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Theorem summolem3 12187
Description: Lemma for summo 12190. (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 8819 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC )  ->  ( m  +  j )  e.  CC )
21adantl 452 . . 3  |-  ( (
ph  /\  ( m  e.  CC  /\  j  e.  CC ) )  -> 
( m  +  j )  e.  CC )
3 addcom 8998 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC )  ->  ( m  +  j )  =  ( j  +  m ) )
43adantl 452 . . 3  |-  ( (
ph  /\  ( m  e.  CC  /\  j  e.  CC ) )  -> 
( m  +  j )  =  ( j  +  m ) )
5 addass 8824 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC  /\  y  e.  CC )  ->  (
( m  +  j )  +  y )  =  ( m  +  ( j  +  y ) ) )
65adantl 452 . . 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 445 . . . 4  |-  ( ph  ->  M  e.  NN )
9 nnuz 10263 . . . 4  |-  NN  =  ( ZZ>= `  1 )
108, 9syl6eleq 2373 . . 3  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
11 ssid 3197 . . . 4  |-  CC  C_  CC
1211a1i 10 . . 3  |-  ( ph  ->  CC  C_  CC )
13 summolem3.6 . . . . . 6  |-  ( ph  ->  f : ( 1 ... M ) -1-1-onto-> A )
14 f1ocnv 5485 . . . . . 6  |-  ( f : ( 1 ... M ) -1-1-onto-> A  ->  `' f : A -1-1-onto-> ( 1 ... M
) )
1513, 14syl 15 . . . . 5  |-  ( ph  ->  `' f : A -1-1-onto-> (
1 ... M ) )
16 summolem3.7 . . . . 5  |-  ( ph  ->  K : ( 1 ... N ) -1-1-onto-> A )
17 f1oco 5496 . . . . 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 642 . . . 4  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) )
19 ovex 5883 . . . . . . . . . 10  |-  ( 1 ... N )  e. 
_V
2019f1oen 6882 . . . . . . . . 9  |-  ( ( `' f  o.  K
) : ( 1 ... N ) -1-1-onto-> ( 1 ... M )  -> 
( 1 ... N
)  ~~  ( 1 ... M ) )
2118, 20syl 15 . . . . . . . 8  |-  ( ph  ->  ( 1 ... N
)  ~~  ( 1 ... M ) )
22 fzfi 11034 . . . . . . . . 9  |-  ( 1 ... N )  e. 
Fin
23 fzfi 11034 . . . . . . . . 9  |-  ( 1 ... M )  e. 
Fin
24 hashen 11346 . . . . . . . . 9  |-  ( ( ( 1 ... N
)  e.  Fin  /\  ( 1 ... M
)  e.  Fin )  ->  ( ( # `  (
1 ... N ) )  =  ( # `  (
1 ... M ) )  <-> 
( 1 ... N
)  ~~  ( 1 ... M ) ) )
2522, 23, 24mp2an 653 . . . . . . . 8  |-  ( (
# `  ( 1 ... N ) )  =  ( # `  (
1 ... M ) )  <-> 
( 1 ... N
)  ~~  ( 1 ... M ) )
2621, 25sylibr 203 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  ( # `  (
1 ... M ) ) )
277simprd 449 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
28 nnnn0 9972 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  NN0 )
29 hashfz1 11345 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
3027, 28, 293syl 18 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
31 nnnn0 9972 . . . . . . . 8  |-  ( M  e.  NN  ->  M  e.  NN0 )
32 hashfz1 11345 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
338, 31, 323syl 18 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
3426, 30, 333eqtr3rd 2324 . . . . . 6  |-  ( ph  ->  M  =  N )
3534oveq2d 5874 . . . . 5  |-  ( ph  ->  ( 1 ... M
)  =  ( 1 ... N ) )
36 f1oeq2 5464 . . . . 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 15 . . . 4  |-  ( ph  ->  ( ( `' f  o.  K ) : ( 1 ... M
)
-1-1-onto-> ( 1 ... M
)  <->  ( `' f  o.  K ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... M
) ) )
3818, 37mpbird 223 . . 3  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... M ) -1-1-onto-> ( 1 ... M ) )
39 elfznn 10819 . . . . . 6  |-  ( m  e.  ( 1 ... M )  ->  m  e.  NN )
