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Theorem prodfap0 11567
Description: The product of finitely many terms apart from zero is apart from zero. (Contributed by Scott Fenton, 14-Jan-2018.) (Revised by Jim Kingdon, 23-Mar-2024.)
Hypotheses
Ref Expression
prodfap0.1  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
prodfap0.2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
prodfap0.3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k ) #  0 )
Assertion
Ref Expression
prodfap0  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N ) #  0 )
Distinct variable groups:    k, F    k, M    k, N    ph, k

Proof of Theorem prodfap0
Dummy variables  n  v  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prodfap0.1 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzfz2 10046 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( M ... N ) )
4 fveq2 5527 . . . . 5  |-  ( m  =  M  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  M
) )
54breq1d 4025 . . . 4  |-  ( m  =  M  ->  (
(  seq M (  x.  ,  F ) `  m ) #  0  <->  (  seq M (  x.  ,  F ) `  M
) #  0 ) )
65imbi2d 230 . . 3  |-  ( m  =  M  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
) #  0 )  <->  ( ph  ->  (  seq M (  x.  ,  F ) `
 M ) #  0 ) ) )
7 fveq2 5527 . . . . 5  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
87breq1d 4025 . . . 4  |-  ( m  =  n  ->  (
(  seq M (  x.  ,  F ) `  m ) #  0  <->  (  seq M (  x.  ,  F ) `  n
) #  0 ) )
98imbi2d 230 . . 3  |-  ( m  =  n  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
) #  0 )  <->  ( ph  ->  (  seq M (  x.  ,  F ) `
 n ) #  0 ) ) )
10 fveq2 5527 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
1110breq1d 4025 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
(  seq M (  x.  ,  F ) `  m ) #  0  <->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) #  0 ) )
1211imbi2d 230 . . 3  |-  ( m  =  ( n  + 
1 )  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
) #  0 )  <->  ( ph  ->  (  seq M (  x.  ,  F ) `
 ( n  + 
1 ) ) #  0 ) ) )
13 fveq2 5527 . . . . 5  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
1413breq1d 4025 . . . 4  |-  ( m  =  N  ->  (
(  seq M (  x.  ,  F ) `  m ) #  0  <->  (  seq M (  x.  ,  F ) `  N
) #  0 ) )
1514imbi2d 230 . . 3  |-  ( m  =  N  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
) #  0 )  <->  ( ph  ->  (  seq M (  x.  ,  F ) `
 N ) #  0 ) ) )
16 eluzfz1 10045 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
17 elfzelz 10039 . . . . . . . 8  |-  ( M  e.  ( M ... N )  ->  M  e.  ZZ )
1817adantl 277 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  M  e.  ZZ )
19 prodfap0.2 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
2019adantlr 477 . . . . . . 7  |-  ( ( ( ph  /\  M  e.  ( M ... N
) )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
21 mulcl 7952 . . . . . . . 8  |-  ( ( k  e.  CC  /\  v  e.  CC )  ->  ( k  x.  v
)  e.  CC )
2221adantl 277 . . . . . . 7  |-  ( ( ( ph  /\  M  e.  ( M ... N
) )  /\  (
k  e.  CC  /\  v  e.  CC )
)  ->  ( k  x.  v )  e.  CC )
2318, 20, 22seq3-1 10474 . . . . . 6  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  (  seq M (  x.  ,  F ) `  M
)  =  ( F `
 M ) )
24 fveq2 5527 . . . . . . . . . 10  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2524breq1d 4025 . . . . . . . . 9  |-  ( k  =  M  ->  (
( F `  k
) #  0  <->  ( F `  M ) #  0 ) )
2625imbi2d 230 . . . . . . . 8  |-  ( k  =  M  ->  (
( ph  ->  ( F `
 k ) #  0 )  <->  ( ph  ->  ( F `  M ) #  0 ) ) )
27 prodfap0.3 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k ) #  0 )
2827expcom 116 . . . . . . . 8  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k ) #  0 ) )
2926, 28vtoclga 2815 . . . . . . 7  |-  ( M  e.  ( M ... N )  ->  ( ph  ->  ( F `  M ) #  0 ) )
3029impcom 125 . . . . . 6  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  ( F `  M ) #  0 )
3123, 30eqbrtrd 4037 . . . . 5  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  (  seq M (  x.  ,  F ) `  M
) #  0 )
3231expcom 116 . . . 4  |-  ( M  e.  ( M ... N )  ->  ( ph  ->  (  seq M
(  x.  ,  F
) `  M ) #  0 ) )
3316, 32syl 14 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ph  ->  (  seq M (  x.  ,  F ) `
 M ) #  0 ) )
34 elfzouz 10165 . . . . . . . . 9  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( ZZ>= `  M )
)
35343ad2ant2 1020 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  n  e.  (
