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Theorem isercolllem2 12155
Description: Lemma for isercoll 12157. (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 10835 . . . . . . . 8  |-  ( x  e.  ( 1 ...
sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  ->  x  e.  NN )
21a1i 10 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( 1 ... sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  ->  x  e.  NN )
)
3 cnvimass 5049 . . . . . . . . 9  |-  ( `' G " ( M ... N ) ) 
C_  dom  G
4 isercoll.g . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> Z )
54adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
--> Z )
6 fdm 5409 . . . . . . . . . 10  |-  ( G : NN --> Z  ->  dom  G  =  NN )
75, 6syl 15 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  dom  G  =  NN )
83, 7syl5sseq 3239 . . . . . . . 8  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  NN )
98sseld 3192 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( `' G "
( M ... N
) )  ->  x  e.  NN ) )
10 id 19 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  x  e.  NN )
11 nnuz 10279 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2386 . . . . . . . . . 10  |-  ( x  e.  NN  ->  x  e.  ( ZZ>= `  1 )
)
13 ltso 8919 . . . . . . . . . . . . . 14  |-  <  Or  RR
1413a1i 10 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  <  Or  RR )
15 fzfid 11051 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( M ... N )  e.  Fin )
16 ffun 5407 . . . . . . . . . . . . . . . . 17  |-  ( G : NN --> Z  ->  Fun  G )
17 funimacnv 5340 . . . . . . . . . . . . . . . . 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 3402 . . . . . . . . . . . . . . . . 17  |-  ( ( M ... N )  i^i  ran  G )  C_  ( M ... N
)
2019a1i 10 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  i^i 
ran  G )  C_  ( M ... N ) )
2118, 20eqsstrd 3225 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  C_  ( M ... N ) )
22 ssfi 7099 . . . . . . . . . . . . . . 15  |-  ( ( ( M ... N
)  e.  Fin  /\  ( G " ( `' G " ( M ... N ) ) )  C_  ( M ... N ) )  -> 
( G " ( `' G " ( M ... N ) ) )  e.  Fin )
2315, 21, 22syl2anc 642 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  e. 
Fin )
24 ssid 3210 . . . . . . . . . . . . . . . . . . . . . 22  |-  NN  C_  NN
25 isercoll.z . . . . . . . . . . . . . . . . . . . . . . 23  |-  Z  =  ( ZZ>= `  M )
26 isercoll.m . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  M  e.  ZZ )
27 isercoll.i . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  < 
( G `  (
k  +  1 ) ) )
2825, 26, 4, 27isercolllem1 12154 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  NN  C_  NN )  ->  ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
2924, 28mpan2 652 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( G  |`  NN ) 
Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
30 ffn 5405 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( G : NN --> Z  ->  G  Fn  NN )
314, 30syl 15 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  G  Fn  NN )
32 fnresdm 5369 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( G  Fn  NN  ->  ( G  |`  NN )  =  G )
3331, 32syl 15 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( G  |`  NN )  =  G )
34 isoeq1 5832 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( G  |`  NN )  =  G  ->  ( ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) )  <->  G  Isom  <  ,  <  ( NN , 
( G " NN ) ) ) )
3533, 34syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( G  |`  NN )  Isom  <  ,  <  ( NN ,  ( G " NN ) )  <->  G  Isom  <  ,  <  ( NN ,  ( G " NN ) ) ) )
3629, 35mpbid 201 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  G  Isom  <  ,  <  ( NN ,  ( G
" NN ) ) )
37 isof1o 5838 . . . . . . . . . . . . . . . . . . . 20  |-  ( G 
Isom  <  ,  <  ( NN ,  ( G " NN ) )  ->  G : NN -1-1-onto-> ( G " NN ) )
3836, 37syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  G : NN -1-1-onto-> ( G " NN ) )
39 f1ocnv 5501 . . . . . . . . . . . . . . . . . . 19  |-  ( G : NN -1-1-onto-> ( G " NN )  ->  `' G :
( G " NN )
-1-1-onto-> NN )
40 f1ofun 5490 . . . . . . . . . . . . . . . . . . 19  |-  ( `' G : ( G
" NN ) -1-1-onto-> NN  ->  Fun  `' G )
4138, 39, 403syl 18 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Fun  `' G )
42 df-f1 5276 . . . . . . . . . . . . . . . . . 18  |-  ( G : NN -1-1-> Z  <->  ( G : NN --> Z  /\  Fun  `' G ) )
434, 41, 42sylanbrc 645 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G : NN -1-1-> Z
)
4443adantr 451 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
-1-1-> Z )
45 nnex 9768 . . . . . . . . . . . . . . . . 17  |-  NN  e.  _V
46 ssexg 4176 . . . . . . . . . . . . . . . . 17  |-  ( ( ( `' G "
( M ... N
) )  C_  NN  /\  NN  e.  _V )  ->  ( `' G "
( M ... N
) )  e.  _V )
478, 45, 46sylancl 643 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  e. 
