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Theorem etransclem10 38221
Description: The given  if term is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem10.n  |-  ( ph  ->  P  e.  NN )
etransclem10.m  |-  ( ph  ->  M  e.  NN0 )
etransclem10.c  |-  ( ph  ->  C : ( 0 ... M ) --> ( 0 ... N ) )
etransclem10.j  |-  ( ph  ->  J  e.  ZZ )
Assertion
Ref Expression
etransclem10  |-  ( ph  ->  if ( ( P  -  1 )  < 
( C `  0
) ,  0 ,  ( ( ( ! `
 ( P  - 
1 ) )  / 
( ! `  (
( P  -  1 )  -  ( C `
 0 ) ) ) )  x.  ( J ^ ( ( P  -  1 )  -  ( C `  0 ) ) ) ) )  e.  ZZ )

Proof of Theorem etransclem10
StepHypRef Expression
1 0zd 10973 . 2  |-  ( (
ph  /\  ( P  -  1 )  < 
( C `  0
) )  ->  0  e.  ZZ )
2 0zd 10973 . . . . . . . . 9  |-  ( ph  ->  0  e.  ZZ )
3 etransclem10.n . . . . . . . . . . 11  |-  ( ph  ->  P  e.  NN )
4 nnm1nn0 10935 . . . . . . . . . . 11  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
53, 4syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( P  -  1 )  e.  NN0 )
65nn0zd 11061 . . . . . . . . 9  |-  ( ph  ->  ( P  -  1 )  e.  ZZ )
7 etransclem10.c . . . . . . . . . . . 12  |-  ( ph  ->  C : ( 0 ... M ) --> ( 0 ... N ) )
8 etransclem10.m . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  NN0 )
9 nn0uz 11217 . . . . . . . . . . . . . 14  |-  NN0  =  ( ZZ>= `  0 )
108, 9syl6eleq 2559 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
11 eluzfz1 11832 . . . . . . . . . . . . 13  |-  ( M  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... M
) )
1210, 11syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  0  e.  ( 0 ... M ) )
137, 12ffvelrnd 6038 . . . . . . . . . . 11  |-  ( ph  ->  ( C `  0
)  e.  ( 0 ... N ) )
1413elfzelzd 37624 . . . . . . . . . 10  |-  ( ph  ->  ( C `  0
)  e.  ZZ )
156, 14zsubcld 11068 . . . . . . . . 9  |-  ( ph  ->  ( ( P  - 
1 )  -  ( C `  0 )
)  e.  ZZ )
162, 6, 153jca 1210 . . . . . . . 8  |-  ( ph  ->  ( 0  e.  ZZ  /\  ( P  -  1 )  e.  ZZ  /\  ( ( P  - 
1 )  -  ( C `  0 )
)  e.  ZZ ) )
1716adantr 472 . . . . . . 7  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( 0  e.  ZZ  /\  ( P  -  1 )  e.  ZZ  /\  ( ( P  - 
1 )  -  ( C `  0 )
)  e.  ZZ ) )
1814zred 11063 . . . . . . . . . 10  |-  ( ph  ->  ( C `  0
)  e.  RR )
1918adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( C `  0
)  e.  RR )
205nn0red 10950 . . . . . . . . . 10  |-  ( ph  ->  ( P  -  1 )  e.  RR )
2120adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( P  -  1 )  e.  RR )
22 simpr 468 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  ->  -.  ( P  -  1 )  <  ( C `
 0 ) )
2319, 21, 22nltled 9802 . . . . . . . 8  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( C `  0
)  <_  ( P  -  1 ) )
2421, 19subge0d 10224 . . . . . . . 8  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( 0  <_  (
( P  -  1 )  -  ( C `
 0 ) )  <-> 
( C `  0
)  <_  ( P  -  1 ) ) )
2523, 24mpbird 240 . . . . . . 7  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
0  <_  ( ( P  -  1 )  -  ( C ` 
0 ) ) )
26 elfzle1 11828 . . . . . . . . . 10  |-  ( ( C `  0 )  e.  ( 0 ... N )  ->  0  <_  ( C `  0
) )
2713, 26syl 17 . . . . . . . . 9  |-  ( ph  ->  0  <_  ( C `  0 ) )
2827adantr 472 . . . . . . . 8  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
0  <_  ( C `  0 ) )
2921, 19subge02d 10226 . . . . . . . 8  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( 0  <_  ( C `  0 )  <->  ( ( P  -  1 )  -  ( C `
 0 ) )  <_  ( P  - 
1 ) ) )
3028, 29mpbid 215 . . . . . . 7  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( P  - 
