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Theorem smupf 12917
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 10168 . . . . 5  |-  0  e.  NN0
2 iftrue 3688 . . . . . 6  |-  ( n  =  0  ->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) )  =  (/) )
3 eqid 2387 . . . . . 6  |-  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) )  =  ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) )
4 0ex 4280 . . . . . 6  |-  (/)  e.  _V
52, 3, 4fvmpt 5745 . . . . 5  |-  ( 0  e.  NN0  ->  ( ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) `
 0 )  =  (/) )
61, 5mp1i 12 . . . 4  |-  ( ph  ->  ( ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ` 
0 )  =  (/) )
7 0elpw 4310 . . . 4  |-  (/)  e.  ~P NN0
86, 7syl6eqel 2475 . . 3  |-  ( ph  ->  ( ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ` 
0 )  e.  ~P NN0 )
9 df-ov 6023 . . . . 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 3750 . . . . . . . . . . 11  |-  ( p  e.  ~P NN0  ->  p 
C_  NN0 )
1110adantr 452 . . . . . . . . . 10  |-  ( ( p  e.  ~P NN0  /\  m  e.  NN0 )  ->  p  C_  NN0 )
12 ssrab2 3371 . . . . . . . . . 10  |-  { n  e.  NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } 
C_  NN0
13 sadcl 12901 . . . . . . . . . 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 644 . . . . . . . . 9  |-  ( ( p  e.  ~P NN0  /\  m  e.  NN0 )  ->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } )  C_  NN0 )
15 nn0ex 10159 . . . . . . . . . 10  |-  NN0  e.  _V
1615elpw2 4305 . . . . . . . . 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 204 . . . . . . . 8  |-  ( ( p  e.  ~P NN0  /\  m  e.  NN0 )  ->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } )  e.  ~P NN0 )
1817rgen2 2745 . . . . . . 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 2387 . . . . . . . 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 6357 . . . . . . 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 200 . . . . . 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 5819 . . . . 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 2457 . . . 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 11 . . 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 10452 . . 3  |-  NN0  =  ( ZZ>= `  0 )
26 0z 10225 . . . 4  |-  0  e.  ZZ
2726a1i 11 . . 3  |-  ( ph  ->  0  e.  ZZ )
28 fvex 5682 . . . 4  |-  ( ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) `
 x )  e. 
_V
2928a1i 11 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( 0  +  1 ) ) )  ->  ( (
n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) `
 x )  e. 
_V )
308, 24, 25, 27, 29seqf2 11269 . 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 5525 . 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 204 1  |-  ( ph  ->  P : NN0 --> ~P NN0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   {crab 2653   _Vcvv 2899    C_ wss 3263   (/)c0 3571   ifcif 3682   ~Pcpw 3742   <.cop 3760    e. cmpt 4207    X. cxp 4816   -->wf 5390   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   0cc0 8923   1c1 8924    + caddc 8926    - cmin 9223   NN0cn0 10153   ZZcz 10214   ZZ>=cuz 10420    seq cseq 11250   sadd csad 12859
This theorem is referenced by:  smupp1  12919  smuval2  12921  smupvallem  12922  smueqlem  12929
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1311  df-tru 1325  df-had 1386  df-cad 1387  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-seq 11251  df-sad 12890
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