Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prodfn0 Structured version   Unicode version

Theorem prodfn0 28633
Description: No term of a non-zero infinite product is zero. (Contributed by Scott Fenton, 14-Jan-2018.)
Hypotheses
Ref Expression
prodfn0.1  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
prodfn0.2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  CC )
prodfn0.3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  =/=  0
)
Assertion
Ref Expression
prodfn0  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  =/=  0 )
Distinct variable groups:    k, F    ph, k    k, M    k, N

Proof of Theorem prodfn0
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prodfn0.1 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzfz2 11694 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
31, 2syl 16 . 2  |-  ( ph  ->  N  e.  ( M ... N ) )
4 fveq2 5866 . . . . 5  |-  ( m  =  M  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  M
) )
54neeq1d 2744 . . . 4  |-  ( m  =  M  ->  (
(  seq M (  x.  ,  F ) `  m )  =/=  0  <->  (  seq M (  x.  ,  F ) `  M )  =/=  0
) )
65imbi2d 316 . . 3  |-  ( m  =  M  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
)  =/=  0 )  <-> 
( ph  ->  (  seq M (  x.  ,  F ) `  M
)  =/=  0 ) ) )
7 fveq2 5866 . . . . 5  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
87neeq1d 2744 . . . 4  |-  ( m  =  n  ->  (
(  seq M (  x.  ,  F ) `  m )  =/=  0  <->  (  seq M (  x.  ,  F ) `  n )  =/=  0
) )
98imbi2d 316 . . 3  |-  ( m  =  n  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
)  =/=  0 )  <-> 
( ph  ->  (  seq M (  x.  ,  F ) `  n
)  =/=  0 ) ) )
10 fveq2 5866 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
1110neeq1d 2744 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
(  seq M (  x.  ,  F ) `  m )  =/=  0  <->  (  seq M (  x.  ,  F ) `  ( n  +  1
) )  =/=  0
) )
1211imbi2d 316 . . 3  |-  ( m  =  ( n  + 
1 )  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
)  =/=  0 )  <-> 
( ph  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =/=  0 ) ) )
13 fveq2 5866 . . . . 5  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
1413neeq1d 2744 . . . 4  |-  ( m  =  N  ->  (
(  seq M (  x.  ,  F ) `  m )  =/=  0  <->  (  seq M (  x.  ,  F ) `  N )  =/=  0
) )
1514imbi2d 316 . . 3  |-  ( m  =  N  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
)  =/=  0 )  <-> 
( ph  ->  (  seq M (  x.  ,  F ) `  N
)  =/=  0 ) ) )
16 eluzfz1 11693 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
17 elfzelz 11688 . . . . . . . 8  |-  ( M  e.  ( M ... N )  ->  M  e.  ZZ )
1817adantl 466 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  M  e.  ZZ )
19 seq1 12088 . . . . . . 7  |-  ( M  e.  ZZ  ->  (  seq M (  x.  ,  F ) `  M
)  =  ( F `
 M ) )
2018, 19syl 16 . . . . . 6  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  (  seq M (  x.  ,  F ) `  M
)  =  ( F `
 M ) )
21 fveq2 5866 . . . . . . . . . 10  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2221neeq1d 2744 . . . . . . . . 9  |-  ( k  =  M  ->  (
( F `  k
)  =/=  0  <->  ( F `  M )  =/=  0 ) )
2322imbi2d 316 . . . . . . . 8  |-  ( k  =  M  ->  (
( ph  ->  ( F `
 k )  =/=  0 )  <->  ( ph  ->  ( F `  M
)  =/=  0 ) ) )
24 prodfn0.3 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  =/=  0
)
2524expcom 435 . . . . . . . 8  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k )  =/=  0
) )
2623, 25vtoclga 3177 . . . . . . 7  |-  ( M  e.  ( M ... N )  ->  ( ph  ->  ( F `  M )  =/=  0
) )
2726impcom 430 . . . . . 6  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  ( F `  M )  =/=  0
)
2820, 27eqnetrd 2760 . . . . 5  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  (  seq M (  x.  ,  F ) `  M
)  =/=  0 )
2928expcom 435 . . . 4  |-  ( M  e.  ( M ... N )  ->  ( ph  ->  (  seq M
(  x.  ,  F
) `  M )  =/=  0 ) )
3016, 29syl 16 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ph  ->  (  seq M (  x.  ,  F ) `
 M )  =/=  0 ) )
31 elfzouz 11801 . . . . . . . . 9  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( ZZ>= `  M )
)
32313ad2ant2 1018 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  n  e.  ( ZZ>= `  M )
)
33 seqp1 12090 . . . . . . . 8  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
3432, 33syl 16 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
35 elfzofz 11811 . . . . . . . . . 10  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( M ... N ) )
