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Theorem summolem3 12539
Description: Lemma for summo 12542. (Contributed by Mario Carneiro, 29-Mar-2014.)
Hypotheses
Ref Expression
summo.1  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
summo.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
summo.3  |-  G  =  ( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
summolem3.4  |-  H  =  ( n  e.  NN  |->  [_ ( K `  n
)  /  k ]_ B )
summolem3.5  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN ) )
summolem3.6  |-  ( ph  ->  f : ( 1 ... M ) -1-1-onto-> A )
summolem3.7  |-  ( ph  ->  K : ( 1 ... N ) -1-1-onto-> A )
Assertion
Ref Expression
summolem3  |-  ( ph  ->  (  seq  1 (  +  ,  G ) `
 M )  =  (  seq  1 (  +  ,  H ) `
 N ) )
Distinct variable groups:    f, k, n, A    f, F, k, n    k, G, n   
k, K, n    k, N, n    ph, k, n    B, f, n    k, M, n
Allowed substitution hints:    ph( f)    B( k)    G( f)    H( f, k, n)    K( f)    M( f)    N( f)

Proof of Theorem summolem3
Dummy variables  i 
j  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addcl 9103 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC )  ->  ( m  +  j )  e.  CC )
21adantl 454 . . 3  |-  ( (
ph  /\  ( m  e.  CC  /\  j  e.  CC ) )  -> 
( m  +  j )  e.  CC )
3 addcom 9283 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC )  ->  ( m  +  j )  =  ( j  +  m ) )
43adantl 454 . . 3  |-  ( (
ph  /\  ( m  e.  CC  /\  j  e.  CC ) )  -> 
( m  +  j )  =  ( j  +  m ) )
5 addass 9108 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC  /\  y  e.  CC )  ->  (
( m  +  j )  +  y )  =  ( m  +  ( j  +  y ) ) )
65adantl 454 . . 3  |-  ( (
ph  /\  ( m  e.  CC  /\  j  e.  CC  /\  y  e.  CC ) )  -> 
( ( m  +  j )  +  y )  =  ( m  +  ( j  +  y ) ) )
7 summolem3.5 . . . . 5  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN ) )
87simpld 447 . . . 4  |-  ( ph  ->  M  e.  NN )
9 nnuz 10552 . . . 4  |-  NN  =  ( ZZ>= `  1 )
108, 9syl6eleq 2532 . . 3  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
11 ssid 3353 . . . 4  |-  CC  C_  CC
1211a1i 11 . . 3  |-  ( ph  ->  CC  C_  CC )
13 summolem3.6 . . . . . 6  |-  ( ph  ->  f : ( 1 ... M ) -1-1-onto-> A )
14 f1ocnv 5716 . . . . . 6  |-  ( f : ( 1 ... M ) -1-1-onto-> A  ->  `' f : A -1-1-onto-> ( 1 ... M
) )
1513, 14syl 16 . . . . 5  |-  ( ph  ->  `' f : A -1-1-onto-> (
1 ... M ) )
16 summolem3.7 . . . . 5  |-  ( ph  ->  K : ( 1 ... N ) -1-1-onto-> A )
17 f1oco 5727 . . . . 5  |-  ( ( `' f : A -1-1-onto-> (
1 ... M )  /\  K : ( 1 ... N ) -1-1-onto-> A )  ->  ( `' f  o.  K
) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) )
1815, 16, 17syl2anc 644 . . . 4  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) )
19 ovex 6135 . . . . . . . . . 10  |-  ( 1 ... N )  e. 
_V
2019f1oen 7157 . . . . . . . . 9  |-  ( ( `' f  o.  K
) : ( 1 ... N ) -1-1-onto-> ( 1 ... M )  -> 
( 1 ... N
)  ~~  ( 1 ... M ) )
2118, 20syl 16 . . . . . . . 8  |-  ( ph  ->  ( 1 ... N
)  ~~  ( 1 ... M ) )
22 fzfi 11342 . . . . . . . . 9  |-  ( 1 ... N )  e. 
Fin
23 fzfi 11342 . . . . . . . . 9  |-  ( 1 ... M )  e. 