4039adantl 452 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  m  e.  NN )
41 f1of 5472 . . . . . . . 8  |-  ( f : ( 1 ... M ) -1-1-onto-> A  ->  f :
( 1 ... M
) --> A )
4213, 41syl 15 . . . . . . 7  |-  ( ph  ->  f : ( 1 ... M ) --> A )
43 ffvelrn 5663 . . . . . . 7  |-  ( ( f : ( 1 ... M ) --> A  /\  m  e.  ( 1 ... M ) )  ->  ( f `  m )  e.  A
)
4442, 43sylan 457 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  (
f `  m )  e.  A )
45 summo.2 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
4645ralrimiva 2626 . . . . . . 7  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
4746adantr 451 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  A. k  e.  A  B  e.  CC )
48 nfcsb1v 3113 . . . . . . . 8  |-  F/_ k [_ ( f `  m
)  /  k ]_ B
4948nfel1 2429 . . . . . . 7  |-  F/ k
[_ ( f `  m )  /  k ]_ B  e.  CC
50 csbeq1a 3089 . . . . . . . 8  |-  ( k  =  ( f `  m )  ->  B  =  [_ ( f `  m )  /  k ]_ B )
5150eleq1d 2349 . . . . . . 7  |-  ( k  =  ( f `  m )  ->  ( B  e.  CC  <->  [_ ( f `
 m )  / 
k ]_ B  e.  CC ) )
5249, 51rspc 2878 . . . . . 6  |-  ( ( f `  m )  e.  A  ->  ( A. k  e.  A  B  e.  CC  ->  [_ ( f `  m
)  /  k ]_ B  e.  CC )
)
5344, 47, 52sylc 56 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  [_ (
f `  m )  /  k ]_ B  e.  CC )
54 fveq2 5525 . . . . . . 7  |-  ( n  =  m  ->  (
f `  n )  =  ( f `  m ) )
5554csbeq1d 3087 . . . . . 6  |-  ( n  =  m  ->  [_ (
f `  n )  /  k ]_ B  =  [_ ( f `  m )  /  k ]_ B )
56 summo.3 . . . . . 6  |-  G  =  ( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
5755, 56fvmptg 5600 . . . . 5  |-  ( ( m  e.  NN  /\  [_ ( f `  m
)  /  k ]_ B  e.  CC )  ->  ( G `  m
)  =  [_ (
f `  m )  /  k ]_ B
)
5840, 53, 57syl2anc 642 . . . 4  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  ( G `  m )  =  [_ ( f `  m )  /  k ]_ B )
5958, 53eqeltrd 2357 . . 3  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  ( G `  m )  e.  CC )
60 f1oeq2 5464 . . . . . . . . . . . 12  |-  ( ( 1 ... M )  =  ( 1 ... N )  ->  ( K : ( 1 ... M ) -1-1-onto-> A  <->  K : ( 1 ... N ) -1-1-onto-> A ) )
6135, 60syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( K : ( 1 ... M ) -1-1-onto-> A  <-> 
K : ( 1 ... N ) -1-1-onto-> A ) )
6216, 61mpbird 223 . . . . . . . . . 10  |-  ( ph  ->  K : ( 1 ... M ) -1-1-onto-> A )
63 f1of 5472 . . . . . . . . . 10  |-  ( K : ( 1 ... M ) -1-1-onto-> A  ->  K :
( 1 ... M
) --> A )
6462, 63syl 15 . . . . . . . . 9  |-  ( ph  ->  K : ( 1 ... M ) --> A )
65 fvco3 5596 . . . . . . . . 9  |-  ( ( K : ( 1 ... M ) --> A  /\  i  e.  ( 1 ... M ) )  ->  ( ( `' f  o.  K
) `  i )  =  ( `' f `
 ( K `  i ) ) )
6664, 65sylan 457 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( `' f  o.  K ) `  i
)  =  ( `' f `  ( K `
 i ) ) )
6766fveq2d 5529 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( ( `' f  o.  K
) `  i )
)  =  ( f `
 ( `' f `
 ( K `  i ) ) ) )
6813adantr 451 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  f : ( 1 ... M ) -1-1-onto-> A )
69 ffvelrn 5663 . . . . . . . . 9  |-  ( ( K : ( 1 ... M ) --> A  /\  i  e.  ( 1 ... M ) )  ->  ( K `  i )  e.  A
)
7064, 69sylan 457 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( K `  i )  e.  A )
71 f1ocnvfv2 5793 . . . . . . . 8  |-  ( ( f : ( 1 ... M ) -1-1-onto-> A  /\  ( K `  i )  e.  A )  -> 
( f `  ( `' f `  ( K `  i )
) )  =  ( K `  i ) )
7268, 70, 71syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( `' f `  ( K `  i ) ) )  =  ( K `  i ) )
