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Theorem etransclem1 38212
Description:  H is a function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem1.x  |-  ( ph  ->  X  C_  CC )
etransclem1.p  |-  ( ph  ->  P  e.  NN )
etransclem1.h  |-  H  =  ( j  e.  ( 0 ... M ) 
|->  ( x  e.  X  |->  ( ( x  -  j ) ^ if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
etransclem1.j  |-  ( ph  ->  J  e.  ( 0 ... M ) )
Assertion
Ref Expression
etransclem1  |-  ( ph  ->  ( H `  J
) : X --> CC )
Distinct variable groups:    x, J    j, M    P, j    j, X, x    ph, x
Allowed substitution hints:    ph( j)    P( x)    H( x, j)    J( j)    M( x)

Proof of Theorem etransclem1
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 etransclem1.x . . . . . 6  |-  ( ph  ->  X  C_  CC )
21sselda 3418 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
3 etransclem1.j . . . . . . . 8  |-  ( ph  ->  J  e.  ( 0 ... M ) )
43elfzelzd 37624 . . . . . . 7  |-  ( ph  ->  J  e.  ZZ )
54zcnd 11064 . . . . . 6  |-  ( ph  ->  J  e.  CC )
65adantr 472 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  J  e.  CC )
72, 6subcld 10005 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
x  -  J )  e.  CC )
8 etransclem1.p . . . . . . 7  |-  ( ph  ->  P  e.  NN )
9 nnm1nn0 10935 . . . . . . 7  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
108, 9syl 17 . . . . . 6  |-  ( ph  ->  ( P  -  1 )  e.  NN0 )
118nnnn0d 10949 . . . . . 6  |-  ( ph  ->  P  e.  NN0 )
1210, 11ifcld 3915 . . . . 5  |-  ( ph  ->  if ( J  =  0 ,  ( P  -  1 ) ,  P )  e.  NN0 )
1312adantr 472 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  if ( J  =  0 ,  ( P  - 
1 ) ,  P
)  e.  NN0 )
147, 13expcld 12454 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( x  -  J
) ^ if ( J  =  0 ,  ( P  -  1 ) ,  P ) )  e.  CC )
15 eqid 2471 . . 3  |-  ( x  e.  X  |->  ( ( x  -  J ) ^ if ( J  =  0 ,  ( P  -  1 ) ,  P ) ) )  =  ( x  e.  X  |->  ( ( x  -  J ) ^ if ( J  =  0 ,  ( P  -  1 ) ,  P ) ) )
1614, 15fmptd 6061 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( ( x  -  J ) ^ if ( J  =  0 ,  ( P  - 
1 ) ,  P
) ) ) : X --> CC )
17 etransclem1.h . . . . . 6  |-  H  =  ( j  e.  ( 0 ... M ) 
|->  ( x  e.  X  |->  ( ( x  -  j ) ^ if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
18 oveq2 6316 . . . . . . . . 9  |-  ( j  =  n  ->  (
x  -  j )  =  ( x  -  n ) )
19 eqeq1 2475 . . . . . . . . . 10  |-  ( j  =  n  ->  (
j  =  0  <->  n  =  0 ) )
2019ifbid 3894 . . . . . . . . 9  |-  ( j  =  n  ->  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  =  if ( n  =  0 ,  ( P  -  1 ) ,  P ) )
2118, 20oveq12d 6326 . . . . . . . 8  |-  ( j  =  n  ->  (
( x  -  j
) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) )  =  ( ( x  -  n ) ^ if ( n  =  0 ,  ( P  -  1 ) ,  P ) ) )
2221mpteq2dv 4483 . . . . . . 7  |-  ( j  =  n  ->  (
