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Theorem etransclem7 38106
Description: The given product is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem7.n  |-  ( ph  ->  P  e.  NN )
etransclem7.c  |-  ( ph  ->  C : ( 0 ... M ) --> ( 0 ... N ) )
etransclem7.j  |-  ( ph  ->  J  e.  ( 0 ... M ) )
Assertion
Ref Expression
etransclem7  |-  ( ph  ->  prod_ j  e.  ( 1 ... M ) if ( P  < 
( C `  j
) ,  0 ,  ( ( ( ! `
 P )  / 
( ! `  ( P  -  ( C `  j ) ) ) )  x.  ( ( J  -  j ) ^ ( P  -  ( C `  j ) ) ) ) )  e.  ZZ )
Distinct variable groups:    j, M    ph, j
Allowed substitution hints:    C( j)    P( j)    J( j)    N( j)

Proof of Theorem etransclem7
StepHypRef Expression
1 fzfid 12186 . 2  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
2 0zd 10949 . . 3  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  P  <  ( C `  j
) )  ->  0  e.  ZZ )
3 0zd 10949 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
0  e.  ZZ )
4 etransclem7.n . . . . . . . . . . 11  |-  ( ph  ->  P  e.  NN )
54nnzd 11039 . . . . . . . . . 10  |-  ( ph  ->  P  e.  ZZ )
65ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  ->  P  e.  ZZ )
75adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  P  e.  ZZ )
8 etransclem7.c . . . . . . . . . . . . . 14  |-  ( ph  ->  C : ( 0 ... M ) --> ( 0 ... N ) )
98adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  C : ( 0 ... M ) --> ( 0 ... N ) )
10 0zd 10949 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( 1 ... M )  ->  0  e.  ZZ )
11 fzp1ss 11847 . . . . . . . . . . . . . . . 16  |-  ( 0  e.  ZZ  ->  (
( 0  +  1 ) ... M ) 
C_  ( 0 ... M ) )
1210, 11syl 17 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( 1 ... M )  ->  (
( 0  +  1 ) ... M ) 
C_  ( 0 ... M ) )
13 id 22 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( 1 ... M )  ->  j  e.  ( 1 ... M
) )
14 1e0p1 11079 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 0  +  1 )
1514oveq1i 6300 . . . . . . . . . . . . . . . 16  |-  ( 1 ... M )  =  ( ( 0  +  1 ) ... M
)
1613, 15syl6eleq 2539 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( 1 ... M )  ->  j  e.  ( ( 0  +  1 ) ... M
) )
1712, 16sseldd 3433 . . . . . . . . . . . . . 14  |-  ( j  e.  ( 1 ... M )  ->  j  e.  ( 0 ... M
) )
1817adantl 468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  j  e.  ( 0 ... M
) )
199, 18ffvelrnd 6023 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( C `  j )  e.  ( 0 ... N
) )
2019elfzelzd 37536 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( C `  j )  e.  ZZ )
217, 20zsubcld 11045 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( P  -  ( C `  j ) )  e.  ZZ )
2221adantr 467 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( P  -  ( C `  j )
)  e.  ZZ )
233, 6, 223jca 1188 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( 0  e.  ZZ  /\  P  e.  ZZ  /\  ( P  -  ( C `  j )
)  e.  ZZ ) )
2420zred 11040 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( C `  j )  e.  RR )
2524adantr 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( C `  j
)  e.  RR )
266zred 11040 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  ->  P  e.  RR )
27 simpr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  ->  -.  P  <  ( C `
 j ) )
2825, 26, 27nltled 9785 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( C `  j
)  <_  P )
2926, 25subge0d 10203 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( 0  <_  ( P  -  ( C `  j ) )  <->  ( C `  j )  <_  P
) )
3028, 29mpbird 236 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
0  <_  ( P  -  ( C `  j ) ) )
31 elfzle1 11802 . . . . . . . . . . 11  |-  ( ( C `  j )  e.  ( 0 ... N )  ->  0  <_  ( C `  j