7367, 72eqtr2d 2316 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( K `  i )  =  ( f `  ( ( `' f  o.  K ) `  i ) ) )
7473csbeq1d 3087 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  [_ ( K `  i )  /  k ]_ B  =  [_ ( f `  ( ( `' f  o.  K ) `  i ) )  / 
k ]_ B )
7574fveq2d 5529 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (  _I  `  [_ ( K `
 i )  / 
k ]_ B )  =  (  _I  `  [_ (
f `  ( ( `' f  o.  K
) `  i )
)  /  k ]_ B ) )
76 elfznn 10819 . . . . . 6  |-  ( i  e.  ( 1 ... M )  ->  i  e.  NN )
7776adantl 452 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  i  e.  NN )
78 fveq2 5525 . . . . . . 7  |-  ( n  =  i  ->  ( K `  n )  =  ( K `  i ) )
7978csbeq1d 3087 . . . . . 6  |-  ( n  =  i  ->  [_ ( K `  n )  /  k ]_ B  =  [_ ( K `  i )  /  k ]_ B )
80 summolem3.4 . . . . . 6  |-  H  =  ( n  e.  NN  |->  [_ ( K `  n
)  /  k ]_ B )
8179, 80fvmpti 5601 . . . . 5  |-  ( i  e.  NN  ->  ( H `  i )  =  (  _I  `  [_ ( K `  i )  /  k ]_ B
) )
8277, 81syl 15 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( H `  i )  =  (  _I  `  [_ ( K `  i )  /  k ]_ B
) )
83 f1of 5472 . . . . . . 7  |-  ( ( `' f  o.  K
) : ( 1 ... M ) -1-1-onto-> ( 1 ... M )  -> 
( `' f  o.  K ) : ( 1 ... M ) --> ( 1 ... M
) )
8438, 83syl 15 . . . . . 6  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... M ) --> ( 1 ... M
) )
85 ffvelrn 5663 . . . . . 6  |-  ( ( ( `' f  o.  K ) : ( 1 ... M ) --> ( 1 ... M
)  /\  i  e.  ( 1 ... M
) )  ->  (
( `' f  o.  K ) `  i
)  e.  ( 1 ... M ) )
8684, 85sylan 457 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( `' f  o.  K ) `  i
)  e.  ( 1 ... M ) )
87 elfznn 10819 . . . . 5  |-  ( ( ( `' f  o.  K ) `  i
)  e.  ( 1 ... M )  -> 
( ( `' f  o.  K ) `  i )  e.  NN )
88 fveq2 5525 . . . . . . 7  |-  ( n  =  ( ( `' f  o.  K ) `
 i )  -> 
( f `  n
)  =  ( f `
 ( ( `' f  o.  K ) `
 i ) ) )
8988csbeq1d 3087 . . . . . 6  |-  ( n  =  ( ( `' f  o.  K ) `
 i )  ->  [_ ( f `  n
)  /  k ]_ B  =  [_ ( f `
 ( ( `' f  o.  K ) `
 i ) )  /  k ]_ B
)
9089, 56fvmpti 5601 . . . . 5  |-  ( ( ( `' f  o.  K ) `  i
)  e.  NN  ->  ( G `  ( ( `' f  o.  K
) `  i )
)  =  (  _I 
`  [_ ( f `  ( ( `' f  o.  K ) `  i ) )  / 
k ]_ B ) )
9186, 87, 903syl 18 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  ( ( `' f  o.  K
) `  i )
)  =  (  _I 
`  [_ ( f `  ( ( `' f  o.  K ) `  i ) )  / 
k ]_ B ) )
9275, 82, 913eqtr4d 2325 . . 3  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( H `  i )  =  ( G `  ( ( `' f  o.  K ) `  i ) ) )
932, 4, 6, 10, 12, 38, 59, 92seqf1o 11087 . 2  |-  ( ph  ->  (  seq  1 (  +  ,  H ) `
 M )  =  (  seq  1 (  +  ,  G ) `
 M ) )
9434fveq2d 5529 . 2  |-  ( ph  ->  (  seq  1 (  +  ,  H ) `
 M )  =  (  seq  1 (  +  ,  H ) `
 N ) )
9593, 94eqtr3d 2317 1  |-  ( ph  ->  (  seq  1 (  +  ,  G ) `
 M )  =  (  seq  1 (  +  ,  H ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   [_csb 3081    C_ wss 3152   ifcif 3565   class class class wbr 4023    e. cmpt 4077    _I cid 4304   `'ccnv 4688    o. ccom 4693   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    ~~ cen 6860   Fincfn 6863   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046   #chash 11337
This theorem is referenced by:  summolem2a  12188  summo  12190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338
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