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Theorem summolem3 12467
Description: Lemma for summo 12470. (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 9032 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC )  ->  ( m  +  j )  e.  CC )
21adantl 453 . . 3  |-  ( (
ph  /\  ( m  e.  CC  /\  j  e.  CC ) )  -> 
( m  +  j )  e.  CC )
3 addcom 9212 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC )  ->  ( m  +  j )  =  ( j  +  m ) )
43adantl 453 . . 3  |-  ( (
ph  /\  ( m  e.  CC  /\  j  e.  CC ) )  -> 
( m  +  j )  =  ( j  +  m ) )
5 addass 9037 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC  /\  y  e.  CC )  ->  (
( m  +  j )  +  y )  =  ( m  +  ( j  +  y ) ) )
65adantl 453 . . 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 446 . . . 4  |-  ( ph  ->  M  e.  NN )
9 nnuz 10481 . . . 4  |-  NN  =  ( ZZ>= `  1 )
108, 9syl6eleq 2498 . . 3  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
11 ssid 3331 . . . 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 5650 . . . . . 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 5661 . . . . 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 643 . . . 4  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) )
19 ovex 6069 . . . . . . . . . 10  |-  ( 1 ... N )  e. 
_V
2019f1oen 7091 . . . . . . . . 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 11270 . . . . . . . . 9  |-  ( 1 ... N )  e. 
Fin
23 fzfi 11270 . . . . . . . . 9  |-  ( 1 ... M )  e. 
Fin
24 hashen 11590 . . . . . . . . 9  |-  ( ( ( 1 ... N
)  e.  Fin  /\  ( 1 ... M
)  e.  Fin )  ->  ( ( # `  (
1 ... N ) )  =  ( # `  (
1 ... M ) )  <-> 
( 1 ... N
)  ~~  ( 1 ... M ) ) )
2522, 23, 24mp2an 654 . . . . . . . 8  |-  ( (
# `  ( 1 ... N ) )  =  ( # `  (
1 ... M ) )  <-> 
( 1 ... N
)  ~~  ( 1 ... M ) )
2621, 25sylibr 204 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  ( # `  (
1 ... M ) ) )
277simprd 450 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
28 nnnn0 10188 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  NN0 )
29 hashfz1 11589 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
3027, 28, 293syl 19 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
31 nnnn0 10188 . . . . . . . 8  |-  ( M  e.  NN  ->  M  e.  NN0 )
32 hashfz1 11589 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
338, 31, 323syl 19 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
3426, 30, 333eqtr3rd 2449 . . . . . 6  |-  ( ph  ->  M  =  N )
3534oveq2d 6060 . . . . 5  |-  ( ph  ->  ( 1 ... M
)  =  ( 1 ... N ) )
36 f1oeq2 5629 . . . . 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 224 . . 3  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... M ) -1-1-onto-> ( 1 ... M ) )
39 elfznn 11040 . . . . . 6  |-  ( m  e.  ( 1 ... M )  ->  m  e.  NN )
4039adantl 453 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  m  e.  NN )
41 f1of 5637 . . . . . . . 8  |-  ( f : ( 1 ... M ) -1-1-onto-> A  ->  f :
( 1 ... M
) --> A )
4213, 41syl 16 . . . . . . 7  |-  ( ph  ->  f : ( 1 ... M ) --> A )
4342ffvelrnda 5833 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  (
f `  m )  e.  A )
44 summo.2 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
4544ralrimiva 2753 . . . . . . 7  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
4645adantr 452 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  A. k  e.  A  B  e.  CC )
47 nfcsb1v 3247 . . . . . . . 8  |-  F/_ k [_ ( f `  m
)  /  k ]_ B
4847nfel1 2554 . . . . . . 7  |-  F/ k
[_ ( f `  m )  /  k ]_ B  e.  CC
49 csbeq1a 3223 . . . . . . . 8  |-  ( k  =  ( f `  m )  ->  B  =  [_ ( f `  m )  /  k ]_ B )
5049eleq1d 2474 . . . . . . 7  |-  ( k  =  ( f `  m )  ->  ( B  e.  CC  <->  [_ ( f `
 m )  / 
k ]_ B  e.  CC ) )
5148, 50rspc 3010 . . . . . 6  |-  ( ( f `  m )  e.  A  ->  ( A. k  e.  A  B  e.  CC  ->  [_ ( f `  m
)  /  k ]_ B  e.  CC )
)
5243, 46, 51sylc 58 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  [_ (
f `  m )  /  k ]_ B  e.  CC )
53 fveq2 5691 . . . . . . 7  |-  ( n  =  m  ->  (
f `  n )  =  ( f `  m ) )
5453csbeq1d 3221 . . . . . 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 5767 . . . . 5  |-  ( ( m  e.  NN  /\  [_ ( f `  m
