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Theorem fprodeq0 11620
Description: Any 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    k, M    k, N    k, Z    ph, k
Allowed substitution hint:    A( k)

Proof of Theorem fprodeq0
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 eluzel2 9531 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  N  e.  ZZ )
21adantl 277 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  ZZ )
32zred 9373 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  RR )
43ltp1d 8885 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  <  ( N  +  1 ) )
5 fzdisj 10049 . . . 4  |-  ( N  <  ( N  + 
1 )  ->  (
( M ... N
)  i^i  ( ( N  +  1 ) ... K ) )  =  (/) )
64, 5syl 14 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( M ... N )  i^i  ( ( N  + 
1 ) ... K
) )  =  (/) )
7 fprodeq0.2 . . . . . . . 8  |-  ( ph  ->  N  e.  Z )
8 eluzel2 9531 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
9 fprodeq0.1 . . . . . . . . 9  |-  Z  =  ( ZZ>= `  M )
108, 9eleq2s 2272 . . . . . . . 8  |-  ( N  e.  Z  ->  M  e.  ZZ )
117, 10syl 14 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
1211adantr 276 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  M  e.  ZZ )
13 eluzelz 9535 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  K  e.  ZZ )
1413adantl 277 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  K  e.  ZZ )
1512, 14, 23jca 1177 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ZZ ) )
16 eluzle 9538 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <_  N )
1716, 9eleq2s 2272 . . . . . . 7  |-  ( N  e.  Z  ->  M  <_  N )
187, 17syl 14 . . . . . 6  |-  ( ph  ->  M  <_  N )
19 eluzle 9538 . . . . . 6  |-  ( K  e.  ( ZZ>= `  N
)  ->  N  <_  K )
2018, 19anim12i 338 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M  <_  N  /\  N  <_  K ) )
21 elfz2 10013 . . . . 5  |-  ( N  e.  ( M ... K )  <->  ( ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  <_  N  /\  N  <_  K ) ) )
2215, 20, 21sylanbrc 417 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  N  e.  ( M ... K ) )
23 fzsplit 10048 . . . 4  |-  ( N  e.  ( M ... K )  ->  ( M ... K )  =  ( ( M ... N )  u.  (
( N  +  1 ) ... K ) ) )
2422, 23syl 14 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M ... K )  =  ( ( M ... N
)  u.  ( ( N  +  1 ) ... K ) ) )
2512, 14fzfigd 10428 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( M ... K )  e.  Fin )
26 elfzelz 10022 . . . . . 6  |-  ( j  e.  ( M ... K )  ->  j  e.  ZZ )
2726adantl 277 . . . . 5  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  N )
)  /\  j  e.  ( M ... K ) )  ->  j  e.  ZZ )
2812adantr 276 . . . . 5  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  N )
)  /\  j  e.  ( M ... K ) )  ->  M  e.  ZZ )
292adantr 276 . . . . 5  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  N )
)  /\  j  e.  ( M ... K ) )  ->  N  e.  ZZ )
30 fzdcel 10037 . . . . 5  |-  ( ( j  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  j  e.  ( M ... N ) )
3127, 28, 29, 30syl3anc 1238 . . . 4  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  N )
)  /\  j  e.  ( M ... K ) )  -> DECID  j  e.  ( M ... N ) )
3231ralrimiva 2550 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  A. j  e.  ( M ... K
)DECID  j  e.  ( M ... N ) )
33 elfzuz 10018 . . . . . 6  |-  ( k  e.  ( M ... K )  ->  k  e.  ( ZZ>= `  M )
)
3433, 9eleqtrrdi 2271 . . . . 5  |-  ( k  e.  ( M ... K )  ->  k  e.  Z )
35 fprodeq0.3 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
3634, 35sylan2 286 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... K ) )  ->  A  e.  CC )
3736adantlr 477 . . 3  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  N )
)  /\  k  e.  ( M ... K ) )  ->  A  e.  CC )
386, 24, 25, 32, 37fprodsplitdc 11599 . 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 ) )
397, 9eleqtrdi 2270 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