Fin
24 hashen 11662 . . . . . . . . 9  |-  ( ( ( 1 ... N
)  e.  Fin  /\  ( 1 ... M
)  e.  Fin )  ->  ( ( # `  (
1 ... N ) )  =  ( # `  (
1 ... M ) )  <-> 
( 1 ... N
)  ~~  ( 1 ... M ) ) )
2522, 23, 24mp2an 655 . . . . . . . 8  |-  ( (
# `  ( 1 ... N ) )  =  ( # `  (
1 ... M ) )  <-> 
( 1 ... N
)  ~~  ( 1 ... M ) )
2621, 25sylibr 205 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  ( # `  (
1 ... M ) ) )
277simprd 451 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
28 nnnn0 10259 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  NN0 )
29 hashfz1 11661 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
3027, 28, 293syl 19 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
31 nnnn0 10259 . . . . . . . 8  |-  ( M  e.  NN  ->  M  e.  NN0 )
32 hashfz1 11661 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
338, 31, 323syl 19 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
3426, 30, 333eqtr3rd 2483 . . . . . 6  |-  ( ph  ->  M  =  N )
3534oveq2d 6126 . . . . 5  |-  ( ph  ->  ( 1 ... M
)  =  ( 1 ... N ) )
36 f1oeq2 5695 . . . . 5  |-  ( ( 1 ... M )  =  ( 1 ... N )  ->  (
( `' f  o.  K ) : ( 1 ... M ) -1-1-onto-> ( 1 ... M )  <-> 
( `' f  o.  K ) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) ) )
3735, 36syl 16 . . . 4  |-  ( ph  ->  ( ( `' f  o.  K ) : ( 1 ... M
)
-1-1-onto-> ( 1 ... M
)  <->  ( `' f  o.  K ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... M
) ) )
3818, 37mpbird 225 . . 3  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... M ) -1-1-onto-> ( 1 ... M ) )
39 elfznn 11111 . . . . . 6  |-  ( m  e.  ( 1 ... M )  ->  m  e.  NN )
4039adantl 454 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  m  e.  NN )
41 f1of 5703 . . . . . . . 8  |-  ( f : ( 1 ... M ) -1-1-onto-> A  ->  f :
( 1 ... M
) --> A )
4213, 41syl 16 . . . . . . 7  |-  ( ph  ->  f : ( 1 ... M ) --> A )
4342ffvelrnda 5899 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  (
f `  m )  e.  A )
44 summo.2 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
4544ralrimiva 2795 . . . . . . 7  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
4645adantr 453 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  A. k  e.  A  B  e.  CC )
47 nfcsb1v 3282 . . . . . . . 8  |-  F/_ k [_ ( f `  m
)  /  k ]_ B
4847nfel1 2588 . . . . . . 7  |-  F/ k
[_ ( f `  m )  /  k ]_ B  e.  CC
49 csbeq1a 3275 . . . . . . . 8  |-  ( k  =  ( f `  m )  ->  B  =  [_ ( f `  m )  /  k ]_ B )
5049eleq1d 2508 . . . . . . 7  |-  ( k  =  ( f `  m )  ->  ( B  e.  CC  <->  [_ ( f `
 m )  / 
k ]_ B  e.  CC ) )
5148, 50rspc 3052 . . . . . 6  |-  ( ( f `  m )  e.  A  ->  ( A. k  e.  A  B  e.  CC  ->  [_ ( f `  m
)  /  k ]_ B  e.  CC )
)
5243, 46, 51sylc 59 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  [_ (
f `  m )  /  k ]_ B  e.  CC )
53 fveq2 5757 . . . . . . 7  |-  ( n  =  m  ->  (
f `  n )  =  ( f `  m ) )
5453csbeq1d 3273 . . . . . 6  |-  ( n  =  m  ->  [_ (
f `  n )  /  k ]_ B  =  [_ ( f `  m )  /  k ]_ B )
55 summo.3 . . . . . 6  |-  G  =  ( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
5654, 55fvmptg 5833 . . . . 5  |-  ( ( m  e.  NN  /\  [_ ( f `  m
)  /  k ]_ B  e.  CC )  ->  ( G `  m
)  =  [_ (
f `  m )  /  k ]_ B
)
5740, 52, 56syl2anc 644 . . . 4  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  ( G `  m )  =  [_ ( f `  m )  /  k ]_ B )
5857, 52eqeltrd 2516 . . 3  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  ( G `  m )  e.  CC )
59 f1oeq2 5695 . . . . . . . . . . . 12  |-  ( ( 1 ... M )  =  ( 1 ... N )  ->  ( K : ( 1 ... M ) -1-1-onto-> A  <->  K : ( 1 ... N ) -1-1-onto-> A ) )
6035, 59syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( K : ( 1 ... M ) -1-1-onto-> A  <-> 
K : ( 1 ... N ) -1-1-onto-> A ) )
6116, 60mpbird 225 . . . . . . . . . 10  |-  ( ph  ->  K : ( 1 ... M ) -1-1-onto-> A )
62 f1of 5703 . . . . . . . . . 10  |-  ( K : ( 1 ... M ) -1-1-onto-> A  ->  K :
( 1 ... M
) --> A )
6361, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  K : ( 1 ... M ) --> A )
64 fvco3 5829 . . . . . . . . 9  |-  ( ( K : ( 1 ... M ) --> A  /\  i  e.  ( 1 ... M ) )  ->  ( ( `' f  o.  K
) `  i )  =  ( `' f `