)  /  k ]_ B  e.  CC )  ->  ( G `  m
)  =  [_ (
f `  m )  /  k ]_ B
)
5740, 52, 56syl2anc 643 . . . 4  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  ( G `  m )  =  [_ ( f `  m )  /  k ]_ B )
5857, 52eqeltrd 2482 . . 3  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  ( G `  m )  e.  CC )
59 f1oeq2 5629 . . . . . . . . . . . 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 224 . . . . . . . . . 10  |-  ( ph  ->  K : ( 1 ... M ) -1-1-onto-> A )
62 f1of 5637 . . . . . . . . . 10  |-  ( K : ( 1 ... M ) -1-1-onto-> A  ->  K :
( 1 ... M
) --> A )
6361, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  K : ( 1 ... M ) --> A )
64 fvco3 5763 . . . . . . . . 9  |-  ( ( K : ( 1 ... M ) --> A  /\  i  e.  ( 1 ... M ) )  ->  ( ( `' f  o.  K
) `  i )  =  ( `' f `
 ( K `  i ) ) )
6563, 64sylan 458 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( `' f  o.  K ) `  i
)  =  ( `' f `  ( K `
 i ) ) )
6665fveq2d 5695 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( ( `' f  o.  K
) `  i )
)  =  ( f `
 ( `' f `
 ( K `  i ) ) ) )
6713adantr 452 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  f : ( 1 ... M ) -1-1-onto-> A )
6863ffvelrnda 5833 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( K `  i )  e.  A )
69 f1ocnvfv2 5978 . . . . . . . 8  |-  ( ( f : ( 1 ... M ) -1-1-onto-> A  /\  ( K `  i )  e.  A )  -> 
( f `  ( `' f `  ( K `  i )
) )  =  ( K `  i ) )
7067, 68, 69syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( `' f `  ( K `  i ) ) )  =  ( K `  i ) )
7166, 70eqtr2d 2441 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( K `  i )  =  ( f `  ( ( `' f  o.  K ) `  i ) ) )
7271csbeq1d 3221 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  [_ ( K `  i )  /  k ]_ B  =  [_ ( f `  ( ( `' f  o.  K ) `  i ) )  / 
k ]_ B )
7372fveq2d 5695 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (  _I  `  [_ ( K `
 i )  / 
k ]_ B )  =  (  _I  `  [_ (
f `  ( ( `' f  o.  K
) `  i )
)  /  k ]_ B ) )
74 elfznn 11040 . . . . . 6  |-  ( i  e.  ( 1 ... M )  ->  i  e.  NN )
7574adantl 453 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  i  e.  NN )
76 fveq2 5691 . . . . . . 7  |-  ( n  =  i  ->  ( K `  n )  =  ( K `  i ) )
7776csbeq1d 3221 . . . . . 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 5768 . . . . 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 5637 . . . . . . 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 5833 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( `' f  o.  K ) `  i
)  e.  ( 1 ... M ) )
84 elfznn 11040 . . . . 5  |-  ( ( ( `' f  o.  K ) `  i
)  e.  ( 1 ... M )  -> 
( ( `' f  o.  K ) `  i )  e.  NN )
85 fveq2 5691 . . . . . . 7  |-  ( n  =  ( ( `' f  o.  K ) `
 i )  -> 
( f `  n
)  =  ( f `
 ( ( `' f  o.  K ) `
 i ) ) )
8685csbeq1d 3221 . . . . . 6  |-  ( n  =  ( ( `' f  o.  K ) `
 i )  ->  [_ ( f `  n
)  /  k ]_ B  =  [_ ( f `
 ( ( `' f  o.  K ) `
 i ) )  /  k ]_ B
)
8786, 55fvmpti 5768 . . . . 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 2450 . . 3  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( H `  i )  =  ( G `  ( ( `' f  o.  K ) `  i ) ) )
902, 4, 6, 10, 12, 38, 58, 89seqf1o 11323 . 2  |-  ( ph  ->  (  seq  1 (  +  ,  H ) `
 M )  =  (  seq  1 (  +  ,  G ) `
 M ) )
9134fveq2d 5695 . 2  |-  ( ph  ->  (  seq  1 (  +  ,  H ) `
 M )  =  (  seq  1 (  +  ,  H ) `
 N ) )
9290, 91eqtr3d 2442 1  |-  ( ph  ->  (  seq  1 (  +  ,  G ) `
 M )  =  (  seq  1 (  +  ,  H ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   [_csb 3215    C_ wss 3284   ifcif 3703   class class class wbr 4176    e. cmpt 4230    _I cid 4457   `'ccnv 4840    o. ccom 4845   -->wf 5413   -1-1-onto->wf1o 5416   ` cfv 5417  (class class class)co 6044    ~~ cen 7069   Fincfn 7072   CCcc 8948   0cc0 8950   1c1 8951    + caddc 8953   NNcn 9960   NN0cn0 10181   ZZcz 10242   ZZ>=cuz 10448   ...cfz 11003    seq cseq 11282   #chash 11577
This theorem is referenced by:  summolem2a  12468  summo  12470
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-fzo 11095  df-seq 11283  df-hash 11578
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