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Theorem prodfn0 25175
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 11021 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
31, 2syl 16 . 2  |-  ( ph  ->  N  e.  ( M ... N ) )
4 fveq2 5687 . . . . 5  |-  ( m  =  M  ->  (  seq  M (  x.  ,  F ) `  m
)  =  (  seq 
M (  x.  ,  F ) `  M
) )
54neeq1d 2580 . . . 4  |-  ( m  =  M  ->  (
(  seq  M (  x.  ,  F ) `  m )  =/=  0  <->  (  seq  M (  x.  ,  F ) `  M )  =/=  0
) )
65imbi2d 308 . . 3  |-  ( m  =  M  ->  (
( ph  ->  (  seq 
M (  x.  ,  F ) `  m
)  =/=  0 )  <-> 
( ph  ->  (  seq 
M (  x.  ,  F ) `  M
)  =/=  0 ) ) )
7 fveq2 5687 . . . . 5  |-  ( m  =  n  ->  (  seq  M (  x.  ,  F ) `  m
)  =  (  seq 
M (  x.  ,  F ) `  n
) )
87neeq1d 2580 . . . 4  |-  ( m  =  n  ->  (
(  seq  M (  x.  ,  F ) `  m )  =/=  0  <->  (  seq  M (  x.  ,  F ) `  n )  =/=  0
) )
98imbi2d 308 . . 3  |-  ( m  =  n  ->  (
( ph  ->  (  seq 
M (  x.  ,  F ) `  m
)  =/=  0 )  <-> 
( ph  ->  (  seq 
M (  x.  ,  F ) `  n
)  =/=  0 ) ) )
10 fveq2 5687 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (  seq  M (  x.  ,  F ) `  m
)  =  (  seq 
M (  x.  ,  F ) `  (
n  +  1 ) ) )
1110neeq1d 2580 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
(  seq  M (  x.  ,  F ) `  m )  =/=  0  <->  (  seq  M (  x.  ,  F ) `  ( n  +  1
) )  =/=  0
) )
1211imbi2d 308 . . 3  |-  ( m  =  ( n  + 
1 )  ->  (
( ph  ->  (  seq 
M (  x.  ,  F ) `  m
)  =/=  0 )  <-> 
( ph  ->  (  seq 
M (  x.  ,  F ) `  (
n  +  1 ) )  =/=  0 ) ) )
13 fveq2 5687 . . . . 5  |-  ( m  =  N  ->  (  seq  M (  x.  ,  F ) `  m
)  =  (  seq 
M (  x.  ,  F ) `  N
) )
1413neeq1d 2580 . . . 4  |-  ( m  =  N  ->  (
(  seq  M (  x.  ,  F ) `  m )  =/=  0  <->  (  seq  M (  x.  ,  F ) `  N )  =/=  0
) )
1514imbi2d 308 . . 3  |-  ( m  =  N  ->  (
( ph  ->  (  seq 
M (  x.  ,  F ) `  m
)  =/=  0 )  <-> 
( ph  ->  (  seq 
M (  x.  ,  F ) `  N
)  =/=  0 ) ) )
16 eluzfz1 11020 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
17 elfzelz 11015 . . . . . . . 8  |-  ( M  e.  ( M ... N )  ->  M  e.  ZZ )
1817adantl 453 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  M  e.  ZZ )
19 seq1 11291 . . . . . . 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 5687 . . . . . . . . . 10  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2221neeq1d 2580 . . . . . . . . 9  |-  ( k  =  M  ->  (
( F `  k
)  =/=  0  <->  ( F `  M )  =/=  0 ) )
2322imbi2d 308 . . . . . . . 8  |-  ( k  =  M  ->  (
( ph  ->  ( F `
 k )  =/=  0 )  <->  ( ph  ->  ( F `  M
)  =/=  0 ) ) )
24 prodfn0.3 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  =/=  0
)
2524expcom 425 . . . . . . . 8  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k )  =/=  0
) )
2623, 25vtoclga 2977 . . . . . . 7  |-  ( M  e.  ( M ... N )  ->  ( ph  ->  ( F `  M )  =/=  0
) )
2726impcom 420 . . . . . 6  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  ( F `  M )  =/=  0
)
2820, 27eqnetrd 2585 . . . . 5  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  (  seq  M (  x.  ,  F
) `  M )  =/=  0 )
2928expcom 425 . . . 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 11099 . . . . . . . . 9  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( ZZ>= `  M )
)
32313ad2ant2 979 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq  M (  x.  ,  F ) `  n )  =/=  0
)  ->  n  e.  ( ZZ>= `  M )
)
33 seqp1 11293 . . . . . . . 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 11109 . . . . . . . . . 10  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( M ... N ) )
36 elfzuz 11011 . . . . . . . . . . . 12  |-  ( n  e.  ( M ... N )  ->  n  e.  ( ZZ>= `  M )
)
3736adantl 453 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  n  e.  ( ZZ>= `  M )
)
38 elfzuz3 11012 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  n )
)
39 fzss2 11048 . . . . . . . . . . . . . . 15  |-  ( N  e.  ( ZZ>= `  n
)  ->  ( M ... n )  C_  ( M ... N ) )
4038, 39syl 16 . . . . . . . . . . . . . 14  |-  ( n  e.  ( M ... N )  ->  ( M ... n )  C_  ( M ... N ) )
