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Theorem isercolllem3 12452
Description: Lemma for isercoll 12453. (Contributed by Mario Carneiro, 6-Apr-2015.)
Hypotheses
Ref Expression
isercoll.z  |-  Z  =  ( ZZ>= `  M )
isercoll.m  |-  ( ph  ->  M  e.  ZZ )
isercoll.g  |-  ( ph  ->  G : NN --> Z )
isercoll.i  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  < 
( G `  (
k  +  1 ) ) )
isercoll.0  |-  ( (
ph  /\  n  e.  ( Z  \  ran  G
) )  ->  ( F `  n )  =  0 )
isercoll.f  |-  ( (
ph  /\  n  e.  Z )  ->  ( F `  n )  e.  CC )
isercoll.h  |-  ( (
ph  /\  k  e.  NN )  ->  ( H `
 k )  =  ( F `  ( G `  k )
) )
Assertion
Ref Expression
isercolllem3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  N )  =  (  seq  1
(  +  ,  H
) `  ( # `  ( G " ( `' G " ( M ... N
) ) ) ) ) )
Distinct variable groups:    k, n, F    k, N, n    ph, k, n    k, G, n    k, H, n    k, M, n   
n, Z
Allowed substitution hint:    Z( k)

Proof of Theorem isercolllem3
StepHypRef Expression
1 addid2 9241 . . 3  |-  ( n  e.  CC  ->  (
0  +  n )  =  n )
21adantl 453 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  CC )  ->  (
0  +  n )  =  n )
3 addid1 9238 . . 3  |-  ( n  e.  CC  ->  (
n  +  0 )  =  n )
43adantl 453 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  CC )  ->  (
n  +  0 )  =  n )
5 addcl 9064 . . 3  |-  ( ( n  e.  CC  /\  k  e.  CC )  ->  ( n  +  k )  e.  CC )
65adantl 453 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  ( n  e.  CC  /\  k  e.  CC ) )  -> 
( n  +  k )  e.  CC )
7 0cn 9076 . . 3  |-  0  e.  CC
87a1i 11 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  0  e.  CC )
9 cnvimass 5216 . . . . 5  |-  ( `' G " ( M ... N ) ) 
C_  dom  G
10 isercoll.g . . . . . . 7  |-  ( ph  ->  G : NN --> Z )
1110adantr 452 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
--> Z )
12 fdm 5587 . . . . . 6  |-  ( G : NN --> Z  ->  dom  G  =  NN )
1311, 12syl 16 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  dom  G  =  NN )
149, 13syl5sseq 3388 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  NN )
15 isercoll.z . . . . 5  |-  Z  =  ( ZZ>= `  M )
16 isercoll.m . . . . 5  |-  ( ph  ->  M  e.  ZZ )
17 isercoll.i . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  < 
( G `  (
k  +  1 ) ) )
1815, 16, 10, 17isercolllem1 12450 . . . 4  |-  ( (
ph  /\  ( `' G " ( M ... N ) )  C_  NN )  ->  ( G  |`  ( `' G "
( M ... N
) ) )  Isom  <  ,  <  ( ( `' G " ( M ... N ) ) ,  ( G "
( `' G "
( M ... N
) ) ) ) )
1914, 18syldan 457 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G  |`  ( `' G "
( M ... N
) ) )  Isom  <  ,  <  ( ( `' G " ( M ... N ) ) ,  ( G "
( `' G "
( M ... N
) ) ) ) )
2015, 16, 10, 17isercolllem2 12451 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) )  =  ( `' G " ( M ... N ) ) )
21 isoeq4 6034 . . . 4  |-  ( ( 1 ... ( # `  ( G " ( `' G " ( M ... N ) ) ) ) )  =  ( `' G "
( M ... N
) )  ->  (
( G  |`  ( `' G " ( M ... N ) ) )  Isom  <  ,  <  ( ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) ,  ( G
" ( `' G " ( M ... N
) ) ) )  <-> 
( G  |`  ( `' G " ( M ... N ) ) )  Isom  <  ,  <  ( ( `' G "
( M ... N
) ) ,  ( G " ( `' G " ( M ... N ) ) ) ) ) )
2220, 21syl 16 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( G  |`  ( `' G " ( M ... N
) ) )  Isom  <  ,  <  ( ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) ) ,  ( G
