Z  ->  dom  G  =  NN )
75, 6syl 16 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  dom  G  =  NN )
83, 7syl5sseq 3399 . . . . . . . 8  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  NN )
98sseld 3350 . . . . . . 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 10888 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2528 . . . . . . . . . 10  |-  ( x  e.  NN  ->  x  e.  ( ZZ>= `  1 )
)
13 ltso 9447 . . . . . . . . . . . . . 14  |-  <  Or  RR
1413a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  <  Or  RR )
15 fzfid 11787 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( M ... N )  e.  Fin )
16 ffun 5556 . . . . . . . . . . . . . . . . 17  |-  ( G : NN --> Z  ->  Fun  G )
17 funimacnv 5485 . . . . . . . . . . . . . . . . 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 3565 . . . . . . . . . . . . . . . 16  |-  ( ( M ... N )  i^i  ran  G )  C_  ( M ... N
)
2018, 19syl6eqss 3401 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  C_  ( M ... N ) )
21 ssfi 7525 . . . . . . . . . . . . . . 15  |-  ( ( ( M ... N
)  e.  Fin  /\  ( G " ( `' G " ( M ... N ) ) )  C_  ( M ... N ) )  -> 
( G " ( `' G " ( M ... N ) ) )  e.  Fin )
2215, 20, 21syl2anc 661 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  e. 
Fin )
23 ssid 3370 . . . . . . . . . . . . . . . . . . . . 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 13134 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  NN  C_  NN )  ->  ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
2823, 27mpan2 671 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  |`  NN ) 
Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
29 ffn 5554 . . . . . . . . . . . . . . . . . . . . 21  |-  ( G : NN --> Z  ->  G  Fn  NN )
30 fnresdm 5515 . . . . . . . . . . . . . . . . . . . . 21  |-  ( G  Fn  NN  ->  ( G  |`  NN )  =  G )
31 isoeq1 6005 . . . . . . . . . . . . . . . . . . . . 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 6011 . . . . . . . . . . . . . . . . . . 19  |-  ( G 
Isom  <  ,  <  ( NN ,  ( G " NN ) )  ->  G : NN -1-1-onto-> ( G " NN ) )
35 f1ocnv 5648 . . . . . . . . . . . . . . . . . . 19  |-  ( G : NN -1-1-onto-> ( G " NN )  ->  `' G :
( G " NN )
-1-1-onto-> NN )
36 f1ofun 5638 . . . . . . . . . . . . . . . . . . 19  |-  ( `' G : ( G
" NN ) -1-1-onto-> NN  ->  Fun  `' G )
3733, 34, 35, 364syl 21 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Fun  `' G )
38 df-f1 5418 . . . . . . . . . . . . . . . . . 18  |-  ( G : NN -1-1-> Z  <->  ( G : NN --> Z  /\  Fun  `' G ) )
394, 37, 38sylanbrc 664 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G : NN -1-1-> Z
)
4039adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
-1-1-> Z )
41 nnex 10320 . . . . . . . . . . . . . . . . 17  |-  NN  e.  _V
42 ssexg 4433 . . . . . . . . . . . . . . . . 17  |-  ( ( ( `' G "
( M ... N
) )  C_  NN  /\  NN  e.  _V )  ->  ( `' G "
( M ... N
) )  e.  _V )
438, 41, 42sylancl 662 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  e. 
_V )
44 f1imaeng 7361 . . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  ~~  ( `' G " ( M ... N ) ) )
4645ensymd 7352 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  ~~  ( G " ( `' G " ( M ... N ) ) ) )
47 enfii 7522 . . . . . . . . . . . . . 14  |-  ( ( ( G " ( `' G " ( M ... N ) ) )  e.  Fin  /\  ( `' G " ( M ... N ) ) 
~~  ( G "
( `' G "
( M ... N
) ) ) )  ->  ( `' G " ( M ... N
) )  e.  Fin )
4822, 46, 47syl2anc 661 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  e. 