x  e.  X  |->  ( ( x  -  j
) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) )  =  ( x  e.  X  |->  ( ( x  -  n
) ^ if ( n  =  0 ,  ( P  -  1 ) ,  P ) ) ) )
2322cbvmptv 4488 . . . . . 6  |-  ( j  e.  ( 0 ... M )  |->  ( x  e.  X  |->  ( ( x  -  j ) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) ) )  =  ( n  e.  ( 0 ... M )  |->  ( x  e.  X  |->  ( ( x  -  n
) ^ if ( n  =  0 ,  ( P  -  1 ) ,  P ) ) ) )
2417, 23eqtri 2493 . . . . 5  |-  H  =  ( n  e.  ( 0 ... M ) 
|->  ( x  e.  X  |->  ( ( x  -  n ) ^ if ( n  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
2524a1i 11 . . . 4  |-  ( ph  ->  H  =  ( n  e.  ( 0 ... M )  |->  ( x  e.  X  |->  ( ( x  -  n ) ^ if ( n  =  0 ,  ( P  -  1 ) ,  P ) ) ) ) )
26 oveq2 6316 . . . . . . 7  |-  ( n  =  J  ->  (
x  -  n )  =  ( x  -  J ) )
27 eqeq1 2475 . . . . . . . 8  |-  ( n  =  J  ->  (
n  =  0  <->  J  =  0 ) )
2827ifbid 3894 . . . . . . 7  |-  ( n  =  J  ->  if ( n  =  0 ,  ( P  - 
1 ) ,  P
)  =  if ( J  =  0 ,  ( P  -  1 ) ,  P ) )
2926, 28oveq12d 6326 . . . . . 6  |-  ( n  =  J  ->  (
( x  -  n
) ^ if ( n  =  0 ,  ( P  -  1 ) ,  P ) )  =  ( ( x  -  J ) ^ if ( J  =  0 ,  ( P  -  1 ) ,  P ) ) )
3029mpteq2dv 4483 . . . . 5  |-  ( n  =  J  ->  (
x  e.  X  |->  ( ( x  -  n
) ^ if ( n  =  0 ,  ( P  -  1 ) ,  P ) ) )  =  ( x  e.  X  |->  ( ( x  -  J
) ^ if ( J  =  0 ,  ( P  -  1 ) ,  P ) ) ) )
3130adantl 473 . . . 4  |-  ( (
ph  /\  n  =  J )  ->  (
x  e.  X  |->  ( ( x  -  n
) ^ if ( n  =  0 ,  ( P  -  1 ) ,  P ) ) )  =  ( x  e.  X  |->  ( ( x  -  J
) ^ if ( J  =  0 ,  ( P  -  1 ) ,  P ) ) ) )
32 cnex 9638 . . . . . 6  |-  CC  e.  _V
3332ssex 4540 . . . . 5  |-  ( X 
C_  CC  ->  X  e. 
_V )
34 mptexg 6151 . . . . 5  |-  ( X  e.  _V  ->  (
x  e.  X  |->  ( ( x  -  J
) ^ if ( J  =  0 ,  ( P  -  1 ) ,  P ) ) )  e.  _V )
351, 33, 343syl 18 . . . 4  |-  ( ph  ->  ( x  e.  X  |->  ( ( x  -  J ) ^ if ( J  =  0 ,  ( P  - 
1 ) ,  P
) ) )  e. 
_V )
3625, 31, 3, 35fvmptd 5969 . . 3  |-  ( ph  ->  ( H `  J
)  =  ( x  e.  X  |->  ( ( x  -  J ) ^ if ( J  =  0 ,  ( P  -  1 ) ,  P ) ) ) )
3736feq1d 5724 . 2  |-  ( ph  ->  ( ( H `  J ) : X --> CC 
<->  ( x  e.  X  |->  ( ( x  -  J ) ^ if ( J  =  0 ,  ( P  - 
1 ) ,  P
) ) ) : X --> CC ) )
3816, 37mpbird 240 1  |-  ( ph  ->  ( H `  J
) : X --> CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    C_ wss 3390   ifcif 3872    |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558    - cmin 9880   NNcn 10631   NN0cn0 10893   ...cfz 11810   ^cexp 12310
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-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-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-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-seq 12252  df-exp 12311
This theorem is referenced by:  etransclem29  38240
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