1 )  -  ( C `  0 )
)  <_  ( P  -  1 ) )
3117, 25, 30jca32 544 . . . . . 6  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( 0  e.  ZZ  /\  ( P  -  1 )  e.  ZZ  /\  ( ( P  -  1 )  -  ( C ` 
0 ) )  e.  ZZ )  /\  (
0  <_  ( ( P  -  1 )  -  ( C ` 
0 ) )  /\  ( ( P  - 
1 )  -  ( C `  0 )
)  <_  ( P  -  1 ) ) ) )
32 elfz2 11817 . . . . . 6  |-  ( ( ( P  -  1 )  -  ( C `
 0 ) )  e.  ( 0 ... ( P  -  1 ) )  <->  ( (
0  e.  ZZ  /\  ( P  -  1
)  e.  ZZ  /\  ( ( P  - 
1 )  -  ( C `  0 )
)  e.  ZZ )  /\  ( 0  <_ 
( ( P  - 
1 )  -  ( C `  0 )
)  /\  ( ( P  -  1 )  -  ( C ` 
0 ) )  <_ 
( P  -  1 ) ) ) )
3331, 32sylibr 217 . . . . 5  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( P  - 
1 )  -  ( C `  0 )
)  e.  ( 0 ... ( P  - 
1 ) ) )
34 permnn 12549 . . . . 5  |-  ( ( ( P  -  1 )  -  ( C `
 0 ) )  e.  ( 0 ... ( P  -  1 ) )  ->  (
( ! `  ( P  -  1 ) )  /  ( ! `
 ( ( P  -  1 )  -  ( C `  0 ) ) ) )  e.  NN )
3533, 34syl 17 . . . 4  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( ! `  ( P  -  1
) )  /  ( ! `  ( ( P  -  1 )  -  ( C ` 
0 ) ) ) )  e.  NN )
3635nnzd 11062 . . 3  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( ! `  ( P  -  1
) )  /  ( ! `  ( ( P  -  1 )  -  ( C ` 
0 ) ) ) )  e.  ZZ )
37 etransclem10.j . . . . 5  |-  ( ph  ->  J  e.  ZZ )
3837adantr 472 . . . 4  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  ->  J  e.  ZZ )
3915adantr 472 . . . . 5  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( P  - 
1 )  -  ( C `  0 )
)  e.  ZZ )
40 elnn0z 10974 . . . . 5  |-  ( ( ( P  -  1 )  -  ( C `
 0 ) )  e.  NN0  <->  ( ( ( P  -  1 )  -  ( C ` 
0 ) )  e.  ZZ  /\  0  <_ 
( ( P  - 
1 )  -  ( C `  0 )
) ) )
4139, 25, 40sylanbrc 677 . . . 4  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( P  - 
1 )  -  ( C `  0 )
)  e.  NN0 )
42 zexpcl 12325 . . . 4  |-  ( ( J  e.  ZZ  /\  ( ( P  - 
1 )  -  ( C `  0 )
)  e.  NN0 )  ->  ( J ^ (
( P  -  1 )  -  ( C `
 0 ) ) )  e.  ZZ )
4338, 41, 42syl2anc 673 . . 3  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( J ^ (
( P  -  1 )  -  ( C `
 0 ) ) )  e.  ZZ )
4436, 43zmulcld 11069 . 2  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( ( ! `
 ( P  - 
1 ) )  / 
( ! `  (
( P  -  1 )  -  ( C `
 0 ) ) ) )  x.  ( J ^ ( ( P  -  1 )  -  ( C `  0 ) ) ) )  e.  ZZ )
451, 44ifclda 3904 1  |-  ( ph  ->  if ( ( P  -  1 )  < 
( C `  0
) ,  0 ,  ( ( ( ! `
 ( P  - 
1 ) )  / 
( ! `  (
( P  -  1 )  -  ( C `
 0 ) ) ) )  x.  ( J ^ ( ( P  -  1 )  -  ( C `  0 ) ) ) ) )  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    e. wcel 1904   ifcif 3872   class class class wbr 4395   -->wf 5585   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   ^cexp 12310   !cfa 12497
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-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  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
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  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-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-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-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-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  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-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526
This theorem is referenced by:  etransclem25  38236  etransclem26  38237  etransclem35  38246  etransclem37  38248
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