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

Theorem smupf 12685
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 9996 . . . . 5  |-  0  e.  NN0
2 iftrue 3584 . . . . . 6  |-  ( n  =  0  ->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) )  =  (/) )
3 eqid 2296 . . . . . 6  |-  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) )  =  ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) )
4 0ex 4166 . . . . . 6  |-  (/)  e.  _V
52, 3, 4fvmpt 5618 . . . . 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 4196 . . . 4  |-  (/)  e.  ~P NN0
86, 7syl6eqel 2384 . . 3  |-  ( ph  ->  ( ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ` 
0 )  e.  ~P NN0 )
9 df-ov 5877 . . . . 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 3646 . . . . . . . . . . 11  |-  ( p  e.  ~P NN0  ->  p 
C_  NN0 )
1110adantr 451 . . . . . . . . . 10  |-  ( ( p  e.  ~P NN0  /\  m  e.  NN0 )  ->  p  C_  NN0 )
12 ssrab2 3271 . . . . . . . . . 10  |-  { n  e.  NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } 
C_  NN0
13 sadcl 12669 . . . . . . . . . 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 9987 . . . . . . . . . 10  |-  NN0  e.  _V
1615elpw2 4191 . . . . . . . . 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 2652 . . . . . . 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 2296 . . . . . . . 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 6207 . . . . . . 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 5687 . . . . 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 2366 . . . 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 10278 . . 3  |-  NN0  =  ( ZZ>= `  0 )
26 0z 10051 . . . 4  |-  0  e.  ZZ
2726a1i 10 . . 3  |-  ( ph  ->  0  e.  ZZ )
28 fvex 5555 . . . 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 11081 . 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 5399 . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ifcif 3578   ~Pcpw 3638   <.cop 3656    e. cmpt 4093    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   0cc0 8753   1c1 8754    + caddc 8756    - cmin 9053   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246    seq cseq 11062   sadd csad 12627
This theorem is referenced by:  smupp1  12687  smuval2  12689  smupvallem  12690  smueqlem  12697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-sad 12658
  Copyright terms: Public domain W3C validator