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Theorem smupf 12669
Description: The sequence of partial sums of the sequence multiplication. (Contributed by Mario Carneiro, 9-Sep-2016.)
Hypotheses
Ref Expression
smuval.a  |-  ( ph  ->  A  C_  NN0 )
smuval.b  |-  ( ph  ->  B  C_  NN0 )
smuval.p  |-  P  =  seq  0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )
Assertion
Ref Expression
smupf  |-  ( ph  ->  P : NN0 --> ~P NN0 )
Distinct variable groups:    m, n, p, A    ph, n    B, m, n, p
Allowed substitution hints:    ph( m, p)    P( m, n, p)

Proof of Theorem smupf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0nn0 9980 . . . . 5  |-  0  e.  NN0
2 iftrue 3571 . . . . . 6  |-  ( n  =  0  ->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) )  =  (/) )
3 eqid 2283 . . . . . 6  |-  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) )  =  ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) )
4 0ex 4150 . . . . . 6  |-  (/)  e.  _V
52, 3, 4fvmpt 5602 . . . . 5  |-  ( 0  e.  NN0  ->  ( ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) `
 0 )  =  (/) )
61, 5mp1i 11 . . . 4  |-  ( ph  ->  ( ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ` 
0 )  =  (/) )
7 0elpw 4180 . . . 4  |-  (/)  e.  ~P NN0
86, 7syl6eqel 2371 . . 3  |-  ( ph  ->  ( ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ` 
0 )  e.  ~P NN0 )
9 df-ov 5861 . . . . 5  |-  ( x ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e. 
NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } ) ) y )  =  ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) ) `  <. x ,  y >.
)
10 elpwi 3633 . . . . . . . . . . 11  |-  ( p  e.  ~P NN0  ->  p 
C_  NN0 )
1110adantr 451 . . . . . . . . . 10  |-  ( ( p  e.  ~P NN0  /\  m  e.  NN0 )  ->  p  C_  NN0 )
12 ssrab2 3258 . . . . . . . . . 10  |-  { n  e.  NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } 
C_  NN0
13 sadcl 12653 . . . . . . . . . 10  |-  ( ( p  C_  NN0  /\  {
n  e.  NN0  | 
( m  e.  A  /\  ( n  -  m
)  e.  B ) }  C_  NN0 )  -> 
( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } )  C_  NN0 )
1411, 12, 13sylancl 643 . . . . . . . . 9  |-  ( ( p  e.  ~P NN0  /\  m  e.  NN0 )  ->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } )  C_  NN0 )
15 nn0ex 9971 . . . . . . . . . 10  |-  NN0  e.  _V
1615elpw2 4175 . . . . . . . . 9  |-  ( ( p sadd  { n  e. 
NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } )  e.  ~P NN0  <->  (
p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } )  C_  NN0 )
1714, 16sylibr 203 . . . . . . . 8  |-  ( ( p  e.  ~P NN0  /\  m  e.  NN0 )  ->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } )  e.  ~P NN0 )
1817rgen2 2639 . . . . . . 7  |-  A. p  e.  ~P  NN0 A. m  e.  NN0  ( p sadd  {
n  e.  NN0  | 
( m  e.  A  /\  ( n  -  m
)  e.  B ) } )  e.  ~P NN0
19 eqid 2283 . . . . . . . 8  |-  ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) )  =  ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e. 
NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } ) )
2019fmpt2 6191 . . . . . . 7  |-  ( A. p  e.  ~P  NN0 A. m  e.  NN0  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } )  e.  ~P NN0 
<->  ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e. 
NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } ) ) : ( ~P NN0  X.  NN0 )
--> ~P NN0 )
2118, 20mpbi 199 . . . . . 6  |-  ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) ) : ( ~P NN0  X.  NN0 ) --> ~P NN0
2221, 7f0cli 5671 . . . . 5  |-  ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) ) `  <. x ,  y >.
)  e.  ~P NN0
239, 22eqeltri 2353 . . . 4  |-  ( x ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e. 
NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } ) ) y )  e.  ~P NN0
2423a1i 10 . . 3  |-  ( (
ph  /\  ( x  e.  ~P NN0  /\  y  e.  _V ) )  -> 
( x ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) ) y )  e.  ~P NN0 )
25 nn0uz 10262 . . 3  |-  NN0  =  ( ZZ>= `  0 )
26 0z 10035 . . . 4  |-  0  e.  ZZ
2726a1i 10 . . 3  |-  ( ph  ->  0  e.  ZZ )
28 fvex 5539 . . . 4  |-  ( ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) `
 x )  e. 
_V
2928a1i 10 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( 0  +  1 ) ) )  ->  ( (
n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) `
 x )  e. 
_V )
308, 24, 25, 27, 29seqf2 11065 . 2  |-  ( ph  ->  seq  0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ) : NN0 --> ~P
NN0 )
31 smuval.p . . 3  |-  P  =  seq  0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )
3231feq1i 5383 . 2  |-  ( P : NN0 --> ~P NN0  <->  seq  0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ) : NN0 --> ~P
NN0 )
3330, 32sylibr 203 1  |-  ( ph  ->  P : NN0 --> ~P NN0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ifcif 3565   ~Pcpw 3625   <.cop 3643    e. cmpt 4077    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   0cc0 8737   1c1 8738    + caddc 8740    - cmin 9037   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230    seq cseq 11046   sadd csad 12611
This theorem is referenced by:  smupp1  12671  smuval2  12673  smupvallem  12674  smueqlem  12681
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-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-xor 1296  df-tru 1310  df-had 1370  df-cad 1371  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-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-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-seq 11047  df-sad 12642
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