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Theorem prodfap0 11501
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 9981 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( M ... N ) )
4 fveq2 5494 . . . . 5  |-  ( m  =  M  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  M
) )
54breq1d 3997 . . . 4  |-  ( m  =  M  ->  (
(  seq M (  x.  ,  F ) `  m ) #  0  <->  (  seq M (  x.  ,  F ) `  M
) #  0 ) )
65imbi2d 229 . . 3  |-  ( m  =  M  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
) #  0 )  <->  ( ph  ->  (  seq M (  x.  ,  F ) `
 M ) #  0 ) ) )
7 fveq2 5494 . . . . 5  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
87breq1d 3997 . . . 4  |-  ( m  =  n  ->  (
(  seq M (  x.  ,  F ) `  m ) #  0  <->  (  seq M (  x.  ,  F ) `  n
) #  0 ) )
98imbi2d 229 . . 3  |-  ( m  =  n  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
) #  0 )  <->  ( ph  ->  (  seq M (  x.  ,  F ) `
 n ) #  0 ) ) )
10 fveq2 5494 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
1110breq1d 3997 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
(  seq M (  x.  ,  F ) `  m ) #  0  <->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) #  0 ) )
1211imbi2d 229 . . 3  |-  ( m  =  ( n  + 
1 )  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
) #  0 )  <->  ( ph  ->  (  seq M (  x.  ,  F ) `
 ( n  + 
1 ) ) #  0 ) ) )
13 fveq2 5494 . . . . 5  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
1413breq1d 3997 . . . 4  |-  ( m  =  N  ->  (
(  seq M (  x.  ,  F ) `  m ) #  0  <->  (  seq M (  x.  ,  F ) `  N
) #  0 ) )
1514imbi2d 229 . . 3  |-  ( m  =  N  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  m
) #  0 )  <->  ( ph  ->  (  seq M (  x.  ,  F ) `
 N ) #  0 ) ) )
16 eluzfz1 9980 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
17 elfzelz 9974 . . . . . . . 8  |-  ( M  e.  ( M ... N )  ->  M  e.  ZZ )
1817adantl 275 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  M  e.  ZZ )
19 prodfap0.2 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
2019adantlr 474 . . . . . . 7  |-  ( ( ( ph  /\  M  e.  ( M ... N
) )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
21 mulcl 7894 . . . . . . . 8  |-  ( ( k  e.  CC  /\  v  e.  CC )  ->  ( k  x.  v
)  e.  CC )
2221adantl 275 . . . . . . 7  |-  ( ( ( ph  /\  M  e.  ( M ... N
) )  /\  (
k  e.  CC  /\  v  e.  CC )
)  ->  ( k  x.  v )  e.  CC )
2318, 20, 22seq3-1 10409 . . . . . 6  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  (  seq M (  x.  ,  F ) `  M
)  =  ( F `
 M ) )
24 fveq2 5494 . . . . . . . . . 10  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2524breq1d 3997 . . . . . . . . 9  |-  ( k  =  M  ->  (
( F `  k
) #  0  <->  ( F `  M ) #  0 ) )
2625imbi2d 229 . . . . . . . 8  |-  ( k  =  M  ->  (
( ph  ->  ( F `
 k ) #  0 )  <->  ( ph  ->  ( F `  M ) #  0 ) ) )
27 prodfap0.3 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k ) #  0 )
2827expcom 115 . . . . . . . 8  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k ) #  0 ) )
2926, 28vtoclga 2796 . . . . . . 7  |-  ( M  e.  ( M ... N )  ->  ( ph  ->  ( F `  M ) #  0 ) )
3029impcom 124 . . . . . 6  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  ( F `  M ) #  0 )
3123, 30eqbrtrd 4009 . . . . 5  |-  ( (
ph  /\  M  e.  ( M ... N ) )  ->  (  seq M (  x.  ,  F ) `  M
) #  0 )
3231expcom 115 . . . 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 10100 . . . . . . . . 9  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( ZZ>= `  M )
)
35343ad2ant2 1014 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  n  e.  (
ZZ>= `  M ) )
36193ad2antl1 1154 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( M..^ N )  /\  (  seq M
(  x.  ,  F
) `  n ) #  0 )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3721adantl 275 . . . . . . . 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 10411 . . . . . . 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 10111 . . . . . . . . . 10  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( M ... N ) )
40 elfzuz 9970 . . . . . . . . . . 11  |-  ( n  e.  ( M ... N )  ->  n  e.  ( ZZ>= `  M )
)
41 eqid 2170 . . . . . . . . . . . . 13  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
421, 16, 173syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  ZZ )
4341, 42, 19prodf 11494 . . . . . . . . . . . 12  |-  ( ph  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
4443ffvelrnda 5629 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  (  seq M (  x.  ,  F ) `  n
)  e.  CC )
4540, 44sylan2 284 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  (  seq M (  x.  ,  F ) `  n
)  e.  CC )
4639, 45sylan2 284 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq M
(  x.  ,  F
) `  n )  e.  CC )
47463adant3 1012 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  (  seq M
(  x.  ,  F
) `  n )  e.  CC )
48 fzofzp1 10176 . . . . . . . . . . 11  |-  ( n  e.  ( M..^ N
)  ->  ( n  +  1 )  e.  ( M ... N
) )
49 fveq2 5494 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
5049eleq1d 2239 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
5150imbi2d 229 . . . . . . . . . . . 12  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  e.  CC )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) ) )
52 elfzuz 9970 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
5319expcom 115 . . . . . . . . . . . . 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 2796 . . . . . . . . . . 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 124 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  e.  CC )
58573adant3 1012 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  ( F `  ( n  +  1
) )  e.  CC )
59 simp3 994 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  (  seq M
(  x.  ,  F
) `  n ) #  0 )
6049breq1d 3997 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
) #  0  <->  ( F `  ( n  +  1 ) ) #  0 ) )
6160imbi2d 229 . . . . . . . . . . . 12  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k ) #  0 )  <->  ( ph  ->  ( F `  ( n  +  1 ) ) #  0 ) ) )
6261, 28vtoclga 2796 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) ) #  0 ) )
6362impcom 124 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  +  1 )  e.  ( M ... N
) )  ->  ( F `  ( n  +  1 ) ) #  0 )
6448, 63sylan2 284 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) ) #  0 )
65643adant3 1012 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  ( F `  ( n  +  1
) ) #  0 )
6647, 58, 59, 65mulap0d 8569 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) #  0 )
6738, 66eqbrtrd 4009 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  F ) `  n ) #  0 )  ->  (  seq M
(  x.  ,  F
) `  ( n  +  1 ) ) #  0 )
68673exp 1197 . . . . 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 10188 . 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 103    /\ w3a 973    = wceq 1348    e. wcel 2141   class class class wbr 3987   ` cfv 5196  (class class class)co 5851   CCcc 7765   0cc0 7767   1c1 7768    + caddc 7770    x. cmul 7772   # cap 8493   ZZcz 9205   ZZ>=cuz 9480   ...cfz 9958  ..^cfzo 10091    seqcseq 10394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-inn 8872  df-n0 9129  df-z 9206  df-uz 9481  df-fz 9959  df-fzo 10092  df-seqfrec 10395
This theorem is referenced by:  prodfrecap  11502  prodfdivap  11503
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