Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fprodeq0 Structured version   Unicode version

Theorem fprodeq0 27399
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 10862 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  N  e.  ZZ )
21adantl 463 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  ZZ )
32zred 10743 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  RR )
43ltp1d 10259 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  <  ( N  +  1 ) )
5 fzdisj 11472 . . . 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 10862 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
9 fprodeq0.1 . . . . . . . . 9  |-  Z  =  ( ZZ>= `  M )
108, 9eleq2s 2533 . . . . . . . 8  |-  ( N  e.  Z  ->  M  e.  ZZ )
117, 10syl 16 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
1211adantr 462 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  M  e.  ZZ )
13 eluzelz 10866 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  K  e.  ZZ )
1413adantl 463 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  ZZ )
1512, 14, 23jca 1163 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ZZ ) )
16 eluzle 10869 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <_  N )
1716, 9eleq2s 2533 . . . . . . 7  |-  ( N  e.  Z  ->  M  <_  N )
187, 17syl 16 . . . . . 6  |-  ( ph  ->  M  <_  N )
19 eluzle 10869 . . . . . 6  |-  ( K  e.  ( ZZ>= `  N
)  ->  N  <_  K )
2018, 19anim12i 563 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M  <_  N  /\  N  <_  K ) )
21 elfz2 11440 . . . . 5  |-  ( N  e.  ( M ... K )  <->  ( ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  <_  N  /\  N  <_  K ) ) )
2215, 20, 21sylanbrc 659 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  ( M ... K ) )
23 fzsplit 11471 . . . 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 11791 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M ... K )  e.  Fin )
26 elfzuz 11445 . . . . . 6  |-  ( k  e.  ( M ... K )  ->  k  e.  ( ZZ>= `  M )
)
2726, 9syl6eleqr 2532 . . . . 5  |-  ( k  e.  ( M ... K )  ->  k  e.  Z )
28 fprodeq0.3 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
2927, 28sylan2 471 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... K ) )  ->  A  e.  CC )
3029adantlr 709 . . 3  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  N )
)  /\  k  e.  ( M ... K ) )  ->  A  e.  CC )
316, 24, 25, 30fprodsplit 27389 . 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 2531 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
33 elfzuz 11445 . . . . . . . 8  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
3433, 9syl6eleqr 2532 . . . . . . 7  |-  ( k  e.  ( M ... N )  ->  k  e.  Z )
3534, 28sylan2 471 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
3632, 35fprodm1s 27393 . . . . 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 3311 . . . . . 6  |-  ( ph  ->  [_ N  /  k ]_ A  =  0
)
3938oveq2d 6106 . . . . 5  |-  ( ph  ->  ( prod_ k  e.  ( M ... ( N  -  1 ) ) A  x.  [_ N  /  k ]_ A
)  =  ( prod_
k  e.  ( M ... ( N  - 
1 ) ) A  x.  0 ) )
40 fzfid 11791 . . . . . . 7  |-  ( ph  ->  ( M ... ( N  -  1 ) )  e.  Fin )
41 elfzuz 11445 . . . . . . . . 9  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  ( ZZ>= `  M )
)
4241, 9syl6eleqr 2532 . . . . . . . 8  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  Z )
4342, 28sylan2 471 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
4440, 43fprodcl 27378 . . . . . 6  |-  ( ph  ->  prod_ k  e.  ( M ... ( N  -  1 ) ) A  e.  CC )
4544mul01d 9564 . . . . 5  |-  ( ph  ->  ( prod_ k  e.  ( M ... ( N  -  1 ) ) A  x.  0 )  =  0 )
4636, 39, 453eqtrd 2477 . . . 4  |-  ( ph  ->  prod_ k  e.  ( M ... N ) A  =  0 )
4746adantr 462 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( M ... N
) A  =  0 )
4847oveq1d 6105 . 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 11791 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( N  +  1 ) ... K )  e. 
Fin )
509peano2uzs 10905 . . . . . . . . 9  |-  ( N  e.  Z  ->  ( N  +  1 )  e.  Z )
517, 50syl 16 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  Z )
52 elfzuz 11445 . . . . . . . 8  |-  ( k  e.  ( ( N  +  1 ) ... K )  ->  k  e.  ( ZZ>= `  ( N  +  1 ) ) )
539uztrn2 10874 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  Z  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
k  e.  Z )
5451, 52, 53syl2an 474 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( N  + 
1 ) ... K
) )  ->  k  e.  Z )
5554adantrl 710 . . . . . 6  |-  ( (
ph  /\  ( K  e.  ( ZZ>= `  N )  /\  k  e.  (
( N  +  1 ) ... K ) ) )  ->  k  e.  Z )
5655, 28syldan 467 . . . . 5  |-  ( (
ph  /\  ( K  e.  ( ZZ>= `  N )  /\  k  e.  (
( N  +  1 ) ... K ) ) )  ->  A  e.  CC )
5756anassrs 643 . . . 4  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  N )
)  /\  k  e.  ( ( N  + 
1 ) ... K
) )  ->  A  e.  CC )
5849, 57fprodcl 27378 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( ( N  + 
1 ) ... K
) A  e.  CC )
5958mul02d 9563 . 2  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( 0  x.  prod_ k  e.  ( ( N  +  1 ) ... K ) A )  =  0 )
6031, 48, 593eqtrd 2477 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 960    = wceq 1364    e. wcel 1761   [_csb 3285    u. cun 3323    i^i cin 3324   (/)c0 3634   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    < clt 9414    <_ cle 9415    - cmin 9591   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   prod_cprod 27331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-prod 27332
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator