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Theorem prodfn0 13705
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 11615 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
31, 2syl 16 . 2  |-  ( ph  ->  N  e.  ( M ... N ) )
4 fveq2 5774 . . . . 5  |-  ( m  =  M  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  M
) )
54neeq1d 2659 . . . 4  |-  ( m  =  M  ->  (
(  seq M (  x.  ,  F ) `  m )  =/=  0  <->  (  seq M (  x.  ,  F ) `  M )  =/=  0
) )
65imbi2d 314 . . 3  |-  ( m  =  M  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
)  =/=  0 )  <-> 
( ph  ->  (  seq M (  x.  ,  F ) `  M
)  =/=  0 ) ) )
7 fveq2 5774 . . . . 5  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
87neeq1d 2659 . . . 4  |-  ( m  =  n  ->  (
(  seq M (  x.  ,  F ) `  m )  =/=  0  <->  (  seq M (  x.  ,  F ) `  n )  =/=  0
) )
98imbi2d 314 . . 3  |-  ( m  =  n  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
)  =/=  0 )  <-> 
( ph  ->  (  seq M (  x.  ,  F ) `  n
)  =/=  0 ) ) )
10 fveq2 5774 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
1110neeq1d 2659 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
(  seq M (  x.  ,  F ) `  m )  =/=  0  <->  (  seq M (  x.  ,  F ) `  ( n  +  1
) )  =/=  0
) )
1211imbi2d 314 . . 3  |-  ( m  =  ( n  + 
1 )  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
)  =/=  0 )  <-> 
( ph  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =/=  0 ) ) )
13 fveq2 5774 . . . . 5  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
1413neeq1d 2659 . . . 4  |-  ( m  =  N  ->  (
(  seq M (  x.  ,  F ) `  m )  =/=  0  <->  (  seq M (  x.  ,  F ) `  N )  =/=  0
) )
1514imbi2d 314 . . 3  |-  ( m  =  N  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
)  =/=  0 )  <-> 
( ph  ->  (  seq M (  x.  ,  F ) `  N
)  =/=  0 ) ) )
16 eluzfz1 11614 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
17 elfzelz 11609 . . . . . . . 8  |-  ( M  e.  ( M ... N )  ->  M  e.  ZZ )
1817adantl 464 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  M  e.  ZZ )
19 seq1 12023 . . . . . . 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 5774 . . . . . . . . . 10  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2221neeq1d 2659 . . . . . . . . 9  |-  ( k  =  M  ->  (
( F `  k
)  =/=  0  <->  ( F `  M )  =/=  0 ) )
2322imbi2d 314 . . . . . . . 8  |-  ( k  =  M  ->  (
( ph  ->  ( F `
 k )  =/=  0 )  <->  ( ph  ->  ( F `  M
)  =/=  0 ) ) )
24 prodfn0.3 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  =/=  0
)
2524expcom 433 . . . . . . . 8  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k )  =/=  0
) )
2623, 25vtoclga 3098 . . . . . . 7  |-  ( M  e.  ( M ... N )  ->  ( ph  ->  ( F `  M )  =/=  0
) )
2726impcom 428 . . . . . 6  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  ( F `  M )  =/=  0
)
2820, 27eqnetrd 2675 . . . . 5  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  (  seq M (  x.  ,  F ) `  M
)  =/=  0 )
2928expcom 433 . . . 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 11726 . . . . . . . . 9  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( ZZ>= `  M )
)
32313ad2ant2 1016 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  n  e.  ( ZZ>= `  M )
)
33 seqp1 12025 . . . . . . . 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 11737 . . . . . . . . . 10  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( M ... N ) )
36 elfzuz 11605 . . . . . . . . . . . 12  |-  ( n  e.  ( M ... N )  ->  n  e.  ( ZZ>= `  M )
)
3736adantl 464 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  n  e.  ( ZZ>= `  M )
)
38 elfzuz3 11606 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  n )
)
39 fzss2 11645 . . . . . . . . . . . . . . 15  |-  ( N  e.  ( ZZ>= `  n
)  ->  ( M ... n )  C_  ( M ... N ) )
4038, 39syl 16 . . . . . . . . . . . . . 14  |-  ( n  e.  ( M ... N )  ->  ( M ... n )  C_  ( M ... N ) )
4140sselda 3417 . . . . . . . . . . . . 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 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( M ... N
)  /\  k  e.  ( M ... n ) ) )  ->  ( F `  k )  e.  CC )
4443anassrs 646 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( M ... N
) )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  e.  CC )
45 mulcl 9487 . . . . . . . . . . . 12  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
4645adantl 464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( M ... N
) )  /\  (
k  e.  CC  /\  x  e.  CC )
)  ->  ( k  x.  x )  e.  CC )
4737, 44, 46seqcl 12030 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  (  seq M (  x.  ,  F ) `  n
)  e.  CC )
4835, 47sylan2 472 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq M
(  x.  ,  F
) `  n )  e.  CC )
49483adant3 1014 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq M (  x.  ,  F ) `  n
)  e.  CC )
50 fzofzp1 11808 . . . . . . . . . . 11  |-  ( n  e.  ( M..^ N
)  ->  ( n  +  1 )  e.  ( M ... N
) )
51 fveq2 5774 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
5251eleq1d 2451 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
5352imbi2d 314 . . . . . . . . . . . 12  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  e.  CC )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) ) )
5442expcom 433 . . . . . . . . . . . 12  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k )  e.  CC ) )
5553, 54vtoclga 3098 . . . . . . . . . . 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 428 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  e.  CC )
58573adant3 1014 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  ( F `  ( n  +  1 ) )  e.  CC )
59 simp3 996 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq M (  x.  ,  F ) `  n
)  =/=  0 )
6051neeq1d 2659 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  =/=  0  <->  ( F `  ( n  +  1 ) )  =/=  0 ) )
6160imbi2d 314 . . . . . . . . . . . 12  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  =/=  0 )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  =/=  0 ) ) )
6261, 25vtoclga 3098 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) )  =/=  0
) )
6362impcom 428 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  +  1 )  e.  ( M ... N
) )  ->  ( F `  ( n  +  1 ) )  =/=  0 )
6450, 63sylan2 472 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  =/=  0
)
65643adant3 1014 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  ( F `  ( n  +  1 ) )  =/=  0
)
6649, 58, 59, 65mulne0d 10118 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) )  =/=  0 )
6734, 66eqnetrd 2675 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n )  =/=  0
)  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =/=  0 )
68673exp 1193 . . . . 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 11823 . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577    C_ wss 3389   ` cfv 5496  (class class class)co 6196   CCcc 9401   0cc0 9403   1c1 9404    + caddc 9406    x. cmul 9408   ZZcz 10781   ZZ>=cuz 11001   ...cfz 11593  ..^cfzo 11717    seqcseq 12010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-seq 12011
This theorem is referenced by:  prodfrec  13706  prodfdiv  13707  fprodn0  13785
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