4140sselda 3308 . . . . . . . . . . . . 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 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( M ... N
)  /\  k  e.  ( M ... n ) ) )  ->  ( F `  k )  e.  CC )
4443anassrs 630 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( M ... N
) )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  e.  CC )
45 mulcl 9030 . . . . . . . . . . . 12  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
4645adantl 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( M ... N
) )  /\  (
k  e.  CC  /\  x  e.  CC )
)  ->  ( k  x.  x )  e.  CC )
4737, 44, 46seqcl 11298 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  (  seq  M (  x.  ,  F
) `  n )  e.  CC )
4835, 47sylan2 461 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq  M
(  x.  ,  F
) `  n )  e.  CC )
49483adant3 977 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq  M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq  M (  x.  ,  F
) `  n )  e.  CC )
50 fzofzp1 11144 . . . . . . . . . . 11  |-  ( n  e.  ( M..^ N
)  ->  ( n  +  1 )  e.  ( M ... N
) )
51 fveq2 5687 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
5251eleq1d 2470 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
5352imbi2d 308 . . . . . . . . . . . 12  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  e.  CC )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) ) )
5442expcom 425 . . . . . . . . . . . 12  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k )  e.  CC ) )
5553, 54vtoclga 2977 . . . . . . . . . . 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 420 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  e.  CC )
58573adant3 977 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq  M (  x.  ,  F ) `  n )  =/=  0
)  ->  ( F `  ( n  +  1 ) )  e.  CC )
59 simp3 959 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq  M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq  M (  x.  ,  F
) `  n )  =/=  0 )
6051neeq1d 2580 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  =/=  0  <->  ( F `  ( n  +  1 ) )  =/=  0 ) )
6160imbi2d 308 . . . . . . . . . . . 12  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  =/=  0 )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  =/=  0 ) ) )
6261, 25vtoclga 2977 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) )  =/=  0
) )
6362impcom 420 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  +  1 )  e.  ( M ... N
) )  ->  ( F `  ( n  +  1 ) )  =/=  0 )
6450, 63sylan2 461 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  =/=  0
)
65643adant3 977 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq  M (  x.  ,  F ) `  n )  =/=  0
)  ->  ( F `  ( n  +  1 ) )  =/=  0
)
6649, 58, 59, 65mulne0d 9630 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq  M (  x.  ,  F ) `  n )  =/=  0
)  ->  ( (  seq  M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) )  =/=  0 )
6734, 66eqnetrd 2585 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq  M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq  M (  x.  ,  F
) `  ( n  +  1 ) )  =/=  0 )
68673exp 1152 . . . . 5  |-  ( ph  ->  ( n  e.  ( M..^ N )  -> 
( (  seq  M
(  x.  ,  F
) `  n )  =/=  0  ->  (  seq 
M (  x.  ,  F ) `  (
n  +  1 ) )  =/=  0 ) ) )
6968com12 29 . . . 4  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( (  seq  M
(  x.  ,  F
) `  n )  =/=  0  ->  (  seq 
M (  x.  ,  F ) `  (
n  +  1 ) )  =/=  0 ) ) )
7069a2d 24 . . 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 11153 . 2  |-  ( N  e.  ( M ... N )  ->  ( ph  ->  (  seq  M
(  x.  ,  F
) `  N )  =/=  0 ) )
723, 71mpcom 34 1  |-  ( ph  ->  (  seq  M (  x.  ,  F ) `
 N )  =/=  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567    C_ wss 3280   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999  ..^cfzo 11090    seq cseq 11278
This theorem is referenced by:  prodfrec  25176  prodfdiv  25177  fprodn0  25256
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-tru 1325  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-fzo 11091  df-seq 11279
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