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Theorem fprodeq0 27486
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 10866 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  N  e.  ZZ )
21adantl 466 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  ZZ )
32zred 10747 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  RR )
43ltp1d 10263 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  <  ( N  +  1 ) )
5 fzdisj 11476 . . . 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 10866 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
9 fprodeq0.1 . . . . . . . . 9  |-  Z  =  ( ZZ>= `  M )
108, 9eleq2s 2535 . . . . . . . 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 10870 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  K  e.  ZZ )
1413adantl 466 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  ZZ )
1512, 14, 23jca 1168 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ZZ ) )
16 eluzle 10873 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <_  N )
1716, 9eleq2s 2535 . . . . . . 7  |-  ( N  e.  Z  ->  M  <_  N )
187, 17syl 16 . . . . . 6  |-  ( ph  ->  M  <_  N )
19 eluzle 10873 . . . . . 6  |-  ( K  e.  ( ZZ>= `  N
)  ->  N  <_  K )
2018, 19anim12i 566 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M  <_  N  /\  N  <_  K ) )
21 elfz2 11444 . . . . 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 11475 . . . 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 11795 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M ... K )  e.  Fin )
26 elfzuz 11449 . . . . . 6  |-  ( k  e.  ( M ... K )  ->  k  e.  ( ZZ>= `  M )
)
2726, 9syl6eleqr 2534 . . . . 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 27476 . 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 2533 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
33 elfzuz 11449 . . . . . . . 8  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
3433, 9syl6eleqr 2534 . . . . . . 7  |-  ( k  e.  ( M ... N )  ->  k  e.  Z )
3534, 28sylan2 474 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
3632, 35fprodm1s 27480 . . . . 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 3314 . . . . . 6  |-  ( ph  ->  [_ N  /  k ]_ A  =  0
)
3938oveq2d 6107 . . . . 5  |-  ( ph  ->  ( prod_ k  e.  ( M ... ( N  -  1 ) ) A  x.  [_ N  /  k ]_ A
)  =  ( prod_
k  e.  ( M ... ( N  - 
1 ) ) A  x.  0 ) )
40 fzfid 11795 . . . . . . 7  |-  ( ph  ->  ( M ... ( N  -  1 ) )  e.  Fin )
41 elfzuz 11449 . . . . . . . . 9  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  ( ZZ>= `  M )
)
4241, 9syl6eleqr 2534 . . . . . . . 8  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  Z )
4342, 28sylan2 474 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
4440, 43fprodcl 27465 . . . . . 6  |-  ( ph  ->  prod_ k  e.  ( M ... ( N  -  1 ) ) A  e.  CC )
4544mul01d 9568 . . . . 5  |-  ( ph  ->  ( prod_ k  e.  ( M ... ( N  -  1 ) ) A  x.  0 )  =  0 )
4636, 39, 453eqtrd 2479 . . . 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 6106 . 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 11795 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( N  +  1 ) ... K )  e. 
Fin )
509peano2uzs 10909 . . . . . . . . 9  |-  ( N  e.  Z  ->  ( N  +  1 )  e.  Z )
517, 50syl 16 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  Z )
52 elfzuz 11449 . . . . . . . 8  |-  ( k  e.  ( ( N  +  1 ) ... K )  ->  k  e.  ( ZZ>= `  ( N  +  1 ) ) )
539uztrn2 10878 . . . . . . . 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 27465 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( ( N  + 
1 ) ... K
) A  e.  CC )
5958mul02d 9567 . 2  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( 0  x.  prod_ k  e.  ( ( N  +  1 ) ... K ) A )  =  0 )
6031, 48, 593eqtrd 2479 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 965    = wceq 1369    e. wcel 1756   [_csb 3288    u. cun 3326    i^i cin 3327   (/)c0 3637   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   CCcc 9280   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    < clt 9418    <_ cle 9419    - cmin 9595   ZZcz 10646   ZZ>=cuz 10861   ...cfz 11437   prod_cprod 27418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-prod 27419
This theorem is referenced by: (None)
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