36 elfzuz 11684 . . . . . . . . . . . 12  |-  ( n  e.  ( M ... N )  ->  n  e.  ( ZZ>= `  M )
)
3736adantl 466 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  n  e.  ( ZZ>= `  M )
)
38 elfzuz3 11685 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  n )
)
39 fzss2 11723 . . . . . . . . . . . . . . 15  |-  ( N  e.  ( ZZ>= `  n
)  ->  ( M ... n )  C_  ( M ... N ) )
4038, 39syl 16 . . . . . . . . . . . . . 14  |-  ( n  e.  ( M ... N )  ->  ( M ... n )  C_  ( M ... N ) )
4140sselda 3504 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( M ... N )  /\  k  e.  ( M ... n ) )  -> 
k  e.  ( M ... N ) )
42 prodfn0.2 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  CC )
4341, 42sylan2 474 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( M ... N
)  /\  k  e.  ( M ... n ) ) )  ->  ( F `  k )  e.  CC )
4443anassrs 648 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( M ... N
) )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  e.  CC )
45 mulcl 9576 . . . . . . . . . . . 12  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
4645adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( M ... N
) )  /\  (
k  e.  CC  /\  x  e.  CC )
)  ->  ( k  x.  x )  e.  CC )
4737, 44, 46seqcl 12095 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  (  seq M (  x.  ,  F ) `  n
)  e.  CC )
4835, 47sylan2 474 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq M
(  x.  ,  F
) `  n )  e.  CC )
49483adant3 1016 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq M (  x.  ,  F ) `  n
)  e.  CC )
50 fzofzp1 11877 . . . . . . . . . . 11  |-  ( n  e.  ( M..^ N
)  ->  ( n  +  1 )  e.  ( M ... N
) )
51 fveq2 5866 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
5251eleq1d 2536 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
5352imbi2d 316 . . . . . . . . . . . 12  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  e.  CC )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) ) )
5442expcom 435 . . . . . . . . . . . 12  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k )  e.  CC ) )
5553, 54vtoclga 3177 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) )  e.  CC ) )
5650, 55syl 16 . . . . . . . . . 10  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) )
5756impcom 430 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  e.  CC )
58573adant3 1016 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  ( F `  ( n  +  1 ) )  e.  CC )
59 simp3 998 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq M (  x.  ,  F ) `  n
)  =/=  0 )
6051neeq1d 2744 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  =/=  0  <->  ( F `  ( n  +  1 ) )  =/=  0 ) )
6160imbi2d 316 . . . . . . . . . . . 12  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  =/=  0 )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  =/=  0 ) ) )
6261, 25vtoclga 3177 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) )  =/=  0
) )
6362impcom 430 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  +  1 )  e.  ( M ... N
) )  ->  ( F `  ( n  +  1 ) )  =/=  0 )
6450, 63sylan2 474 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  =/=  0
)
65643adant3 1016 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  ( F `  ( n  +  1 ) )  =/=  0
)
6649, 58, 59, 65mulne0d 10201 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) )  =/=  0 )
6734, 66eqnetrd 2760 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =/=  0 )
68673exp 1195 . . . . 5  |-  ( ph  ->  ( n  e.  ( M..^ N )  -> 
( (  seq M
(  x.  ,  F
) `  n )  =/=  0  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =/=  0 ) ) )
6968com12 31 . . . 4  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  n )  =/=  0  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =/=  0 ) ) )
7069a2d 26 . . 3  |-  ( n  e.  ( M..^ N
)  ->  ( ( ph  ->  (  seq M
(  x.  ,  F
) `  n )  =/=  0 )  ->  ( ph  ->  (  seq M
(  x.  ,  F
) `  ( n  +  1 ) )  =/=  0 ) ) )
716, 9, 12, 15, 30, 70fzind2 11892 . 2  |-  ( N  e.  ( M ... N )  ->  ( ph  ->  (  seq M
(  x.  ,  F
) `  N )  =/=  0 ) )
723, 71mpcom 36 1  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3476   ` cfv 5588  (class class class)co 6284   CCcc 9490   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497   ZZcz 10864   ZZ>=cuz 11082   ...cfz 11672  ..^cfzo 11792    seqcseq 12075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-seq 12076
This theorem is referenced by:  prodfrec  28634  prodfdiv  28635  fprodn0  28714
  Copyright terms: Public domain W3C validator