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Theorem clwlkisclwwlklem2fv2 24985
Description: Lemma 4b for clwlkisclwwlklem2a 24987. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
Hypothesis
Ref Expression
clwlkisclwwlklem2.f  |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) )  |->  if ( x  <  ( (
# `  P )  -  2 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) } ) ,  ( `' E `  { ( P `  x ) ,  ( P ` 
0 ) } ) ) )
Assertion
Ref Expression
clwlkisclwwlklem2fv2  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  ( F `  ( ( # `
 P )  - 
2 ) )  =  ( `' E `  { ( P `  ( ( # `  P
)  -  2 ) ) ,  ( P `
 0 ) } ) )
Distinct variable groups:    x, E    x, P
Allowed substitution hint:    F( x)

Proof of Theorem clwlkisclwwlklem2fv2
StepHypRef Expression
1 clwlkisclwwlklem2.f . . 3  |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) )  |->  if ( x  <  ( (
# `  P )  -  2 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) } ) ,  ( `' E `  { ( P `  x ) ,  ( P ` 
0 ) } ) ) )
21a1i 11 . 2  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  F  =  ( x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) 
|->  if ( x  < 
( ( # `  P
)  -  2 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) } ) ,  ( `' E `  { ( P `  x ) ,  ( P ` 
0 ) } ) ) ) )
3 simpr 459 . . . . . . . . . 10  |-  ( ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  /\  x  =  ( ( # `  P )  -  2 ) )  ->  x  =  ( ( # `  P )  -  2 ) )
4 nn0z 10883 . . . . . . . . . . . . . 14  |-  ( (
# `  P )  e.  NN0  ->  ( # `  P
)  e.  ZZ )
5 2z 10892 . . . . . . . . . . . . . 14  |-  2  e.  ZZ
64, 5jctir 536 . . . . . . . . . . . . 13  |-  ( (
# `  P )  e.  NN0  ->  ( ( # `
 P )  e.  ZZ  /\  2  e.  ZZ ) )
7 zsubcl 10902 . . . . . . . . . . . . 13  |-  ( ( ( # `  P
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  P
)  -  2 )  e.  ZZ )
86, 7syl 16 . . . . . . . . . . . 12  |-  ( (
# `  P )  e.  NN0  ->  ( ( # `
 P )  - 
2 )  e.  ZZ )
98adantr 463 . . . . . . . . . . 11  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  (
( # `  P )  -  2 )  e.  ZZ )
109adantr 463 . . . . . . . . . 10  |-  ( ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  /\  x  =  ( ( # `  P )  -  2 ) )  ->  (
( # `  P )  -  2 )  e.  ZZ )
113, 10eqeltrd 2542 . . . . . . . . 9  |-  ( ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  /\  x  =  ( ( # `  P )  -  2 ) )  ->  x  e.  ZZ )
1211ex 432 . . . . . . . 8  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  (
x  =  ( (
# `  P )  -  2 )  ->  x  e.  ZZ )
)
13 zre 10864 . . . . . . . . . . 11  |-  ( x  e.  ZZ  ->  x  e.  RR )
14 nn0re 10800 . . . . . . . . . . . . 13  |-  ( (
# `  P )  e.  NN0  ->  ( # `  P
)  e.  RR )
15 2re 10601 . . . . . . . . . . . . . 14  |-  2  e.  RR
1615a1i 11 . . . . . . . . . . . . 13  |-  ( (
# `  P )  e.  NN0  ->  2  e.  RR )
1714, 16resubcld 9983 . . . . . . . . . . . 12  |-  ( (
# `  P )  e.  NN0  ->  ( ( # `
 P )  - 
2 )  e.  RR )
1817adantr 463 . . . . . . . . . . 11  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  (
( # `  P )  -  2 )  e.  RR )
19 lttri3 9657 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  ( ( # `  P
)  -  2 )  e.  RR )  -> 
( x  =  ( ( # `  P
)  -  2 )  <-> 
( -.  x  < 
( ( # `  P
)  -  2 )  /\  -.  ( (
# `  P )  -  2 )  < 
x ) ) )
2013, 18, 19syl2anr 476 . . . . . . . . . 10  |-  ( ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  /\  x  e.  ZZ )  ->  (
x  =  ( (
# `  P )  -  2 )  <->  ( -.  x  <  ( ( # `  P )  -  2 )  /\  -.  (
( # `  P )  -  2 )  < 
x ) ) )
21 simpl 455 . . . . . . . . . 10  |-  ( ( -.  x  <  (
( # `  P )  -  2 )  /\  -.  ( ( # `  P
)  -  2 )  <  x )  ->  -.  x  <  ( (
# `  P )  -  2 ) )
2220, 21syl6bi 228 . . . . . . . . 9  |-  ( ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  /\  x  e.  ZZ )  ->  (
x  =  ( (
# `  P )  -  2 )  ->  -.  x  <  ( (
# `  P )  -  2 ) ) )
2322ex 432 . . . . . . . 8  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  (
x  e.  ZZ  ->  ( x  =  ( (
# `  P )  -  2 )  ->  -.  x  <  ( (
# `  P )  -  2 ) ) ) )
2412, 23syld 44 . . . . . . 7  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  (
x  =  ( (
# `  P )  -  2 )  -> 
( x  =  ( ( # `  P
)  -  2 )  ->  -.  x  <  ( ( # `  P
)  -  2 ) ) ) )
2524com13 80 . . . . . 6  |-  ( x  =  ( ( # `  P )  -  2 )  ->  ( x  =  ( ( # `  P )  -  2 )  ->  ( (
( # `  P )  e.  NN0  /\  2  <_  ( # `  P