_V )
48 f1imaeng 6937 . . . . . . . . . . . . . . . 16  |-  ( ( G : NN -1-1-> Z  /\  ( `' G "
( M ... N
) )  C_  NN  /\  ( `' G "
( M ... N
) )  e.  _V )  ->  ( G "
( `' G "
( M ... N
) ) )  ~~  ( `' G " ( M ... N ) ) )
4944, 8, 47, 48syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  ~~  ( `' G " ( M ... N ) ) )
50 ensym 6926 . . . . . . . . . . . . . . 15  |-  ( ( G " ( `' G " ( M ... N ) ) )  ~~  ( `' G " ( M ... N ) )  ->  ( `' G " ( M ... N
) )  ~~  ( G " ( `' G " ( M ... N
) ) ) )
5149, 50syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  ~~  ( G " ( `' G " ( M ... N ) ) ) )
52 enfii 7096 . . . . . . . . . . . . . 14  |-  ( ( ( G " ( `' G " ( M ... N ) ) )  e.  Fin  /\  ( `' G " ( M ... N ) ) 
~~  ( G "
( `' G "
( M ... N
) ) ) )  ->  ( `' G " ( M ... N
) )  e.  Fin )
5323, 51, 52syl2anc 642 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  e. 
Fin )
54 1nn 9773 . . . . . . . . . . . . . . . 16  |-  1  e.  NN
5554a1i 10 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  NN )
56 ffvelrn 5679 . . . . . . . . . . . . . . . . . . 19  |-  ( ( G : NN --> Z  /\  1  e.  NN )  ->  ( G `  1
)  e.  Z )
574, 54, 56sylancl 643 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G `  1
)  e.  Z )
5857, 25syl6eleq 2386 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( G `  1
)  e.  ( ZZ>= `  M ) )
5958adantr 451 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( ZZ>= `  M )
)
60 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  N  e.  ( ZZ>= `  ( G `  1 ) ) )
61 elfzuzb 10808 . . . . . . . . . . . . . . . 16  |-  ( ( G `  1 )  e.  ( M ... N )  <->  ( ( G `  1 )  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  ( G ` 
1 ) ) ) )
6259, 60, 61sylanbrc 645 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( M ... N
) )
635, 30syl 15 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G  Fn  NN )
64 elpreima 5661 . . . . . . . . . . . . . . . 16  |-  ( G  Fn  NN  ->  (
1  e.  ( `' G " ( M ... N ) )  <-> 
( 1  e.  NN  /\  ( G `  1
)  e.  ( M ... N ) ) ) )
6563, 64syl 15 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1  e.  ( `' G " ( M ... N
) )  <->  ( 1  e.  NN  /\  ( G `  1 )  e.  ( M ... N
) ) ) )
6655, 62, 65mpbir2and 888 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  ( `' G " ( M ... N ) ) )
67 ne0i 3474 . . . . . . . . . . . . . 14  |-  ( 1  e.  ( `' G " ( M ... N
) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
6866, 67syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
69 nnssre 9766 . . . . . . . . . . . . . 14  |-  NN  C_  RR
708, 69syl6ss 3204 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  RR )
71 fisupcl 7234 . . . . . . . . . . . . 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
) ) )
7214, 53, 68, 70, 71syl13anc 1184 . . . . . . . . . . . 12  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  ( `' G " ( M ... N ) ) )
738, 72sseldd 3194 . . . . . . . . . . 11  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  NN )
7473nnzd 10132 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  ZZ )
75 elfz5 10806 . . . . . . . . . 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 ,  <  ) ) )
7612, 74, 75syl2anr 464 . . . . . . . . 9  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <->  x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  ) ) )
77 elpreima 5661 . . . . . . . . . . . . . . . . . 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 ) ) ) )
7863, 77syl 15 . . . . . . . . . . . . . . . . 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 ) ) ) )
7972, 78mpbid 201 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN  /\  ( G `  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  e.  ( M ... N
) ) )
8079simprd 449 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  ( M ... N ) )
81 elfzle2 10816 . . . . . . . . . . . . . . 15  |-  ( ( G `  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  ) )  e.  ( M ... N