" ( `' G " ( M ... N
) ) ) )  <-> 
( G  |`  ( `' G " ( M ... N ) ) )  Isom  <  ,  <  ( ( `' G "
( M ... N
) ) ,  ( G " ( `' G " ( M ... N ) ) ) ) ) )
2319, 22mpbird 224 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G  |`  ( `' G "
( M ... N
) ) )  Isom  <  ,  <  ( ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) ) ,  ( G
" ( `' G " ( M ... N
) ) ) ) )
249a1i 11 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  dom  G )
25 dfss1 3537 . . . . 5  |-  ( ( `' G " ( M ... N ) ) 
C_  dom  G  <->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =  ( `' G "
( M ... N
) ) )
2624, 25sylib 189 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =  ( `' G "
( M ... N
) ) )
27 1nn 10003 . . . . . . 7  |-  1  e.  NN
2827a1i 11 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  NN )
29 ffvelrn 5860 . . . . . . . . . 10  |-  ( ( G : NN --> Z  /\  1  e.  NN )  ->  ( G `  1
)  e.  Z )
3010, 27, 29sylancl 644 . . . . . . . . 9  |-  ( ph  ->  ( G `  1
)  e.  Z )
3130, 15syl6eleq 2525 . . . . . . . 8  |-  ( ph  ->  ( G `  1
)  e.  ( ZZ>= `  M ) )
3231adantr 452 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( ZZ>= `  M )
)
33 simpr 448 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  N  e.  ( ZZ>= `  ( G `  1 ) ) )
34 elfzuzb 11045 . . . . . . 7  |-  ( ( G `  1 )  e.  ( M ... N )  <->  ( ( G `  1 )  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  ( G ` 
1 ) ) ) )
3532, 33, 34sylanbrc 646 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( M ... N
) )
36 ffn 5583 . . . . . . 7  |-  ( G : NN --> Z  ->  G  Fn  NN )
37 elpreima 5842 . . . . . . 7  |-  ( G  Fn  NN  ->  (
1  e.  ( `' G " ( M ... N ) )  <-> 
( 1  e.  NN  /\  ( G `  1
)  e.  ( M ... N ) ) ) )
3811, 36, 373syl 19 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1  e.  ( `' G " ( M ... N
) )  <->  ( 1  e.  NN  /\  ( G `  1 )  e.  ( M ... N
) ) ) )
3928, 35, 38mpbir2and 889 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  ( `' G " ( M ... N ) ) )
40 ne0i 3626 . . . . 5  |-  ( 1  e.  ( `' G " ( M ... N
) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
4139, 40syl 16 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
4226, 41eqnetrd 2616 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =/=  (/) )
43 imadisj 5215 . . . 4  |-  ( ( G " ( `' G " ( M ... N ) ) )  =  (/)  <->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =  (/) )
4443necon3bii 2630 . . 3  |-  ( ( G " ( `' G " ( M ... N ) ) )  =/=  (/)  <->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =/=  (/) )
4542, 44sylibr 204 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  =/=  (/) )
46 ffun 5585 . . . 4  |-  ( G : NN --> Z  ->  Fun  G )
47 funimacnv 5517 . . . 4  |-  ( Fun 
G  ->  ( G " ( `' G "
( M ... N
) ) )  =  ( ( M ... N )  i^i  ran  G ) )
4811, 46, 473syl 19 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  =  ( ( M ... N )  i^i  ran  G ) )
49 inss1 3553 . . . 4  |-  ( ( M ... N )  i^i  ran  G )  C_  ( M ... N
)
5049a1i 11 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  i^i 
ran  G )  C_  ( M ... N ) )
5148, 50eqsstrd 3374 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  C_  ( M ... N ) )
52 simpl 444 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ph )
53 elfzuz 11047 . . . 4  |-  ( n  e.  ( M ... N )  ->  n  e.  ( ZZ>= `  M )
