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Theorem isercolllem3 12347
Description: Lemma for isercoll 12348. (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 9142 . . 3  |-  ( n  e.  CC  ->  (
0  +  n )  =  n )
21adantl 452 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  CC )  ->  (
0  +  n )  =  n )
3 addid1 9139 . . 3  |-  ( n  e.  CC  ->  (
n  +  0 )  =  n )
43adantl 452 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  CC )  ->  (
n  +  0 )  =  n )
5 addcl 8966 . . 3  |-  ( ( n  e.  CC  /\  k  e.  CC )  ->  ( n  +  k )  e.  CC )
65adantl 452 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  ( n  e.  CC  /\  k  e.  CC ) )  -> 
( n  +  k )  e.  CC )
7 0cn 8978 . . 3  |-  0  e.  CC
87a1i 10 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  0  e.  CC )
9 cnvimass 5136 . . . . 5  |-  ( `' G " ( M ... N ) ) 
C_  dom  G
10 isercoll.g . . . . . . 7  |-  ( ph  ->  G : NN --> Z )
1110adantr 451 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  G : NN
--> Z )
12 fdm 5499 . . . . . 6  |-  ( G : NN --> Z  ->  dom  G  =  NN )
1311, 12syl 15 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  dom  G  =  NN )
149, 13syl5sseq 3312 . . . 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 12345 . . . 4  |-  ( (
ph  /\  ( `' G " ( M ... N ) )  C_  NN )  ->  ( G  |`  ( `' G "
( M ... N
) ) )  Isom  <  ,  <  ( ( `' G " ( M ... N ) ) ,  ( G "
( `' G "
( M ... N
) ) ) ) )
1914, 18syldan 456 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G  |`  ( `' G "
( M ... N
) ) )  Isom  <  ,  <  ( ( `' G " ( M ... N ) ) ,  ( G "
( `' G "
( M ... N
) ) ) ) )
2015, 16, 10, 17isercolllem2 12346 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) )  =  ( `' G " ( M ... N ) ) )
21 isoeq4 5942 . . . 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 15 . . 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 223 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G  |`  ( `' G "
( M ... N
) ) )  Isom  <  ,  <  ( ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) ) ,  ( G
" ( `' G " ( M ... N
) ) ) ) )
249a1i 10 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  C_  dom  G )
25 dfss1 3461 . . . . 5  |-  ( ( `' G " ( M ... N ) ) 
C_  dom  G  <->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =  ( `' G "
( M ... N
) ) )
2624, 25sylib 188 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =  ( `' G "
( M ... N
) ) )
27 1nn 9904 . . . . . . 7  |-  1  e.  NN
2827a1i 10 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  NN )
29 ffvelrn 5770 . . . . . . . . . 10  |-  ( ( G : NN --> Z  /\  1  e.  NN )  ->  ( G `  1
)  e.  Z )
3010, 27, 29sylancl 643 . . . . . . . . 9  |-  ( ph  ->  ( G `  1
)  e.  Z )
3130, 15syl6eleq 2456 . . . . . . . 8  |-  ( ph  ->  ( G `  1
)  e.  ( ZZ>= `  M ) )
3231adantr 451 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( ZZ>= `  M )
)
33 simpr 447 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  N  e.  ( ZZ>= `  ( G `  1 ) ) )
34 elfzuzb 10945 . . . . . . 7  |-  ( ( G `  1 )  e.  ( M ... N )  <->  ( ( G `  1 )  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  ( G ` 
1 ) ) ) )
3532, 33, 34sylanbrc 645 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G `  1 )  e.  ( M ... N
) )
36 ffn 5495 . . . . . . 7  |-  ( G : NN --> Z  ->  G  Fn  NN )
37 elpreima 5752 . . . . . . 7  |-  ( G  Fn  NN  ->  (
1  e.  ( `' G " ( M ... N ) )  <-> 
( 1  e.  NN  /\  ( G `  1
)  e.  ( M ... N ) ) ) )
3811, 36, 373syl 18 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1  e.  ( `' G " ( M ... N
) )  <->  ( 1  e.  NN  /\  ( G `  1 )  e.  ( M ... N
) ) ) )
3928, 35, 38mpbir2and 888 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  ( `' G " ( M ... N ) ) )
40 ne0i 3549 . . . . 5  |-  ( 1  e.  ( `' G " ( M ... N
) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
4139, 40syl 15 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
4226, 41eqnetrd 2547 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =/=  (/) )
