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Theorem prodfn0 27548
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 11571 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
31, 2syl 16 . 2  |-  ( ph  ->  N  e.  ( M ... N ) )
4 fveq2 5794 . . . . 5  |-  ( m  =  M  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  M
) )
54neeq1d 2726 . . . 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 5794 . . . . 5  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
87neeq1d 2726 . . . 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 5794 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
1110neeq1d 2726 . . . 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 5794 . . . . 5  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
1413neeq1d 2726 . . . 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 11570 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
17 elfzelz 11565 . . . . . . . 8  |-  ( M  e.  ( M ... N )  ->  M  e.  ZZ )
1817adantl 466 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  M  e.  ZZ )
19 seq1 11931 . . . . . . 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 5794 . . . . . . . . . 10  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2221neeq1d 2726 . . . . . . . . 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 3136 . . . . . . 7  |-  ( M  e.  ( M ... N )  ->  ( ph  ->  ( F `  M )  =/=  0
) )
2726impcom 430 . . . . . 6  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  ( F `  M )  =/=  0
)
2820, 27eqnetrd 2742 . . . . 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 11669 . . . . . . . . 9  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( ZZ>= `  M )
)
32313ad2ant2 1010 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  n  e.  ( ZZ>= `  M )
)
33 seqp1 11933 . . . . . . . 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 11679 . . . . . . . . . 10  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( M ... N ) )
36 elfzuz 11561 . . . . . . . . . . . 12  |-  ( n  e.  ( M ... N )  ->  n  e.  ( ZZ>= `  M )
)
3736adantl 466 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  n  e.  ( ZZ>= `  M )
)
38 elfzuz3 11562 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  n )
)
39 fzss2 11610 . . . . . . . . . . . . . . 15  |-  ( N  e.  ( ZZ>= `  n
)  ->  ( M ... n )  C_  ( M ... N ) )
4038, 39syl 16 . . . . . . . . . . . . . 14  |-  ( n  e.  ( M ... N )  ->  ( M ... n )  C_  ( M ... N ) )
4140sselda 3459 . . . . . . . . . . . . 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 9472 . . . . . . . . . . . 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 11938 . . . . . . . . . 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 1008 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq M (  x.  ,  F ) `  n
)  e.  CC )
50 fzofzp1 11736 . . . . . . . . . . 11  |-  ( n  e.  ( M..^ N
)  ->  ( n  +  1 )  e.  ( M ... N
) )
51 fveq2 5794 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
5251eleq1d 2521 . . . . . . . . . . . . 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 3136 . . . . . . . . . . 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 1008 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  ( F `  ( n  +  1 ) )  e.  CC )
59 simp3 990 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq M (  x.  ,  F ) `  n
)  =/=  0 )
6051neeq1d 2726 . . . . . . . . . . . . 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 3136 . . . . . . . . . . 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 1008 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  ( F `  ( n  +  1 ) )  =/=  0
)
6649, 58, 59, 65mulne0d 10094 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) )  =/=  0 )
6734, 66eqnetrd 2742 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =/=  0 )
68673exp 1187 . . . . 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 11749 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2645    C_ wss 3431   ` cfv 5521  (class class class)co 6195   CCcc 9386   0cc0 9388   1c1 9389    + caddc 9391    x. cmul 9393   ZZcz 10752   ZZ>=cuz 10967   ...cfz 11549  ..^cfzo 11660    seqcseq 11918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-fzo 11661  df-seq 11919
This theorem is referenced by:  prodfrec  27549  prodfdiv  27550  fprodn0  27629
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