)  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
8280, 81syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
8382adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <_  N )
84 uzssz 10263 . . . . . . . . . . . . . . . . 17  |-  ( ZZ>= `  M )  C_  ZZ
8525, 84eqsstri 3221 . . . . . . . . . . . . . . . 16  |-  Z  C_  ZZ
86 zssre 10047 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
8785, 86sstri 3201 . . . . . . . . . . . . . . 15  |-  Z  C_  RR
88 ffvelrn 5679 . . . . . . . . . . . . . . . 16  |-  ( ( G : NN --> Z  /\  x  e.  NN )  ->  ( G `  x
)  e.  Z )
895, 88sylan 457 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  Z )
9087, 89sseldi 3191 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  RR )
91 ffvelrn 5679 . . . . . . . . . . . . . . . . 17  |-  ( ( G : NN --> Z  /\  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  e.  Z )
925, 73, 91syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  Z )
9392adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  Z )
9487, 93sseldi 3191 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  e.  RR )
95 eluzelz 10254 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ( ZZ>= `  ( G `  1 )
)  ->  N  e.  ZZ )
9695ad2antlr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  N  e.  ZZ )
9786, 96sseldi 3191 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  N  e.  RR )
98 letr 8930 . . . . . . . . . . . . . 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 )
)
9990, 94, 97, 98syl3anc 1182 . . . . . . . . . . . . 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 )
)
10083, 99mpan2d 655 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
( G `  x
)  <_  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  ->  ( G `  x )  <_  N
) )
10136ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  G  Isom  <  ,  <  ( NN ,  ( G " NN ) ) )
10269a1i 10 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  NN  C_  RR )
103 ressxr 8892 . . . . . . . . . . . . . 14  |-  RR  C_  RR*
104102, 103syl6ss 3204 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  NN  C_ 
RR* )
105 imassrn 5041 . . . . . . . . . . . . . . . 16  |-  ( G
" NN )  C_  ran  G
1064ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  G : NN --> Z )
107 frn 5411 . . . . . . . . . . . . . . . . 17  |-  ( G : NN --> Z  ->  ran  G  C_  Z )
108106, 107syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ran  G 
C_  Z )
109105, 108syl5ss 3203 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  Z )
110109, 87syl6ss 3204 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  RR )
111110, 103syl6ss 3204 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G " NN )  C_  RR* )
112 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  x  e.  NN )
11373adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN )
114 leisorel 11414 . . . . . . . . . . . . 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 ,  <  ) ) ) )
115101, 104, 111, 112, 113, 114syl122anc 1191 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  <->  ( G `  x )  <_  ( G `  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) ) )
11689, 25syl6eleq 2386 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  ( G `  x )  e.  ( ZZ>= `  M )
)
117 elfz5 10806 . . . . . . . . . . . . 13  |-  ( ( ( G `  x
)  e.  ( ZZ>= `  M )  /\  N  e.  ZZ )  ->  (
( G `  x
)  e.  ( M ... N )  <->  ( G `  x )  <_  N
) )
118116, 96, 117syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
( G `  x
)  e.  ( M ... N )  <->  ( G `  x )  <_  N
) )
119100, 115, 1183imtr4d 259 . . . . . . . . . . 11  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  ->  ( G `  x )  e.  ( M ... N
) ) )
120 elpreima 5661 . . . . . . . . . . . . 13  |-  ( G  Fn  NN  ->  (
x  e.  ( `' G " ( M ... N ) )  <-> 
( x  e.  NN  /\  ( G `  x
)  e.  ( M ... N ) ) ) )
121120baibd 875 . . . . . . . . . . . 12  |-  ( ( G  Fn  NN  /\  x  e.  NN )  ->  ( x  e.  ( `' G " ( M ... N ) )  <-> 
