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Theorem fprodeq0 25252
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 10449 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  N  e.  ZZ )
21adantl 453 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  ZZ )
32zred 10331 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  RR )
43ltp1d 9897 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  <  ( N  +  1 ) )
5 fzdisj 11034 . . . 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 10449 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
9 fprodeq0.1 . . . . . . . . 9  |-  Z  =  ( ZZ>= `  M )
108, 9eleq2s 2496 . . . . . . . 8  |-  ( N  e.  Z  ->  M  e.  ZZ )
117, 10syl 16 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
1211adantr 452 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  M  e.  ZZ )
13 eluzelz 10452 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  K  e.  ZZ )
1413adantl 453 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  ZZ )
1512, 14, 23jca 1134 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ZZ ) )
16 eluzle 10454 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <_  N )
1716, 9eleq2s 2496 . . . . . . 7  |-  ( N  e.  Z  ->  M  <_  N )
187, 17syl 16 . . . . . 6  |-  ( ph  ->  M  <_  N )
19 eluzle 10454 . . . . . 6  |-  ( K  e.  ( ZZ>= `  N
)  ->  N  <_  K )
2018, 19anim12i 550 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M  <_  N  /\  N  <_  K ) )
21 elfz2 11006 . . . . 5  |-  ( N  e.  ( M ... K )  <->  ( ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  <_  N  /\  N  <_  K ) ) )
2215, 20, 21sylanbrc 646 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  ( M ... K ) )
23 fzsplit 11033 . . . 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 11267 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M ... K )  e.  Fin )
26 elfzuz 11011 . . . . . 6  |-  ( k  e.  ( M ... K )  ->  k  e.  ( ZZ>= `  M )
)
2726, 9syl6eleqr 2495 . . . . 5  |-  ( k  e.  ( M ... K )  ->  k  e.  Z )
28 fprodeq0.3 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
2927, 28sylan2 461 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... K ) )  ->  A  e.  CC )
3029adantlr 696 . . 3  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  N )
)  /\  k  e.  ( M ... K ) )  ->  A  e.  CC )
316, 24, 25, 30fprodsplit 25242 . 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 2494 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
33 elfzuz 11011 . . . . . . . 8  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
3433, 9syl6eleqr 2495 . . . . . . 7  |-  ( k  e.  ( M ... N )  ->  k  e.  Z )
3534, 28sylan2 461 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
3632, 35fprodm1s 25246 . . . . 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 3253 . . . . . 6  |-  ( ph  ->  [_ N  /  k ]_ A  =  0
)
3938oveq2d 6056 . . . . 5  |-  ( ph  ->  ( prod_ k  e.  ( M ... ( N  -  1 ) ) A  x.  [_ N  /  k ]_ A
)  =  ( prod_
k  e.  ( M ... ( N  - 
1 ) ) A  x.  0 ) )
40 fzfid 11267 . . . . . . 7  |-  ( ph  ->  ( M ... ( N  -  1 ) )  e.  Fin )
41 elfzuz 11011 . . . . . . . . 9  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  ( ZZ>= `  M )
)
4241, 9syl6eleqr 2495 . . . . . . . 8  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  Z )
4342, 28sylan2 461 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
4440, 43fprodcl 25231 . . . . . 6  |-  ( ph  ->  prod_ k  e.  ( M ... ( N  -  1 ) ) A  e.  CC )
4544mul01d 9221 . . . . 5  |-  ( ph  ->  ( prod_ k  e.  ( M ... ( N  -  1 ) ) A  x.  0 )  =  0 )
4636, 39, 453eqtrd 2440 . . . 4  |-  ( ph  ->  prod_ k  e.  ( M ... N ) A  =  0 )
4746adantr 452 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( M ... N
) A  =  0 )
4847oveq1d 6055 . 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 11267 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( N  +  1 ) ... K )  e. 
Fin )
509peano2uzs 10487 . . . . . . . . 9  |-  ( N  e.  Z  ->  ( N  +  1 )  e.  Z )
517, 50syl 16 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  Z )
52 elfzuz 11011 . . . . . . . 8  |-  ( k  e.  ( ( N  +  1 ) ... K )  ->  k  e.  ( ZZ>= `  ( N  +  1 ) ) )
539uztrn2 10459 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  Z  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
k  e.  Z )
5451, 52, 53syl2an 464 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( N  + 
1 ) ... K
) )  ->  k  e.  Z )
5554adantrl 697 . . . . . 6  |-  ( (
ph  /\  ( K  e.  ( ZZ>= `  N )  /\  k  e.  (
( N  +  1 ) ... K ) ) )  ->  k  e.  Z )
5655, 28syldan 457 . . . . 5  |-  ( (
ph  /\  ( K  e.  ( ZZ>= `  N )  /\  k  e.  (
( N  +  1 ) ... K ) ) )  ->  A  e.  CC )
5756anassrs 630 . . . 4  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  N )
)  /\  k  e.  ( ( N  + 
1 ) ... K
) )  ->  A  e.  CC )
5849, 57fprodcl 25231 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( ( N  + 
1 ) ... K
) A  e.  CC )
5958mul02d 9220 . 2  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( 0  x.  prod_ k  e.  ( ( N  +  1 ) ... K ) A )  =  0 )
6031, 48, 593eqtrd 2440 1  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( M ... K
) A  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   [_csb 3211    u. cun 3278    i^i cin 3279   (/)c0 3588   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   prod_cprod 25184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-prod 25185
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