) )  ->  -.  x  <  ( ( # `  P )  -  2 ) ) ) )
2625pm2.43i 47 . . . . 5  |-  ( x  =  ( ( # `  P )  -  2 )  ->  ( (
( # `  P )  e.  NN0  /\  2  <_  ( # `  P
) )  ->  -.  x  <  ( ( # `  P )  -  2 ) ) )
2726impcom 428 . . . 4  |-  ( ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  /\  x  =  ( ( # `  P )  -  2 ) )  ->  -.  x  <  ( ( # `  P )  -  2 ) )
2827iffalsed 3940 . . 3  |-  ( ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  /\  x  =  ( ( # `  P )  -  2 ) )  ->  if ( x  <  ( (
# `  P )  -  2 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) } ) ,  ( `' E `  { ( P `  x ) ,  ( P ` 
0 ) } ) )  =  ( `' E `  { ( P `  x ) ,  ( P ` 
0 ) } ) )
29 fveq2 5848 . . . . . 6  |-  ( x  =  ( ( # `  P )  -  2 )  ->  ( P `  x )  =  ( P `  ( (
# `  P )  -  2 ) ) )
3029adantl 464 . . . . 5  |-  ( ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  /\  x  =  ( ( # `  P )  -  2 ) )  ->  ( P `  x )  =  ( P `  ( ( # `  P
)  -  2 ) ) )
3130preq1d 4101 . . . 4  |-  ( ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  /\  x  =  ( ( # `  P )  -  2 ) )  ->  { ( P `  x ) ,  ( P ` 
0 ) }  =  { ( P `  ( ( # `  P
)  -  2 ) ) ,  ( P `
 0 ) } )
3231fveq2d 5852 . . 3  |-  ( ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  /\  x  =  ( ( # `  P )  -  2 ) )  ->  ( `' E `  { ( P `  x ) ,  ( P ` 
0 ) } )  =  ( `' E `  { ( P `  ( ( # `  P
)  -  2 ) ) ,  ( P `
 0 ) } ) )
3328, 32eqtrd 2495 . 2  |-  ( ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  /\  x  =  ( ( # `  P )  -  2 ) )  ->  if ( x  <  ( (
# `  P )  -  2 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) } ) ,  ( `' E `  { ( P `  x ) ,  ( P ` 
0 ) } ) )  =  ( `' E `  { ( P `  ( (
# `  P )  -  2 ) ) ,  ( P ` 
0 ) } ) )
346adantr 463 . . . . 5  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  (
( # `  P )  e.  ZZ  /\  2  e.  ZZ ) )
3534, 7syl 16 . . . 4  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  (
( # `  P )  -  2 )  e.  ZZ )
3614, 16subge0d 10138 . . . . 5  |-  ( (
# `  P )  e.  NN0  ->  ( 0  <_  ( ( # `  P )  -  2 )  <->  2  <_  ( # `
 P ) ) )
3736biimpar 483 . . . 4  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  0  <_  ( ( # `  P
)  -  2 ) )
38 elnn0z 10873 . . . 4  |-  ( ( ( # `  P
)  -  2 )  e.  NN0  <->  ( ( (
# `  P )  -  2 )  e.  ZZ  /\  0  <_ 
( ( # `  P
)  -  2 ) ) )
3935, 37, 38sylanbrc 662 . . 3  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  (
( # `  P )  -  2 )  e. 
NN0 )
40 nn0ge2m1nn 10857 . . 3  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  (
( # `  P )  -  1 )  e.  NN )
41 1red 9600 . . . 4  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  1  e.  RR )
4215a1i 11 . . . 4  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  2  e.  RR )
4314adantr 463 . . . 4  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  ( # `
 P )  e.  RR )
44 1lt2 10698 . . . . 5  |-  1  <  2
4544a1i 11 . . . 4  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  1  <  2 )
4641, 42, 43, 45ltsub2dd 10161 . . 3  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  (
( # `  P )  -  2 )  < 
( ( # `  P
)  -  1 ) )
47 elfzo0 11840 . . 3  |-  ( ( ( # `  P
)  -  2 )  e.  ( 0..^ ( ( # `  P
)  -  1 ) )  <->  ( ( (
# `  P )  -  2 )  e. 
NN0  /\  ( ( # `
 P )  - 
1 )  e.  NN  /\  ( ( # `  P
)  -  2 )  <  ( ( # `  P )  -  1 ) ) )
4839, 40, 46, 47syl3anbrc 1178 . 2  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  (
( # `  P )  -  2 )  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )
49 fvex 5858 . . 3  |-  ( `' E `  { ( P `  ( (
# `  P )  -  2 ) ) ,  ( P ` 
0 ) } )  e.  _V
5049a1i 11 . 2  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  ( `' E `  { ( P `  ( (
# `  P )  -  2 ) ) ,  ( P ` 
0 ) } )  e.  _V )
512, 33, 48, 50fvmptd 5936 1  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  ( F `  ( ( # `
 P )  - 
2 ) )  =  ( `' E `  { ( P `  ( ( # `  P
)  -  2 ) ) ,  ( P `
 0 ) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   ifcif 3929   {cpr 4018   class class class wbr 4439    |-> cmpt 4497   `'ccnv 4987   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617    <_ cle 9618    - cmin 9796   NNcn 10531   2c2 10581   NN0cn0 10791   ZZcz 10860  ..^cfzo 11799   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800
This theorem is referenced by:  clwlkisclwwlklem2a4  24986
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