)
5453, 15syl6eleqr 2526 . . 3  |-  ( n  e.  ( M ... N )  ->  n  e.  Z )
55 isercoll.f . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( F `  n )  e.  CC )
5652, 54, 55syl2an 464 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  ( M ... N
) )  ->  ( F `  n )  e.  CC )
5748difeq2d 3457 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  =  ( ( M ... N
)  \  ( ( M ... N )  i^i 
ran  G ) ) )
58 difin 3570 . . . . . 6  |-  ( ( M ... N ) 
\  ( ( M ... N )  i^i 
ran  G ) )  =  ( ( M ... N )  \  ran  G )
5957, 58syl6eq 2483 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  =  ( ( M ... N
)  \  ran  G ) )
6054ssriv 3344 . . . . . 6  |-  ( M ... N )  C_  Z
61 ssdif 3474 . . . . . 6  |-  ( ( M ... N ) 
C_  Z  ->  (
( M ... N
)  \  ran  G ) 
C_  ( Z  \  ran  G ) )
6260, 61mp1i 12 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \  ran  G )  C_  ( Z  \  ran  G ) )
6359, 62eqsstrd 3374 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  C_  ( Z  \  ran  G ) )
6463sselda 3340 . . 3  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  ( ( M ... N )  \  ( G " ( `' G " ( M ... N
) ) ) ) )  ->  n  e.  ( Z  \  ran  G
) )
65 isercoll.0 . . . 4  |-  ( (
ph  /\  n  e.  ( Z  \  ran  G
) )  ->  ( F `  n )  =  0 )
6665adantlr 696 . . 3  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  ( Z  \  ran  G ) )  ->  ( F `  n )  =  0 )
6764, 66syldan 457 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  ( ( M ... N )  \  ( G " ( `' G " ( M ... N
) ) ) ) )  ->  ( F `  n )  =  0 )
68 elfznn 11072 . . . 4  |-  ( k  e.  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) )  ->  k  e.  NN )
69 isercoll.h . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( H `
 k )  =  ( F `  ( G `  k )
) )
7052, 68, 69syl2an 464 . . 3  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  ( H `  k )  =  ( F `  ( G `  k ) ) )
7120eleq2d 2502 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) )  <->  k  e.  ( `' G " ( M ... N ) ) ) )
7271biimpa 471 . . . . 5  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  k  e.  ( `' G "
( M ... N
) ) )
73 fvres 5737 . . . . 5  |-  ( k  e.  ( `' G " ( M ... N
) )  ->  (
( G  |`  ( `' G " ( M ... N ) ) ) `  k )  =  ( G `  k ) )
7472, 73syl 16 . . . 4  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  (
( G  |`  ( `' G " ( M ... N ) ) ) `  k )  =  ( G `  k ) )
7574fveq2d 5724 . . 3  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  ( F `  ( ( G  |`  ( `' G " ( M ... N
) ) ) `  k ) )  =  ( F `  ( G `  k )
) )
7670, 75eqtr4d 2470 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  ( H `  k )  =  ( F `  ( ( G  |`  ( `' G " ( M ... N ) ) ) `  k ) ) )
772, 4, 6, 8, 23, 45, 51, 56, 67, 76seqcoll2 11705 1  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  N )  =  (  seq  1
(  +  ,  H
) `  ( # `  ( G " ( `' G " ( M ... N
) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440    Fn wfn 5441   -->wf 5442   ` cfv 5446    Isom wiso 5447  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    < clt 9112   NNcn 9992   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035    seq cseq 11315   #chash 11610
This theorem is referenced by:  isercoll  12453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-seq 11316  df-hash 11611
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