MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  smupf Unicode version

Theorem smupf 12945
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 10192 . . . . 5  |-  0  e.  NN0
2 iftrue 3705 . . . . . 6  |-  ( n  =  0  ->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) )  =  (/) )
3 eqid 2404 . . . . . 6  |-  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) )  =  ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) )
4 0ex 4299 . . . . . 6  |-  (/)  e.  _V
52, 3, 4fvmpt 5765 . . . . 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 4329 . . . 4  |-  (/)  e.  ~P NN0
86, 7syl6eqel 2492 . . 3  |-  ( ph  ->  ( ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ` 
0 )  e.  ~P NN0 )
9 df-ov 6043 . . . . 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 3767 . . . . . . . . . . 11  |-  ( p  e.  ~P NN0  ->  p 
C_  NN0 )
1110adantr 452 . . . . . . . . . 10  |-  ( ( p  e.  ~P NN0  /\  m  e.  NN0 )  ->  p  C_  NN0 )
12 ssrab2 3388 . . . . . . . . . 10  |-  { n  e.  NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } 
C_  NN0
13 sadcl 12929 . . . . . . . . . 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 10183 . . . . . . . . . 10  |-  NN0  e.  _V
1615elpw2 4324 . . . . . . . . 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 2762 . . . . . . 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 2404 . . . . . . . 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 6377 . . . . . . 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 5839 . . . . 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 2474 . . . 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 10476 . . 3  |-  NN0  =  ( ZZ>= `  0 )
26 0z 10249 . . . 4  |-  0  e.  ZZ
2726a1i 11 . . 3  |-  ( ph  ->  0  e.  ZZ )
28 fvex 5701 . . . 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 11297 . 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 5544 . 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 1721   A.wral 2666   {crab 2670   _Vcvv 2916    C_ wss 3280   (/)c0 3588   ifcif 3699   ~Pcpw 3759   <.cop 3777    e. cmpt 4226    X. cxp 4835   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   0cc0 8946   1c1 8947    + caddc 8949    - cmin 9247   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444    seq cseq 11278   sadd csad 12887
This theorem is referenced by:  smupp1  12947  smuval2  12949  smupvallem  12950  smueqlem  12957
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-seq 11279  df-sad 12918
  Copyright terms: Public domain W3C validator