Fin )
49 1nn 10325 . . . . . . . . . . . . . . . 16  |-  1  e.  NN
5049a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  NN )
51 ffvelrn 5836 . . . . . . . . . . . . . . . . . . 19  |-  ( ( G : NN --> Z  /\  1  e.  NN )  ->  ( G `  1
)  e.  Z )
524, 49, 51sylancl 662 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G `  1
)  e.  Z )
5352, 24syl6eleq 2528 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( G `  1
)  e.  ( ZZ>= `  M ) )
5453adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( ZZ>= `  M )
)
55 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  N  e.  ( ZZ>= `  ( G `  1 ) ) )
56 elfzuzb 11439 . . . . . . . . . . . . . . . 16  |-  ( ( G `  1 )  e.  ( M ... N )  <->  ( ( G `  1 )  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  ( G ` 
1 ) ) ) )
5754, 55, 56sylanbrc 664 . . . . . . . . . . . . . . 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 5818 . . . . . . . . . . . . . . . 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 913 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  ( `' G " ( M ... N ) ) )
62 ne0i 3638 . . . . . . . . . . . . . 14  |-  ( 1  e.  ( `' G " ( M ... N
) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
6361, 62syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
64 nnssre 10318 . . . . . . . . . . . . . 14  |-  NN  C_  RR
658, 64syl6ss 3363 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  RR )
66 fisupcl 7709 . . . . . . . . . . . . 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 1220 . . . . . . . . . . . 12  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  ( `' G " ( M ... N ) ) )
688, 67sseldd 3352 . . . . . . . . . . 11  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  NN )
6968nnzd 10738 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  ZZ )
70 elfz5 11437 . . . . . . . . . 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 478 . . . . . . . . 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 5818 . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  ( M ... N ) )
76 elfzle2 11447 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
79 uzssz 10872 . . . . . . . . . . . . . . . . 17  |-  ( ZZ>= `  M )  C_  ZZ
8024, 79eqsstri 3381 . . . . . . . . . . . . . . . 16  |-  Z  C_  ZZ
81 zssre 10645 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
8280, 81sstri 3360 . . . . . . . . . . . . . . 15  |-  Z  C_  RR
835ffvelrnda 5838 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  Z )
8482, 83sseldi 3349 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  RR )
855, 68ffvelrnd 5839 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  Z )
8685adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  Z )
8782, 86sseldi 3349 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  RR )
88 eluzelz 10862 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ( ZZ>= `  ( G `  1 )
)  ->  N  e.  ZZ )
8988ad2antlr 726 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  N  e.  ZZ )
9081, 89sseldi 3349 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  N  e.  RR )
91 letr 9460 . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . 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 674 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
( G `  x
)  <_  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  ->  ( G `  x )  <_  N
) )
9433ad2antrr 725 . . . . . . . . . . . . 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 9419 . . . . . . . . . . . . . 14  |-  RR  C_  RR*
9795, 96syl6ss 3363 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  NN  C_ 
RR* )
98 imassrn 5175 . . . . . . . . . . . . . . . 16  |-  ( G
" NN )  C_  ran  G
994ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  G : NN --> Z )
100 frn 5560 . . . . . . . . . . . . . . . . 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 3362 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  Z )
103102, 82syl6ss 3363 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  RR )
104103, 96syl6ss 3363 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  RR* )
105 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  x  e.  NN )
10668adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN )
107 leisorel 12205 . . . . . . . . . . . . 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 1227 . . . . . . . . . . . 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 2528 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  ( ZZ>= `  M )
)
110 elfz5 11437 . . . . . . . . . . . . 13  |-  ( ( ( G `  x
)  e.  ( ZZ>= `  M )  /\  N  e.  ZZ )  ->  (
( G `  x
)  e.  ( M ... N )  <->  ( G `  x )  <_  N
) )
111109, 89, 110syl2anc 661 . . . . . . . . . . . 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 5818 . . . . . . . . . . . . 13  |-  ( G  Fn  NN  ->  (
x  e.  ( `' G " ( M ... N ) )  <-> 
( x  e.  NN  /\  ( G `  x
)  e.  ( M ... N ) ) ) )
114113baibd 900 . . . . . . . . . . . 12  |-  ( ( G  Fn  NN  /\  x  e.  NN )  ->  ( x  e.  ( `' G " ( M ... N ) )  <-> 
( G `  x
)  e.  ( M ... N ) ) )
11558, 114sylan 471 . . . . . . . . . . 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 10270 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  E. x  e.  RR  A. y  e.  ( `' G "
( M ... N
) ) y  <_  x )
119 suprub 10283 . . . . . . . . . . . . 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 434 . . . . . . . . . . . 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 1218 . . . . . . . . . . 11  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( `' G "
( M ... N
) )  ->  x  <_  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )
122121adantr 465 . . . . . . . . . 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 434 . . . . . . 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 354 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( 1 ... sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
127126eqrdv 2436 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  =  ( `' G " ( M ... N ) ) )
128127fveq2d 5690 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  (
# `  ( `' G " ( M ... N ) ) ) )
12968nnnn0d 10628 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  NN0 )
130 hashfz1 12109 . . . . 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 12110 . . . . . 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 661 . . . . 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 2478 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  =  (
# `  ( G " ( `' G "
( M ... N
) ) ) ) )
136135oveq2d 6102 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  =  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) ) )
137136, 127eqtr3d 2472 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   E.wrex 2711   _Vcvv 2967    i^i cin 3322    C_ wss 3323   (/)c0 3632   class class class wbr 4287    Or wor 4635   `'ccnv 4834   dom cdm 4835   ran crn 4836    |` cres 4837   "cima 4838   Fun wfun 5407    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413    Isom wiso 5414  (class class class)co 6086    ~~ cen 7299   Fincfn 7302   supcsup 7682   RRcr 9273   1c1 9275    + caddc 9277   RR*cxr 9409    < clt 9410    <_ cle 9411   NNcn 10314   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429   #chash 12095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-hash 12096
This theorem is referenced by:  isercolllem3  13136
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