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Theorem isercolllem3 13140
Description: Lemma for isercoll 13141. (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 9548 . . 3  |-  ( n  e.  CC  ->  (
0  +  n )  =  n )
21adantl 463 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  CC )  ->  (
0  +  n )  =  n )
3 addid1 9545 . . 3  |-  ( n  e.  CC  ->  (
n  +  0 )  =  n )
43adantl 463 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  CC )  ->  (
n  +  0 )  =  n )
5 addcl 9360 . . 3  |-  ( ( n  e.  CC  /\  k  e.  CC )  ->  ( n  +  k )  e.  CC )
65adantl 463 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  ( n  e.  CC  /\  k  e.  CC ) )  -> 
( n  +  k )  e.  CC )
7 0cnd 9375 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  0  e.  CC )
8 cnvimass 5186 . . . . 5  |-  ( `' G " ( M ... N ) ) 
C_  dom  G
9 isercoll.g . . . . . . 7  |-  ( ph  ->  G : NN --> Z )
109adantr 462 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
--> Z )
11 fdm 5560 . . . . . 6  |-  ( G : NN --> Z  ->  dom  G  =  NN )
1210, 11syl 16 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  dom  G  =  NN )
138, 12syl5sseq 3401 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  NN )
14 isercoll.z . . . . 5  |-  Z  =  ( ZZ>= `  M )
15 isercoll.m . . . . 5  |-  ( ph  ->  M  e.  ZZ )
16 isercoll.i . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  < 
( G `  (
k  +  1 ) ) )
1714, 15, 9, 16isercolllem1 13138 . . . 4  |-  ( (
ph  /\  ( `' G " ( M ... N ) )  C_  NN )  ->  ( G  |`  ( `' G "
( M ... N
) ) )  Isom  <  ,  <  ( ( `' G " ( M ... N ) ) ,  ( G "
( `' G "
( M ... N
) ) ) ) )
1813, 17syldan 467 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G  |`  ( `' G "
( M ... N
) ) )  Isom  <  ,  <  ( ( `' G " ( M ... N ) ) ,  ( G "
( `' G "
( M ... N
) ) ) ) )
1914, 15, 9, 16isercolllem2 13139 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) )  =  ( `' G " ( M ... N ) ) )
20 isoeq4 6010 . . . 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 ) ) ) ) ) )
2119, 20syl 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 ) ) ) ) ) )
2218, 21mpbird 232 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G  |`  ( `' G "
( M ... N
) ) )  Isom  <  ,  <  ( ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) ) ,  ( G
" ( `' G " ( M ... N
) ) ) ) )
238a1i 11 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  dom  G )
24 dfss1 3552 . . . . 5  |-  ( ( `' G " ( M ... N ) ) 
C_  dom  G  <->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =  ( `' G "
( M ... N
) ) )
2523, 24sylib 196 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =  ( `' G "
( M ... N
) ) )
26 1nn 10329 . . . . . . 7  |-  1  e.  NN
2726a1i 11 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  NN )
28 ffvelrn 5838 . . . . . . . . . 10  |-  ( ( G : NN --> Z  /\  1  e.  NN )  ->  ( G `  1
)  e.  Z )
299, 26, 28sylancl 657 . . . . . . . . 9  |-  ( ph  ->  ( G `  1
)  e.  Z )
3029, 14syl6eleq 2531 . . . . . . . 8  |-  ( ph  ->  ( G `  1
)  e.  ( ZZ>= `  M ) )
3130adantr 462 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( ZZ>= `  M )
)
32 simpr 458 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  N  e.  ( ZZ>= `  ( G `  1 ) ) )
33 elfzuzb 11443 . . . . . . 7  |-  ( ( G `  1 )  e.  ( M ... N )  <->  ( ( G `  1 )  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  ( G ` 
1 ) ) ) )
3431, 32, 33sylanbrc 659 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( M ... N
) )
35 ffn 5556 . . . . . . 7  |-  ( G : NN --> Z  ->  G  Fn  NN )
36 elpreima 5820 . . . . . . 7  |-  ( G  Fn  NN  ->  (
1  e.  ( `' G " ( M ... N ) )  <-> 
( 1  e.  NN  /\  ( G `  1
)  e.  ( M ... N ) ) ) )
3710, 35, 363syl 20 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1  e.  ( `' G " ( M ... N
) )  <->  ( 1  e.  NN  /\  ( G `  1 )  e.  ( M ... N
) ) ) )
3827, 34, 37mpbir2and 908 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  ( `' G " ( M ... N ) ) )
39 ne0i 3640 . . . . 5  |-  ( 1  e.  ( `' G " ( M ... N
) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
4038, 39syl 16 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
4125, 40eqnetrd 2624 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =/=  (/) )
42 imadisj 5185 . . . 4  |-  ( ( G " ( `' G " ( M ... N ) ) )  =  (/)  <->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =  (/) )
