Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  etransclem10 Structured version   Visualization version   Unicode version

Theorem etransclem10 38109
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 10949 . 2  |-  ( (
ph  /\  ( P  -  1 )  < 
( C `  0
) )  ->  0  e.  ZZ )
2 0zd 10949 . . . . . . . . 9  |-  ( ph  ->  0  e.  ZZ )
3 etransclem10.n . . . . . . . . . . 11  |-  ( ph  ->  P  e.  NN )
4 nnm1nn0 10911 . . . . . . . . . . 11  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
53, 4syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( P  -  1 )  e.  NN0 )
65nn0zd 11038 . . . . . . . . 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 11193 . . . . . . . . . . . . . 14  |-  NN0  =  ( ZZ>= `  0 )
108, 9syl6eleq 2539 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
11 eluzfz1 11806 . . . . . . . . . . . . 13  |-  ( M  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... M
) )
1210, 11syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  0  e.  ( 0 ... M ) )
137, 12ffvelrnd 6023 . . . . . . . . . . 11  |-  ( ph  ->  ( C `  0
)  e.  ( 0 ... N ) )
1413elfzelzd 37536 . . . . . . . . . 10  |-  ( ph  ->  ( C `  0
)  e.  ZZ )
156, 14zsubcld 11045 . . . . . . . . 9  |-  ( ph  ->  ( ( P  - 
1 )  -  ( C `  0 )
)  e.  ZZ )
162, 6, 153jca 1188 . . . . . . . 8  |-  ( ph  ->  ( 0  e.  ZZ  /\  ( P  -  1 )  e.  ZZ  /\  ( ( P  - 
1 )  -  ( C `  0 )
)  e.  ZZ ) )
1716adantr 467 . . . . . . 7  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( 0  e.  ZZ  /\  ( P  -  1 )  e.  ZZ  /\  ( ( P  - 
1 )  -  ( C `  0 )
)  e.  ZZ ) )
1814zred 11040 . . . . . . . . . 10  |-  ( ph  ->  ( C `  0
)  e.  RR )
1918adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( C `  0
)  e.  RR )
205nn0red 10926 . . . . . . . . . 10  |-  ( ph  ->  ( P  -  1 )  e.  RR )
2120adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( P  -  1 )  e.  RR )
22 simpr 463 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  ->  -.  ( P  -  1 )  <  ( C `
 0 ) )
2319, 21, 22nltled 9785 . . . . . . . 8  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( C `  0
)  <_  ( P  -  1 ) )
2421, 19subge0d 10203 . . . . . . . 8  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( 0  <_  (
( P  -  1 )  -  ( C `
 0 ) )  <-> 
( C `  0
)  <_  ( P  -  1 ) ) )
2523, 24mpbird 236 . . . . . . 7  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
0  <_  ( ( P  -  1 )  -  ( C ` 
0 ) ) )
26 elfzle1 11802 . . . . . . . . . 10  |-  ( ( C `  0 )  e.  ( 0 ... N )  ->  0  <_  ( C `  0
) )
2713, 26syl 17 . . . . . . . . 9  |-  ( ph  ->  0  <_  ( C `  0 ) )
2827adantr 467 . . . . . . . 8  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
0  <_  ( C `  0 ) )
2921, 19subge02d 10205 . . . . . . . 8  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( 0  <_  ( C `  0 )  <->  ( ( P  -  1 )  -  ( C `
 0 ) )  <_  ( P  - 
1 ) ) )
3028, 29mpbid 214 . . . . . . 7  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( P  - 
1 )  -  ( C `  0 )
)  <_  ( P  -  1 ) )
3117, 25, 30jca32 538 . . . . . 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 11791 . . . . . 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 216 . . . . 5  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( P  - 
1 )  -  ( C `  0 )
)  e.  ( 0 ... ( P  - 
1 ) ) )
34 permnn 12511 . . . . 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 11039 . . 3  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( ! `  ( P  -  1
) )  /  ( ! `  ( ( P  -  1 )  -  ( C ` 
0 ) ) ) )  e.  ZZ )
37 etransclem10.j . . . . 5  |-  ( ph  ->  J  e.  ZZ )
3837adantr 467 . . . 4  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  ->  J  e.  ZZ )
3915adantr 467 . . . . 5  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( P  - 
1 )  -  ( C `  0 )
)  e.  ZZ )
40 elnn0z 10950 . . . . 5  |-  ( ( ( P  -  1 )  -  ( C `
 0 ) )  e.  NN0  <->  ( ( ( P  -  1 )  -  ( C ` 
0 ) )  e.  ZZ  /\  0  <_ 
( ( P  - 
1 )  -  ( C `  0 )
) ) )
4139, 25, 40sylanbrc 670 . . . 4  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( P  - 
1 )  -  ( C `  0 )
)  e.  NN0 )
42 zexpcl 12287 . . . 4  |-  ( ( J  e.  ZZ  /\  ( ( P  - 
1 )  -  ( C `  0 )
)  e.  NN0 )  ->  ( J ^ (
( P  -  1 )  -  ( C `
 0 ) ) )  e.  ZZ )
4338, 41, 42syl2anc 667 . . 3  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( J ^ (
( P  -  1 )  -  ( C `
 0 ) ) )  e.  ZZ )
4436, 43zmulcld 11046 . 2  |-  ( (
ph  /\  -.  ( P  -  1 )  <  ( C ` 
0 ) )  -> 
( ( ( ! `
 ( P  - 
1 ) )  / 
( ! `  (
( P  -  1 )  -  ( C `
 0 ) ) ) )  x.  ( J ^ ( ( P  -  1 )  -  ( C `  0 ) ) ) )  e.  ZZ )
451, 44ifclda 3913 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 371    /\ w3a 985    e. wcel 1887   ifcif 3881   class class class wbr 4402   -->wf 5578   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784   ^cexp 12272   !cfa 12459
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-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  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
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  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-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-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-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-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  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-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488
This theorem is referenced by:  etransclem25  38124  etransclem26  38125  etransclem35  38134  etransclem37  38136
  Copyright terms: Public domain W3C validator