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Theorem isercolllem3 12460
Description: Lemma for isercoll 12461. (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 9249 . . 3  |-  ( n  e.  CC  ->  (
0  +  n )  =  n )
21adantl 453 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  CC )  ->  (
0  +  n )  =  n )
3 addid1 9246 . . 3  |-  ( n  e.  CC  ->  (
n  +  0 )  =  n )
43adantl 453 . 2  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  /\  n  e.  CC )  ->  (
n  +  0 )  =  n )
5 addcl 9072 . . 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 9084 . . 3  |-  0  e.  CC
87a1i 11 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  0  e.  CC )
9 cnvimass 5224 . . . . 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 5595 . . . . . 6  |-  ( G : NN --> Z  ->  dom  G  =  NN )
1311, 12syl 16 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  dom  G  =  NN )
149, 13syl5sseq 3396 . . . 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 12458 . . . 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 12459 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( 1 ... ( # `  ( G " ( `' G " ( M ... N
) ) ) ) )  =  ( `' G " ( M ... N ) ) )
21 isoeq4 6042 . . . 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 3545 . . . . 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 10011 . . . . . . 7  |-  1  e.  NN
2827a1i 11 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  1  e.  NN )
29 ffvelrn 5868 . . . . . . . . . 10  |-  ( ( G : NN --> Z  /\  1  e.  NN )  ->  ( G `  1
)  e.  Z )
3010, 27, 29sylancl 644 . . . . . . . . 9  |-  ( ph  ->  ( G `  1
)  e.  Z )
3130, 15syl6eleq 2526 . . . . . . . 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 11053 . . . . . . 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 5591 . . . . . . 7  |-  ( G : NN --> Z  ->  G  Fn  NN )
37 elpreima 5850 . . . . . . 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 3634 . . . . 5  |-  ( 1  e.  ( `' G " ( M ... N
) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
4139, 40syl 16 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( `' G " ( M ... N ) )  =/=  (/) )
4226, 41eqnetrd 2619 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =/=  (/) )
43 imadisj 5223 . . . 4  |-  ( ( G " ( `' G " ( M ... N ) ) )  =  (/)  <->  ( dom  G  i^i  ( `' G " ( M ... N
) ) )  =  (/) )
4443necon3bii 2633 . . 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 5593 . . . 4  |-  ( G : NN --> Z  ->  Fun  G )
47 funimacnv 5525 . . . 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 3561 . . . 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 3382 . 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 11055 . . . 4  |-  ( n  e.  ( M ... N )  ->  n  e.  ( ZZ>= `  M )
)
5453, 15syl6eleqr 2527 . . 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 3465 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  =  ( ( M ... N
)  \  ( ( M ... N )  i^i 
ran  G ) ) )
58 difin 3578 . . . . . 6  |-  ( ( M ... N ) 
\  ( ( M ... N )  i^i 
ran  G ) )  =  ( ( M ... N )  \  ran  G )
5957, 58syl6eq 2484 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  =  ( ( M ... N
)  \  ran  G ) )
6054ssriv 3352 . . . . . 6  |-  ( M ... N )  C_  Z
61 ssdif 3482 . . . . . 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 3382 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  ( ( M ... N )  \ 
( G " ( `' G " ( M ... N ) ) ) )  C_  ( Z  \  ran  G ) )
6463sselda 3348 . . 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 11080 . . . 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 2503 . . . . . 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 5745 . . . . 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 5732 . . 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 2471 . 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 11713 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 2599    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   class class class wbr 4212   `'ccnv 4877   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881   Fun wfun 5448    Fn wfn 5449   -->wf 5450   ` cfv 5454    Isom wiso 5455  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120   NNcn 10000   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043    seq cseq 11323   #chash 11618
This theorem is referenced by:  isercoll  12461
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-seq 11324  df-hash 11619
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