 ( K `  i ) ) )
6563, 64sylan 459 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( `' f  o.  K ) `  i
)  =  ( `' f `  ( K `
 i ) ) )
6665fveq2d 5761 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( ( `' f  o.  K
) `  i )
)  =  ( f `
 ( `' f `
 ( K `  i ) ) ) )
6713adantr 453 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  f : ( 1 ... M ) -1-1-onto-> A )
6863ffvelrnda 5899 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( K `  i )  e.  A )
69 f1ocnvfv2 6044 . . . . . . . 8  |-  ( ( f : ( 1 ... M ) -1-1-onto-> A  /\  ( K `  i )  e.  A )  -> 
( f `  ( `' f `  ( K `  i )
) )  =  ( K `  i ) )
7067, 68, 69syl2anc 644 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( `' f `  ( K `  i ) ) )  =  ( K `  i ) )
7166, 70eqtr2d 2475 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( K `  i )  =  ( f `  ( ( `' f  o.  K ) `  i ) ) )
7271csbeq1d 3273 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  [_ ( K `  i )  /  k ]_ B  =  [_ ( f `  ( ( `' f  o.  K ) `  i ) )  / 
k ]_ B )
7372fveq2d 5761 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (  _I  `  [_ ( K `
 i )  / 
k ]_ B )  =  (  _I  `  [_ (
f `  ( ( `' f  o.  K
) `  i )
)  /  k ]_ B ) )
74 elfznn 11111 . . . . . 6  |-  ( i  e.  ( 1 ... M )  ->  i  e.  NN )
7574adantl 454 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  i  e.  NN )
76 fveq2 5757 . . . . . . 7  |-  ( n  =  i  ->  ( K `  n )  =  ( K `  i ) )
7776csbeq1d 3273 . . . . . 6  |-  ( n  =  i  ->  [_ ( K `  n )  /  k ]_ B  =  [_ ( K `  i )  /  k ]_ B )
78 summolem3.4 . . . . . 6  |-  H  =  ( n  e.  NN  |->  [_ ( K `  n
)  /  k ]_ B )
7977, 78fvmpti 5834 . . . . 5  |-  ( i  e.  NN  ->  ( H `  i )  =  (  _I  `  [_ ( K `  i )  /  k ]_ B
) )
8075, 79syl 16 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( H `  i )  =  (  _I  `  [_ ( K `  i )  /  k ]_ B
) )
81 f1of 5703 . . . . . . 7  |-  ( ( `' f  o.  K
) : ( 1 ... M ) -1-1-onto-> ( 1 ... M )  -> 
( `' f  o.  K ) : ( 1 ... M ) --> ( 1 ... M
) )
8238, 81syl 16 . . . . . 6  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... M ) --> ( 1 ... M
) )
8382ffvelrnda 5899 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( `' f  o.  K ) `  i
)  e.  ( 1 ... M ) )
84 elfznn 11111 . . . . 5  |-  ( ( ( `' f  o.  K ) `  i
)  e.  ( 1 ... M )  -> 
( ( `' f  o.  K ) `  i )  e.  NN )
85 fveq2 5757 . . . . . . 7  |-  ( n  =  ( ( `' f  o.  K ) `
 i )  -> 
( f `  n
)  =  ( f `
 ( ( `' f  o.  K ) `
 i ) ) )
8685csbeq1d 3273 . . . . . 6  |-  ( n  =  ( ( `' f  o.  K ) `
 i )  ->  [_ ( f `  n
)  /  k ]_ B  =  [_ ( f `
 ( ( `' f  o.  K ) `
 i ) )  /  k ]_ B
)
8786, 55fvmpti 5834 . . . . 5  |-  ( ( ( `' f  o.  K ) `  i
)  e.  NN  ->  ( G `  ( ( `' f  o.  K
) `  i )
)  =  (  _I 
`  [_ ( f `  ( ( `' f  o.  K ) `  i ) )  / 
k ]_ B ) )
8883, 84, 873syl 19 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  ( ( `' f  o.  K
) `  i )
)  =  (  _I 
`  [_ ( f `  ( ( `' f  o.  K ) `  i ) )  / 
k ]_ B ) )
8973, 80, 883eqtr4d 2484 . . 3  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( H `  i )  =  ( G `  ( ( `' f  o.  K ) `  i ) ) )
902, 4, 6, 10, 12, 38, 58, 89seqf1o 11395 . 2  |-  ( ph  ->  (  seq  1 (  +  ,  H ) `
 M )  =  (  seq  1 (  +  ,  G ) `
 M ) )
9134fveq2d 5761 . 2  |-  ( ph  ->  (  seq  1 (  +  ,  H ) `
 M )  =  (  seq  1 (  +  ,  H ) `
 N ) )
9290, 91eqtr3d 2476 1  |-  ( ph  ->  (  seq  1 (  +  ,  G ) `
 M )  =  (  seq  1 (  +  ,  H ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   A.wral 2711   [_csb 3267    C_ wss 3306   ifcif 3763   class class class wbr 4237    e. cmpt 4291    _I cid 4522   `'ccnv 4906    o. ccom 4911   -->wf 5479   -1-1-onto->wf1o 5482   ` cfv 5483  (class class class)co 6110    ~~ cen 7135   Fincfn 7138   CCcc 9019   0cc0 9021   1c1 9022    + caddc 9024   NNcn 10031   NN0cn0 10252   ZZcz 10313   ZZ>=cuz 10519   ...cfz 11074    seq cseq 11354   #chash 11649
This theorem is referenced by:  summolem2a  12540  summo  12542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-card 7857  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-fzo 11167  df-seq 11355  df-hash 11650
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