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Theorem fprodeq0 13790
Description: Anything finite product containing a zero term is itself zero. (Contributed by Scott Fenton, 27-Dec-2017.)
Hypotheses
Ref Expression
fprodeq0.1  |-  Z  =  ( ZZ>= `  M )
fprodeq0.2  |-  ( ph  ->  N  e.  Z )
fprodeq0.3  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
fprodeq0.4  |-  ( (
ph  /\  k  =  N )  ->  A  =  0 )
Assertion
Ref Expression
fprodeq0  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( M ... K
) A  =  0 )
Distinct variable groups:    k, K    ph, k    k, M    k, N    k, Z
Allowed substitution hint:    A( k)

Proof of Theorem fprodeq0
StepHypRef Expression
1 eluzel2 11111 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  N  e.  ZZ )
21adantl 466 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  ZZ )
32zred 10990 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  RR )
43ltp1d 10496 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  <  ( N  +  1 ) )
5 fzdisj 11737 . . . 4  |-  ( N  <  ( N  + 
1 )  ->  (
( M ... N
)  i^i  ( ( N  +  1 ) ... K ) )  =  (/) )
64, 5syl 16 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( M ... N )  i^i  ( ( N  + 
1 ) ... K
) )  =  (/) )
7 fprodeq0.2 . . . . . . . 8  |-  ( ph  ->  N  e.  Z )
8 eluzel2 11111 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
9 fprodeq0.1 . . . . . . . . 9  |-  Z  =  ( ZZ>= `  M )
108, 9eleq2s 2565 . . . . . . . 8  |-  ( N  e.  Z  ->  M  e.  ZZ )
117, 10syl 16 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
1211adantr 465 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  M  e.  ZZ )
13 eluzelz 11115 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  K  e.  ZZ )
1413adantl 466 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  ZZ )
1512, 14, 23jca 1176 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ZZ ) )
16 eluzle 11118 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <_  N )
1716, 9eleq2s 2565 . . . . . . 7  |-  ( N  e.  Z  ->  M  <_  N )
187, 17syl 16 . . . . . 6  |-  ( ph  ->  M  <_  N )
19 eluzle 11118 . . . . . 6  |-  ( K  e.  ( ZZ>= `  N
)  ->  N  <_  K )
2018, 19anim12i 566 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M  <_  N  /\  N  <_  K ) )
21 elfz2 11704 . . . . 5  |-  ( N  e.  ( M ... K )  <->  ( ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  <_  N  /\  N  <_  K ) ) )
2215, 20, 21sylanbrc 664 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  ( M ... K ) )
23 fzsplit 11736 . . . 4  |-  ( N  e.  ( M ... K )  ->  ( M ... K )  =  ( ( M ... N )  u.  (
( N  +  1 ) ... K ) ) )
2422, 23syl 16 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M ... K )  =  ( ( M ... N
)  u.  ( ( N  +  1 ) ... K ) ) )
25 fzfid 12085 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M ... K )  e.  Fin )
26 elfzuz 11709 . . . . . 6  |-  ( k  e.  ( M ... K )  ->  k  e.  ( ZZ>= `  M )
)
2726, 9syl6eleqr 2556 . . . . 5  |-  ( k  e.  ( M ... K )  ->  k  e.  Z )
28 fprodeq0.3 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
2927, 28sylan2 474 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... K ) )  ->  A  e.  CC )
3029adantlr 714 . . 3  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  N )
)  /\  k  e.  ( M ... K ) )  ->  A  e.  CC )
316, 24, 25, 30fprodsplit 13781 . 2  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( M ... K
) A  =  (
prod_ k  e.  ( M ... N ) A  x.  prod_ k  e.  ( ( N  +  1 ) ... K ) A ) )
