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Theorem fperiodmul 37610
Description: A function with period T is also periodic with period multiple of T. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fperiodmul.f  |-  ( ph  ->  F : RR --> CC )
fperiodmul.t  |-  ( ph  ->  T  e.  RR )
fperiodmul.n  |-  ( ph  ->  N  e.  ZZ )
fperiodmul.x  |-  ( ph  ->  X  e.  RR )
fperiodmul.per  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
Assertion
Ref Expression
fperiodmul  |-  ( ph  ->  ( F `  ( X  +  ( N  x.  T ) ) )  =  ( F `  X ) )
Distinct variable groups:    x, F    x, N    x, T    x, X    ph, x

Proof of Theorem fperiodmul
StepHypRef Expression
1 fperiodmul.f . . . 4  |-  ( ph  ->  F : RR --> CC )
21adantr 472 . . 3  |-  ( (
ph  /\  N  e.  NN0 )  ->  F : RR
--> CC )
3 fperiodmul.t . . . 4  |-  ( ph  ->  T  e.  RR )
43adantr 472 . . 3  |-  ( (
ph  /\  N  e.  NN0 )  ->  T  e.  RR )
5 simpr 468 . . 3  |-  ( (
ph  /\  N  e.  NN0 )  ->  N  e.  NN0 )
6 fperiodmul.x . . . 4  |-  ( ph  ->  X  e.  RR )
76adantr 472 . . 3  |-  ( (
ph  /\  N  e.  NN0 )  ->  X  e.  RR )
8 fperiodmul.per . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
98adantlr 729 . . 3  |-  ( ( ( ph  /\  N  e.  NN0 )  /\  x  e.  RR )  ->  ( F `  ( x  +  T ) )  =  ( F `  x
) )
102, 4, 5, 7, 9fperiodmullem 37609 . 2  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( F `  ( X  +  ( N  x.  T ) ) )  =  ( F `  X ) )
116recnd 9687 . . . . . . 7  |-  ( ph  ->  X  e.  CC )
12 fperiodmul.n . . . . . . . . 9  |-  ( ph  ->  N  e.  ZZ )
1312zcnd 11064 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
143recnd 9687 . . . . . . . 8  |-  ( ph  ->  T  e.  CC )
1513, 14mulcld 9681 . . . . . . 7  |-  ( ph  ->  ( N  x.  T
)  e.  CC )
1611, 15subnegd 10012 . . . . . 6  |-  ( ph  ->  ( X  -  -u ( N  x.  T )
)  =  ( X  +  ( N  x.  T ) ) )
1713, 14mulneg1d 10092 . . . . . . . 8  |-  ( ph  ->  ( -u N  x.  T )  =  -u ( N  x.  T
) )
1817eqcomd 2477 . . . . . . 7  |-  ( ph  -> 
-u ( N  x.  T )  =  (
-u N  x.  T
) )
1918oveq2d 6324 . . . . . 6  |-  ( ph  ->  ( X  -  -u ( N  x.  T )
)  =  ( X  -  ( -u N  x.  T ) ) )
2016, 19eqtr3d 2507 . . . . 5  |-  ( ph  ->  ( X  +  ( N  x.  T ) )  =  ( X  -  ( -u N  x.  T ) ) )
2120fveq2d 5883 . . . 4  |-  ( ph  ->  ( F `  ( X  +  ( N  x.  T ) ) )  =  ( F `  ( X  -  ( -u N  x.  T ) ) ) )
2221adantr 472 . . 3  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  ( F `  ( X  +  ( N  x.  T ) ) )  =  ( F `  ( X  -  ( -u N  x.  T ) ) ) )
231adantr 472 . . . 4  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  F : RR --> CC )
243adantr 472 . . . 4  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  T  e.  RR )
25 znnn0nn 11070 . . . . . 6  |-  ( ( N  e.  ZZ  /\  -.  N  e.  NN0 )  ->  -u N  e.  NN )
2612, 25sylan 479 . . . . 5  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  -u N  e.  NN )
2726nnnn0d 10949 . . . 4  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  -u N  e.  NN0 )
286adantr 472 . . . . 5  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  X  e.  RR )
2912adantr 472 . . . . . . . 8  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  N  e.  ZZ )
3029zred 11063 . . . . . . 7  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  N  e.  RR )
3130renegcld 10067 . . . . . 6  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  -u N  e.  RR )
3231, 24remulcld 9689 . . . . 5  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  ( -u N  x.  T )  e.  RR )
3328, 32resubcld 10068 . . . 4  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  ( X  -  ( -u N  x.  T ) )  e.  RR )
348adantlr 729 . . . 4  |-  ( ( ( ph  /\  -.  N  e.  NN0 )  /\  x  e.  RR )  ->  ( F `  (
x  +  T ) )  =  ( F `
 x ) )
3523, 24, 27, 33, 34fperiodmullem 37609 . . 3  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  ( F `  ( ( X  -  ( -u N  x.  T ) )  +  ( -u N  x.  T ) ) )  =  ( F `  ( X  -  ( -u N  x.  T ) ) ) )
3628recnd 9687 . . . . 5  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  X  e.  CC )
3730recnd 9687 . . . . . . 7  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  N  e.  CC )
3837negcld 9992 . . . . . 6  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  -u N  e.  CC )
3924recnd 9687 . . . . . 6  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  T  e.  CC )
4038, 39mulcld 9681 . . . . 5  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  ( -u N  x.  T )  e.  CC )
4136, 40npcand 10009 . . . 4  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  (
( X  -  ( -u N  x.  T ) )  +  ( -u N  x.  T )
)  =  X )
4241fveq2d 5883 . . 3  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  ( F `  ( ( X  -  ( -u N  x.  T ) )  +  ( -u N  x.  T ) ) )  =  ( F `  X ) )
4322, 35, 423eqtr2d 2511 . 2  |-  ( (
ph  /\  -.  N  e.  NN0 )  ->  ( F `  ( X  +  ( N  x.  T ) ) )  =  ( F `  X ) )
4410, 43pm2.61dan 808 1  |-  ( ph  ->  ( F `  ( X  +  ( N  x.  T ) ) )  =  ( F `  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556    + caddc 9560    x. cmul 9562    - cmin 9880   -ucneg 9881   NNcn 10631   NN0cn0 10893   ZZcz 10961
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-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-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
This theorem is referenced by:  fourierdlem89  38171  fourierdlem90  38172  fourierdlem91  38173  fourierdlem94  38176  fourierdlem97  38179  fourierdlem113  38195
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