ZZ>= `  M ) )
36193ad2antl1 1160 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( M..^ N )  /\  (  seq M
(  x.  ,  F
) `  n ) #  0 )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3721adantl 277 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( M..^ N )  /\  (  seq M
(  x.  ,  F
) `  n ) #  0 )  /\  (
k  e.  CC  /\  v  e.  CC )
)  ->  ( k  x.  v )  e.  CC )
3835, 36, 37seq3p1 10476 . . . . . . 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 ) ) ) )
39 elfzofz 10176 . . . . . . . . . 10  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( M ... N ) )
40 elfzuz 10035 . . . . . . . . . . 11  |-  ( n  e.  ( M ... N )  ->  n  e.  ( ZZ>= `  M )
)
41 eqid 2187 . . . . . . . . . . . . 13  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
421, 16, 173syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  ZZ )
4341, 42, 19prodf 11560 . . . . . . . . . . . 12  |-  ( ph  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
4443ffvelcdmda 5664 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  (  seq M (  x.  ,  F ) `  n
)  e.  CC )
4540, 44sylan2 286 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  (  seq M (  x.  ,  F ) `  n
)  e.  CC )
4639, 45sylan2 286 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq M
(  x.  ,  F
) `  n )  e.  CC )
47463adant3 1018 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  (  seq M
(  x.  ,  F
) `  n )  e.  CC )
48 fzofzp1 10241 . . . . . . . . . . 11  |-  ( n  e.  ( M..^ N
)  ->  ( n  +  1 )  e.  ( M ... N
) )
49 fveq2 5527 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
5049eleq1d 2256 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
5150imbi2d 230 . . . . . . . . . . . 12  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  e.  CC )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) ) )
52 elfzuz 10035 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
5319expcom 116 . . . . . . . . . . . . 13  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( F `  k
)  e.  CC ) )
5452, 53syl 14 . . . . . . . . . . . 12  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k )  e.  CC ) )
5551, 54vtoclga 2815 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) )  e.  CC ) )
5648, 55syl 14 . . . . . . . . . 10  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) )
5756impcom 125 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  e.  CC )
58573adant3 1018 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  ( F `  ( n  +  1
) )  e.  CC )
59 simp3 1000 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  (  seq M
(  x.  ,  F
) `  n ) #  0 )
6049breq1d 4025 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
) #  0  <->  ( F `  ( n  +  1 ) ) #  0 ) )
6160imbi2d 230 . . . . . . . . . . . 12  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k ) #  0 )  <->  ( ph  ->  ( F `  ( n  +  1 ) ) #  0 ) ) )
6261, 28vtoclga 2815 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) ) #  0 ) )
6362impcom 125 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  +  1 )  e.  ( M ... N
) )  ->  ( F `  ( n  +  1 ) ) #  0 )
6448, 63sylan2 286 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) ) #  0 )
65643adant3 1018 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  ( F `  ( n  +  1
) ) #  0 )
6647, 58, 59, 65mulap0d 8629 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) #  0 )
6738, 66eqbrtrd 4037 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  (  seq M
(  x.  ,  F
) `  ( n  +  1 ) ) #  0 )
68673exp 1203 . . . . 5  |-  ( ph  ->  ( n  e.  ( M..^ N )  -> 
( (  seq M
(  x.  ,  F
) `  n ) #  0  ->  (  seq M
(  x.  ,  F
) `  ( n  +  1 ) ) #  0 ) ) )
6968com12 30 . . . 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, 33, 70fzind2 10253 . 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 set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 979    = wceq 1363    e. wcel 2158   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   CCcc 7823   0cc0 7825   1c1 7826    + caddc 7828    x. cmul 7830   # cap 8552   ZZcz 9267   ZZ>=cuz 9542   ...cfz 10022  ..^cfzo 10156    seqcseq 10459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-inn 8934  df-n0 9191  df-z 9268  df-uz 9543  df-fz 10023  df-fzo 10157  df-seqfrec 10460
This theorem is referenced by:  prodfrecap  11568  prodfdivap  11569
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