) )
3219, 31syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  0  <_  ( C `  j
) )
3332adantr 467 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
0  <_  ( C `  j ) )
3426, 25subge02d 10205 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( 0  <_  ( C `  j )  <->  ( P  -  ( C `
 j ) )  <_  P ) )
3533, 34mpbid 214 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( P  -  ( C `  j )
)  <_  P )
3623, 30, 35jca32 538 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( ( 0  e.  ZZ  /\  P  e.  ZZ  /\  ( P  -  ( C `  j ) )  e.  ZZ )  /\  (
0  <_  ( P  -  ( C `  j ) )  /\  ( P  -  ( C `  j )
)  <_  P )
) )
37 elfz2 11791 . . . . . . 7  |-  ( ( P  -  ( C `
 j ) )  e.  ( 0 ... P )  <->  ( (
0  e.  ZZ  /\  P  e.  ZZ  /\  ( P  -  ( C `  j ) )  e.  ZZ )  /\  (
0  <_  ( P  -  ( C `  j ) )  /\  ( P  -  ( C `  j )
)  <_  P )
) )
3836, 37sylibr 216 . . . . . 6  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( P  -  ( C `  j )
)  e.  ( 0 ... P ) )
39 permnn 12511 . . . . . 6  |-  ( ( P  -  ( C `
 j ) )  e.  ( 0 ... P )  ->  (
( ! `  P
)  /  ( ! `
 ( P  -  ( C `  j ) ) ) )  e.  NN )
4038, 39syl 17 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( ( ! `  P )  /  ( ! `  ( P  -  ( C `  j ) ) ) )  e.  NN )
4140nnzd 11039 . . . 4  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( ( ! `  P )  /  ( ! `  ( P  -  ( C `  j ) ) ) )  e.  ZZ )
42 etransclem7.j . . . . . . . . 9  |-  ( ph  ->  J  e.  ( 0 ... M ) )
4342elfzelzd 37536 . . . . . . . 8  |-  ( ph  ->  J  e.  ZZ )
4443adantr 467 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  J  e.  ZZ )
45 elfzelz 11800 . . . . . . . 8  |-  ( j  e.  ( 1 ... M )  ->  j  e.  ZZ )
4645adantl 468 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  j  e.  ZZ )
4744, 46zsubcld 11045 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( J  -  j )  e.  ZZ )
4847adantr 467 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( J  -  j
)  e.  ZZ )
49 elnn0z 10950 . . . . . 6  |-  ( ( P  -  ( C `
 j ) )  e.  NN0  <->  ( ( P  -  ( C `  j ) )  e.  ZZ  /\  0  <_ 
( P  -  ( C `  j )
) ) )
5022, 30, 49sylanbrc 670 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( P  -  ( C `  j )
)  e.  NN0 )
51 zexpcl 12287 . . . . 5  |-  ( ( ( J  -  j
)  e.  ZZ  /\  ( P  -  ( C `  j )
)  e.  NN0 )  ->  ( ( J  -  j ) ^ ( P  -  ( C `  j ) ) )  e.  ZZ )
5248, 50, 51syl2anc 667 . . . 4  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( ( J  -  j ) ^ ( P  -  ( C `  j ) ) )  e.  ZZ )
5341, 52zmulcld 11046 . . 3  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( ( ( ! `
 P )  / 
( ! `  ( P  -  ( C `  j ) ) ) )  x.  ( ( J  -  j ) ^ ( P  -  ( C `  j ) ) ) )  e.  ZZ )
542, 53ifclda 3913 . 2  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  if ( P  <  ( C `
 j ) ,  0 ,  ( ( ( ! `  P
)  /  ( ! `
 ( P  -  ( C `  j ) ) ) )  x.  ( ( J  -  j ) ^ ( P  -  ( C `  j ) ) ) ) )  e.  ZZ )
551, 54fprodzcl 14008 1  |-  ( ph  ->  prod_ j  e.  ( 1 ... M ) if ( P  < 
( C `  j
) ,  0 ,  ( ( ( ! `
 P )  / 
( ! `  ( P  -  ( C `  j ) ) ) )  x.  ( ( J  -  j ) ^ ( P  -  ( C `  j ) ) ) ) )  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 985    e. wcel 1887    C_ wss 3404   ifcif 3881   class class class wbr 4402   -->wf 5578   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   NNcn 10609   NN0cn0 10869   ZZcz 10937   ...cfz 11784   ^cexp 12272   !cfa 12459   prod_cprod 13959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-prod 13960
This theorem is referenced by:  etransclem15  38114  etransclem28  38127
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