40 elfzuz 10018 . . . . . . . 8  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
4140, 9eleqtrrdi 2271 . . . . . . 7  |-  ( k  e.  ( M ... N )  ->  k  e.  Z )
4241, 35sylan2 286 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
4339, 42fprodm1s 11604 . . . . 5  |-  ( ph  ->  prod_ k  e.  ( M ... N ) A  =  ( prod_
k  e.  ( M ... ( N  - 
1 ) ) A  x.  [_ N  / 
k ]_ A ) )
44 fprodeq0.4 . . . . . . 7  |-  ( (
ph  /\  k  =  N )  ->  A  =  0 )
457, 44csbied 3103 . . . . . 6  |-  ( ph  ->  [_ N  /  k ]_ A  =  0
)
4645oveq2d 5890 . . . . 5  |-  ( ph  ->  ( prod_ k  e.  ( M ... ( N  -  1 ) ) A  x.  [_ N  /  k ]_ A
)  =  ( prod_
k  e.  ( M ... ( N  - 
1 ) ) A  x.  0 ) )
47 eluzelz 9535 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
4839, 47syl 14 . . . . . . . . 9  |-  ( ph  ->  N  e.  ZZ )
49 peano2zm 9289 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
5048, 49syl 14 . . . . . . . 8  |-  ( ph  ->  ( N  -  1 )  e.  ZZ )
5111, 50fzfigd 10428 . . . . . . 7  |-  ( ph  ->  ( M ... ( N  -  1 ) )  e.  Fin )
52 elfzuz 10018 . . . . . . . . 9  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  ( ZZ>= `  M )
)
5352, 9eleqtrrdi 2271 . . . . . . . 8  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  Z )
5453, 35sylan2 286 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
5551, 54fprodcl 11610 . . . . . 6  |-  ( ph  ->  prod_ k  e.  ( M ... ( N  -  1 ) ) A  e.  CC )
5655mul01d 8348 . . . . 5  |-  ( ph  ->  ( prod_ k  e.  ( M ... ( N  -  1 ) ) A  x.  0 )  =  0 )
5743, 46, 563eqtrd 2214 . . . 4  |-  ( ph  ->  prod_ k  e.  ( M ... N ) A  =  0 )
5857adantr 276 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( M ... N
) A  =  0 )
5958oveq1d 5889 . 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 ) )
602peano2zd 9376 . . . . 5  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( N  +  1 )  e.  ZZ )
6160, 14fzfigd 10428 . . . 4  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( ( N  +  1 ) ... K )  e. 
Fin )
629peano2uzs 9582 . . . . . . . . 9  |-  ( N  e.  Z  ->  ( N  +  1 )  e.  Z )
637, 62syl 14 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  Z )
64 elfzuz 10018 . . . . . . . 8  |-  ( k  e.  ( ( N  +  1 ) ... K )  ->  k  e.  ( ZZ>= `  ( N  +  1 ) ) )
659uztrn2 9543 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  Z  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
k  e.  Z )
6663, 64, 65syl2an 289 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( N  + 
1 ) ... K
) )  ->  k  e.  Z )
6766adantrl 478 . . . . . 6  |-  ( (
ph  /\  ( K  e.  ( ZZ>= `  N )  /\  k  e.  (
( N  +  1 ) ... K ) ) )  ->  k  e.  Z )
6867, 35syldan 282 . . . . 5  |-  ( (
ph  /\  ( K  e.  ( ZZ>= `  N )  /\  k  e.  (
( N  +  1 ) ... K ) ) )  ->  A  e.  CC )
6968anassrs 400 . . . 4  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  N )
)  /\  k  e.  ( ( N  + 
1 ) ... K
) )  ->  A  e.  CC )
7061, 69fprodcl 11610 . . 3  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( ( N  + 
1 ) ... K
) A  e.  CC )
7170mul02d 8347 . 2  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  ( 0  x.  prod_ k  e.  ( ( N  +  1 ) ... K ) A )  =  0 )
7238, 59, 713eqtrd 2214 1  |-  ( (
ph  /\  K  e.  ( ZZ>= `  N )
)  ->  prod_ k  e.  ( M ... K
) A  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2148   [_csb 3057    u. cun 3127    i^i cin 3128   (/)c0 3422   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   CCcc 7808   0cc0 7810   1c1 7811    + caddc 7813    x. cmul 7815    < clt 7990    <_ cle 7991    - cmin 8126   ZZcz 9251   ZZ>=cuz 9526   ...cfz 10006   prod_cprod 11553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-fz 10007  df-fzo 10140  df-seqfrec 10443  df-exp 10517  df-ihash 10751  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-clim 11282  df-proddc 11554
This theorem is referenced by: (None)
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