4342necon3bii 2638 . . 3  |-  ( ( G " ( `' G " ( M ... N ) ) )  =/=  (/)  <->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =/=  (/) )
4441, 43sylibr 212 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  =/=  (/) )
45 ffun 5558 . . . 4  |-  ( G : NN --> Z  ->  Fun  G )
46 funimacnv 5487 . . . 4  |-  ( Fun 
G  ->  ( G " ( `' G "
( M ... N
) ) )  =  ( ( M ... N )  i^i  ran  G ) )
4710, 45, 463syl 20 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  =  ( ( M ... N )  i^i  ran  G ) )
48 inss1 3567 . . . 4  |-  ( ( M ... N )  i^i  ran  G )  C_  ( M ... N
)
4948a1i 11 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  i^i 
ran  G )  C_  ( M ... N ) )
5047, 49eqsstrd 3387 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  C_  ( M ... N ) )
51 simpl 454 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ph )
52 elfzuz 11445 . . . 4  |-  ( n  e.  ( M ... N )  ->  n  e.  ( ZZ>= `  M )
)
5352, 14syl6eleqr 2532 . . 3  |-  ( n  e.  ( M ... N )  ->  n  e.  Z )
54 isercoll.f . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( F `  n )  e.  CC )
5551, 53, 54syl2an 474 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  ( M ... N
) )  ->  ( F `  n )  e.  CC )
5647difeq2d 3471 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  =  ( ( M ... N
)  \  ( ( M ... N )  i^i 
ran  G ) ) )
57 difin 3584 . . . . . 6  |-  ( ( M ... N ) 
\  ( ( M ... N )  i^i 
ran  G ) )  =  ( ( M ... N )  \  ran  G )
5856, 57syl6eq 2489 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  =  ( ( M ... N
)  \  ran  G ) )
5953ssriv 3357 . . . . . 6  |-  ( M ... N )  C_  Z
60 ssdif 3488 . . . . . 6  |-  ( ( M ... N ) 
C_  Z  ->  (
( M ... N
)  \  ran  G ) 
C_  ( Z  \  ran  G ) )
6159, 60mp1i 12 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \  ran  G )  C_  ( Z  \  ran  G ) )
6258, 61eqsstrd 3387 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  C_  ( Z  \  ran  G ) )
6362sselda 3353 . . 3  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  ( ( M ... N )  \  ( G " ( `' G " ( M ... N
) ) ) ) )  ->  n  e.  ( Z  \  ran  G
) )
64 isercoll.0 . . . 4  |-  ( (
ph  /\  n  e.  ( Z  \  ran  G
) )  ->  ( F `  n )  =  0 )
6564adantlr 709 . . 3  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  ( Z  \  ran  G ) )  ->  ( F `  n )  =  0 )
6663, 65syldan 467 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  ( ( M ... N )  \  ( G " ( `' G " ( M ... N
) ) ) ) )  ->  ( F `  n )  =  0 )
67 elfznn 11474 . . . 4  |-  ( k  e.  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) )  ->  k  e.  NN )
68 isercoll.h . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( H `
 k )  =  ( F `  ( G `  k )
) )
6951, 67, 68syl2an 474 . . 3  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  ( H `  k )  =  ( F `  ( G `  k ) ) )
7019eleq2d 2508 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) )  <->  k  e.  ( `' G " ( M ... N ) ) ) )
7170biimpa 481 . . . . 5  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  k  e.  ( `' G "
( M ... N
) ) )
72 fvres 5701 . . . . 5  |-  ( k  e.  ( `' G " ( M ... N
) )  ->  (
( G  |`  ( `' G " ( M ... N ) ) ) `  k )  =  ( G `  k ) )
7371, 72syl 16 . . . 4  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  (
( G  |`  ( `' G " ( M ... N ) ) ) `  k )  =  ( G `  k ) )
7473fveq2d 5692 . . 3  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  ( F `  ( ( G  |`  ( `' G " ( M ... N
) ) ) `  k ) )  =  ( F `  ( G `  k )
) )
7569, 74eqtr4d 2476 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  ( H `  k )  =  ( F `  ( ( G  |`  ( `' G " ( M ... N ) ) ) `  k ) ) )
762, 4, 6, 7, 22, 44, 50, 55, 66, 75seqcoll2 12213 1  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  (  seq M (  +  ,  F ) `  N
)  =  (  seq 1 (  +  ,  H ) `  ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604    \ cdif 3322    i^i cin 3324    C_ wss 3325   (/)c0 3634   class class class wbr 4289   `'ccnv 4835   dom cdm 4836   ran crn 4837    |` cres 4838   "cima 4839   Fun wfun 5409    Fn wfn 5410   -->wf 5411   ` cfv 5415    Isom wiso 5416  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    < clt 9414   NNcn 10318   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433    seqcseq 11802   #chash 12099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803  df-hash 12100
This theorem is referenced by:  isercoll  13141
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