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Theorem isercolllem3 12415
Description: Lemma for isercoll 12416. (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 9205 . . 3  |-  ( n  e.  CC  ->  (
0  +  n )  =  n )
21adantl 453 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  CC )  ->  (
0  +  n )  =  n )
3 addid1 9202 . . 3  |-  ( n  e.  CC  ->  (
n  +  0 )  =  n )
43adantl 453 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  CC )  ->  (
n  +  0 )  =  n )
5 addcl 9028 . . 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 9040 . . 3  |-  0  e.  CC
87a1i 11 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  0  e.  CC )
9 cnvimass 5183 . . . . 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 5554 . . . . . 6  |-  ( G : NN --> Z  ->  dom  G  =  NN )
1311, 12syl 16 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  dom  G  =  NN )
149, 13syl5sseq 3356 . . . 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 12413 . . . 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 12414 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) )  =  ( `' G " ( M ... N ) ) )
21 isoeq4 6001 . . . 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 3505 . . . . 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 9967 . . . . . . 7  |-  1  e.  NN
2827a1i 11 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  NN )
29 ffvelrn 5827 . . . . . . . . . 10  |-  ( ( G : NN --> Z  /\  1  e.  NN )  ->  ( G `  1
)  e.  Z )
3010, 27, 29sylancl 644 . . . . . . . . 9  |-  ( ph  ->  ( G `  1
)  e.  Z )
3130, 15syl6eleq 2494 . . . . . . . 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 11009 . . . . . . 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 5550 . . . . . . 7  |-  ( G : NN --> Z  ->  G  Fn  NN )
37 elpreima 5809 . . . . . . 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 3594 . . . . 5  |-  ( 1  e.  ( `' G " ( M ... N
) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
4139, 40syl 16 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
4226, 41eqnetrd 2585 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =/=  (/) )
43 imadisj 5182 . . . 4  |-  ( ( G " ( `' G " ( M ... N ) ) )  =  (/)  <->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =  (/) )
4443necon3bii 2599 . . 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 5552 . . . 4  |-  ( G : NN --> Z  ->  Fun  G )
47 funimacnv 5484 . . . 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 3521 . . . 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 3342 . 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 11011 . . . 4  |-  ( n  e.  ( M ... N )  ->  n  e.  ( ZZ>= `  M )
)
5453, 15syl6eleqr 2495 . . 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 3425 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  =  ( ( M ... N
)  \  ( ( M ... N )  i^i 
ran  G ) ) )
58 difin 3538 . . . . . 6  |-  ( ( M ... N ) 
\  ( ( M ... N )  i^i 
ran  G ) )  =  ( ( M ... N )  \  ran  G )
5957, 58syl6eq 2452 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  =  ( ( M ... N
)  \  ran  G ) )
6054ssriv 3312 . . . . . 6  |-  ( M ... N )  C_  Z
61 ssdif 3442 . . . . . 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 3342 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  C_  ( Z  \  ran  G ) )
6463sselda 3308 . . 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 11036 . . . 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 2471 . . . . . 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 5704 . . . . 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 5691 . . 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 2439 . 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 11668 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 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   class class class wbr 4172   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413    Isom wiso 5414  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    < clt 9076   NNcn 9956   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278   #chash 11573
This theorem is referenced by:  isercoll  12416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-seq 11279  df-hash 11574
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