43 imadisj 5135 . . . 4  |-  ( ( G " ( `' G " ( M ... N ) ) )  =  (/)  <->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =  (/) )
4443necon3bii 2561 . . 3  |-  ( ( G " ( `' G " ( M ... N ) ) )  =/=  (/)  <->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =/=  (/) )
4542, 44sylibr 203 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  =/=  (/) )
46 ffun 5497 . . . 4  |-  ( G : NN --> Z  ->  Fun  G )
47 funimacnv 5429 . . . 4  |-  ( Fun 
G  ->  ( G " ( `' G "
( M ... N
) ) )  =  ( ( M ... N )  i^i  ran  G ) )
4811, 46, 473syl 18 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  =  ( ( M ... N )  i^i  ran  G ) )
49 inss1 3477 . . . 4  |-  ( ( M ... N )  i^i  ran  G )  C_  ( M ... N
)
5049a1i 10 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  i^i 
ran  G )  C_  ( M ... N ) )
5148, 50eqsstrd 3298 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( G " ( `' G "
( M ... N
) ) )  C_  ( M ... N ) )
52 simpl 443 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ph )
53 elfzuz 10947 . . . 4  |-  ( n  e.  ( M ... N )  ->  n  e.  ( ZZ>= `  M )
)
5453, 15syl6eleqr 2457 . . 3  |-  ( n  e.  ( M ... N )  ->  n  e.  Z )
55 isercoll.f . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( F `  n )  e.  CC )
5652, 54, 55syl2an 463 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  ( M ... N
) )  ->  ( F `  n )  e.  CC )
5748difeq2d 3381 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  =  ( ( M ... N
)  \  ( ( M ... N )  i^i 
ran  G ) ) )
58 difin 3494 . . . . . 6  |-  ( ( M ... N ) 
\  ( ( M ... N )  i^i 
ran  G ) )  =  ( ( M ... N )  \  ran  G )
5957, 58syl6eq 2414 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  =  ( ( M ... N
)  \  ran  G ) )
6054ssriv 3270 . . . . . 6  |-  ( M ... N )  C_  Z
61 ssdif 3398 . . . . . 6  |-  ( ( M ... N ) 
C_  Z  ->  (
( M ... N
)  \  ran  G ) 
C_  ( Z  \  ran  G ) )
6260, 61mp1i 11 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \  ran  G )  C_  ( Z  \  ran  G ) )
6359, 62eqsstrd 3298 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  C_  ( Z  \  ran  G ) )
6463sselda 3266 . . 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 695 . . 3  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  ( Z  \  ran  G ) )  ->  ( F `  n )  =  0 )
6764, 66syldan 456 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  ( ( M ... N )  \  ( G " ( `' G " ( M ... N
) ) ) ) )  ->  ( F `  n )  =  0 )
68 elfznn 10972 . . . 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 463 . . 3  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  ( H `  k )  =  ( F `  ( G `  k ) ) )
7120eleq2d 2433 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) )  <->  k  e.  ( `' G " ( M ... N ) ) ) )
7271biimpa 470 . . . . 5  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  k  e.  ( `' G "
( M ... N
) ) )
73 fvres 5649 . . . . 5  |-  ( k  e.  ( `' G " ( M ... N
) )  ->  (
( G  |`  ( `' G " ( M ... N ) ) ) `  k )  =  ( G `  k ) )
7472, 73syl 15 . . . 4  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  k  e.  ( 1 ... ( # `
 ( G "
( `' G "
( M ... N
) ) ) ) ) )  ->  (
( G  |`  ( `' G " ( M ... N ) ) ) `  k )  =  ( G `  k ) )
7574fveq2d 5636 . . 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 2401 . 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 11600 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 176    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529    \ cdif 3235    i^i cin 3237    C_ wss 3238   (/)c0 3543   class class class wbr 4125   `'ccnv 4791   dom cdm 4792   ran crn 4793    |` cres 4794   "cima 4795   Fun wfun 5352    Fn wfn 5353   -->wf 5354   ` cfv 5358    Isom wiso 5359  (class class class)co 5981   CCcc 8882   0cc0 8884   1c1 8885    + caddc 8887    < clt 9014   NNcn 9893   ZZcz 10175   ZZ>=cuz 10381   ...cfz 10935    seq cseq 11210   #chash 11505
This theorem is referenced by:  isercoll  12348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-seq 11211  df-hash 11506
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