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Theorem etransclem7 38218
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 12224 . 2  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
2 0zd 10973 . . 3  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  P  <  ( C `  j
) )  ->  0  e.  ZZ )
3 0zd 10973 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
0  e.  ZZ )
4 etransclem7.n . . . . . . . . . . 11  |-  ( ph  ->  P  e.  NN )
54nnzd 11062 . . . . . . . . . 10  |-  ( ph  ->  P  e.  ZZ )
65ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  ->  P  e.  ZZ )
75adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  P  e.  ZZ )
8 etransclem7.c . . . . . . . . . . . . . 14  |-  ( ph  ->  C : ( 0 ... M ) --> ( 0 ... N ) )
98adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  C : ( 0 ... M ) --> ( 0 ... N ) )
10 0zd 10973 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( 1 ... M )  ->  0  e.  ZZ )
11 fzp1ss 11873 . . . . . . . . . . . . . . . 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 11102 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 0  +  1 )
1514oveq1i 6318 . . . . . . . . . . . . . . . 16  |-  ( 1 ... M )  =  ( ( 0  +  1 ) ... M
)
1613, 15syl6eleq 2559 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( 1 ... M )  ->  j  e.  ( ( 0  +  1 ) ... M
) )
1712, 16sseldd 3419 . . . . . . . . . . . . . 14  |-  ( j  e.  ( 1 ... M )  ->  j  e.  ( 0 ... M
) )
1817adantl 473 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  j  e.  ( 0 ... M
) )
199, 18ffvelrnd 6038 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( C `  j )  e.  ( 0 ... N
) )
2019elfzelzd 37624 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( C `  j )  e.  ZZ )
217, 20zsubcld 11068 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( P  -  ( C `  j ) )  e.  ZZ )
2221adantr 472 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( P  -  ( C `  j )
)  e.  ZZ )
233, 6, 223jca 1210 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( 0  e.  ZZ  /\  P  e.  ZZ  /\  ( P  -  ( C `  j )
)  e.  ZZ ) )
2420zred 11063 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( C `  j )  e.  RR )
2524adantr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( C `  j
)  e.  RR )
266zred 11063 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  ->  P  e.  RR )
27 simpr 468 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  ->  -.  P  <  ( C `
 j ) )
2825, 26, 27nltled 9802 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( C `  j
)  <_  P )
2926, 25subge0d 10224 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( 0  <_  ( P  -  ( C `  j ) )  <->  ( C `  j )  <_  P
) )
3028, 29mpbird 240 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
0  <_  ( P  -  ( C `  j ) ) )
31 elfzle1 11828 . . . . . . . . . . 11  |-  ( ( C `  j )  e.  ( 0 ... N )  ->  0  <_  ( C `  j
) )
3219, 31syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  0  <_  ( C `  j
) )
3332adantr 472 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
0  <_  ( C `  j ) )
3426, 25subge02d 10226 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( 0  <_  ( C `  j )  <->  ( P  -  ( C `
 j ) )  <_  P ) )
3533, 34mpbid 215 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( P  -  ( C `  j )
)  <_  P )
3623, 30, 35jca32 544 . . . . . . 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 11817 . . . . . . 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 217 . . . . . 6  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( P  -  ( C `  j )
)  e.  ( 0 ... P ) )
39 permnn 12549 . . . . . 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 11062 . . . 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 37624 . . . . . . . 8  |-  ( ph  ->  J  e.  ZZ )
4443adantr 472 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  J  e.  ZZ )
45 elfzelz 11826 . . . . . . . 8  |-  ( j  e.  ( 1 ... M )  ->  j  e.  ZZ )
4645adantl 473 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  j  e.  ZZ )
4744, 46zsubcld 11068 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( J  -  j )  e.  ZZ )
4847adantr 472 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( J  -  j
)  e.  ZZ )
49 elnn0z 10974 . . . . . 6  |-  ( ( P  -  ( C `
 j ) )  e.  NN0  <->  ( ( P  -  ( C `  j ) )  e.  ZZ  /\  0  <_ 
( P  -  ( C `  j )
) ) )
5022, 30, 49sylanbrc 677 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( P  -  ( C `  j )
)  e.  NN0 )
51 zexpcl 12325 . . . . 5  |-  ( ( ( J  -  j
)  e.  ZZ  /\  ( P  -  ( C `  j )
)  e.  NN0 )  ->  ( ( J  -  j ) ^ ( P  -  ( C `  j ) ) )  e.  ZZ )
5248, 50, 51syl2anc 673 . . . 4  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( ( J  -  j ) ^ ( P  -  ( C `  j ) ) )  e.  ZZ )
5341, 52zmulcld 11069 . . 3  |-  ( ( ( ph  /\  j  e.  ( 1 ... M
) )  /\  -.  P  <  ( C `  j ) )  -> 
( ( ( ! `
 P )  / 
( ! `  ( P  -  ( C `  j ) ) ) )  x.  ( ( J  -  j ) ^ ( P  -  ( C `  j ) ) ) )  e.  ZZ )
542, 53ifclda 3904 . 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 14085 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 376    /\ w3a 1007    e. wcel 1904    C_ wss 3390   ifcif 3872   class class class wbr 4395   -->wf 5585   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   NN0cn0 10893   ZZcz 10961   ...cfz 11810   ^cexp 12310   !cfa 12497   prod_cprod 14036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-prod 14037
This theorem is referenced by:  etransclem15  38226  etransclem28  38239
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