( G `  x
)  e.  ( M ... N ) ) )
12263, 121sylan 457 . . . . . . . . . . 11  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( `' G " ( M ... N ) )  <-> 
( G `  x
)  e.  ( M ... N ) ) )
123119, 122sylibrd 225 . . . . . . . . . 10  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  ->  x  e.  ( `' G "
( M ... N
) ) ) )
124 fimaxre2 9718 . . . . . . . . . . . . 13  |-  ( ( ( `' G "
( M ... N
) )  C_  RR  /\  ( `' G "
( M ... N
) )  e.  Fin )  ->  E. x  e.  RR  A. y  e.  ( `' G " ( M ... N ) ) y  <_  x )
12570, 53, 124syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  E. x  e.  RR  A. y  e.  ( `' G "
( M ... N
) ) y  <_  x )
126 suprub 9731 . . . . . . . . . . . . 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 ,  <  ) )
127126ex 423 . . . . . . . . . . . 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 ,  <  ) ) )
12870, 68, 125, 127syl3anc 1182 . . . . . . . . . . 11  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( `' G "
( M ... N
) )  ->  x  <_  sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )
129128adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( `' G " ( M ... N ) )  ->  x  <_  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) ) )
130123, 129impbid 183 . . . . . . . . 9  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  <_  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
13176, 130bitrd 244 . . . . . . . 8  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  x  e.  NN )  ->  (
x  e.  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
132131ex 423 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  NN  ->  ( x  e.  ( 1 ... sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  <->  x  e.  ( `' G " ( M ... N ) ) ) ) )
1332, 9, 132pm5.21ndd 343 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( x  e.  ( 1 ... sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )  <->  x  e.  ( `' G " ( M ... N ) ) ) )
134133eqrdv 2294 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  =  ( `' G " ( M ... N ) ) )
135134fveq2d 5545 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  (
# `  ( `' G " ( M ... N ) ) ) )
13673nnnn0d 10034 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  e.  NN0 )
137 hashfz1 11361 . . . . 5  |-  ( sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  )  e.  NN0  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )
138136, 137syl 15 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  (
1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) ) )  =  sup ( ( `' G " ( M ... N
) ) ,  RR ,  <  ) )
139 hashen 11362 . . . . . 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 ) ) ) ) )
14053, 23, 139syl2anc 642 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( # `
 ( `' G " ( M ... N
) ) )  =  ( # `  ( G " ( `' G " ( M ... N
) ) ) )  <-> 
( `' G "
( M ... N
) )  ~~  ( G " ( `' G " ( M ... N
) ) ) ) )
14151, 140mpbird 223 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( # `  ( `' G " ( M ... N ) ) )  =  ( # `  ( G " ( `' G " ( M ... N ) ) ) ) )
142135, 138, 1413eqtr3d 2336 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  sup (
( `' G "
( M ... N
) ) ,  RR ,  <  )  =  (
# `  ( G " ( `' G "
( M ... N
) ) ) ) )
143142oveq2d 5890 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... sup ( ( `' G " ( M ... N ) ) ,  RR ,  <  ) )  =  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) ) )
144143, 134eqtr3d 2330 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   class class class wbr 4039    Or wor 4329   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272  (class class class)co 5874    ~~ cen 6876   Fincfn 6879   supcsup 7209   RRcr 8752   1c1 8754    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   #chash 11353
This theorem is referenced by:  isercolllem3  12156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354
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