327, 9syl6eleq 2555 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
33 elfzuz 11709 . . . . . . . 8  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
3433, 9syl6eleqr 2556 . . . . . . 7  |-  ( k  e.  ( M ... N )  ->  k  e.  Z )
3534, 28sylan2 474 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
3632, 35fprodm1s 13785 . . . . 5  |-  ( ph  ->  prod_ k  e.  ( M ... N ) A  =  ( prod_
k  e.  ( M ... ( N  - 
1 ) ) A  x.  [_ N  / 
k ]_ A ) )
37 fprodeq0.4 . . . . . . 7  |-  ( (
ph  /\  k  =  N )  ->  A  =  0 )
387, 37csbied 3457 . . . . . 6  |-  ( ph  ->  [_ N  /  k ]_ A  =  0
)
3938oveq2d 6312 . . . . 5  |-  ( ph  ->  ( prod_ k  e.  ( M ... ( N  -  1 ) ) A  x.  [_ N  /  k ]_ A
)  =  ( prod_
k  e.  ( M ... ( N  - 
1 ) ) A  x.  0 ) )
40 fzfid 12085 . . . . . . 7  |-  ( ph  ->  ( M ... ( N  -  1 ) )  e.  Fin )
41 elfzuz 11709 . . . . . . . . 9  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  ( ZZ>= `  M )
)
4241, 9syl6eleqr 2556 . . . . . . . 8  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  Z )
4342, 28sylan2 474 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
4440, 43fprodcl 13770 . . . . . 6  |-  ( ph  ->  prod_ k  e.  ( M ... ( N  -  1 ) ) A  e.  CC )
4544mul01d 9796 . . . . 5  |-  ( ph  ->  ( prod_ k  e.  ( M ... ( N  -  1 ) ) A  x.  0 )  =  0 )
4636, 39, 453eqtrd 2502 . . . 4  |-  ( ph  ->  prod_ k  e.  ( M ... N ) A  =  0 )
4746adantr 465 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( M ... N
) A  =  0 )
4847oveq1d 6311 . 2  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( prod_ k  e.  ( M ... N ) A  x.  prod_ k  e.  ( ( N  +  1 ) ... K ) A )  =  ( 0  x.  prod_ k  e.  ( ( N  +  1 ) ... K ) A ) )
49 fzfid 12085 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( N  +  1 ) ... K )  e. 
Fin )
509peano2uzs 11160 . . . . . . . . 9  |-  ( N  e.  Z  ->  ( N  +  1 )  e.  Z )
517, 50syl 16 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  Z )
52 elfzuz 11709 . . . . . . . 8  |-  ( k  e.  ( ( N  +  1 ) ... K )  ->  k  e.  ( ZZ>= `  ( N  +  1 ) ) )
539uztrn2 11123 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  Z  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
k  e.  Z )
5451, 52, 53syl2an 477 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( N  + 
1 ) ... K
) )  ->  k  e.  Z )
5554adantrl 715 . . . . . 6  |-  ( (
ph  /\  ( K  e.  ( ZZ>= `  N )  /\  k  e.  (
( N  +  1 ) ... K ) ) )  ->  k  e.  Z )
5655, 28syldan 470 . . . . 5  |-  ( (
ph  /\  ( K  e.  ( ZZ>= `  N )  /\  k  e.  (
( N  +  1 ) ... K ) ) )  ->  A  e.  CC )
5756anassrs 648 . . . 4  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  N )
)  /\  k  e.  ( ( N  + 
1 ) ... K
) )  ->  A  e.  CC )
5849, 57fprodcl 13770 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( ( N  + 
1 ) ... K
) A  e.  CC )
5958mul02d 9795 . 2  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( 0  x.  prod_ k  e.  ( ( N  +  1 ) ... K ) A )  =  0 )
6031, 48, 593eqtrd 2502 1  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( M ... K
) A  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   [_csb 3430    u. cun 3469    i^i cin 3470   (/)c0 3793   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697   prod_cprod 13723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11821  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-prod 13724
This theorem is referenced by:  bcc0  31407
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