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Theorem expgrowthi 36752
Description: Exponential growth and decay model. See expgrowth 36754 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.)
Hypotheses
Ref Expression
expgrowthi.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
expgrowthi.k  |-  ( ph  ->  K  e.  CC )
expgrowthi.y0  |-  ( ph  ->  C  e.  CC )
expgrowthi.yt  |-  Y  =  ( t  e.  S  |->  ( C  x.  ( exp `  ( K  x.  t ) ) ) )
Assertion
Ref Expression
expgrowthi  |-  ( ph  ->  ( S  _D  Y
)  =  ( ( S  X.  { K } )  oF  x.  Y ) )
Distinct variable groups:    t, C    t, K    t, S
Allowed substitution hints:    ph( t)    Y( t)

Proof of Theorem expgrowthi
Dummy variables  y  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 expgrowthi.yt . . . . 5  |-  Y  =  ( t  e.  S  |->  ( C  x.  ( exp `  ( K  x.  t ) ) ) )
2 oveq2 6316 . . . . . . . 8  |-  ( t  =  y  ->  ( K  x.  t )  =  ( K  x.  y ) )
32fveq2d 5883 . . . . . . 7  |-  ( t  =  y  ->  ( exp `  ( K  x.  t ) )  =  ( exp `  ( K  x.  y )
) )
43oveq2d 6324 . . . . . 6  |-  ( t  =  y  ->  ( C  x.  ( exp `  ( K  x.  t
) ) )  =  ( C  x.  ( exp `  ( K  x.  y ) ) ) )
54cbvmptv 4488 . . . . 5  |-  ( t  e.  S  |->  ( C  x.  ( exp `  ( K  x.  t )
) ) )  =  ( y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y ) ) ) )
61, 5eqtri 2493 . . . 4  |-  Y  =  ( y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y ) ) ) )
76oveq2i 6319 . . 3  |-  ( S  _D  Y )  =  ( S  _D  (
y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y
) ) ) ) )
8 expgrowthi.s . . . . 5  |-  ( ph  ->  S  e.  { RR ,  CC } )
9 elpri 3976 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
10 eleq2 2538 . . . . . . . . . 10  |-  ( S  =  RR  ->  (
y  e.  S  <->  y  e.  RR ) )
11 recn 9647 . . . . . . . . . 10  |-  ( y  e.  RR  ->  y  e.  CC )
1210, 11syl6bi 236 . . . . . . . . 9  |-  ( S  =  RR  ->  (
y  e.  S  -> 
y  e.  CC ) )
13 eleq2 2538 . . . . . . . . . 10  |-  ( S  =  CC  ->  (
y  e.  S  <->  y  e.  CC ) )
1413biimpd 212 . . . . . . . . 9  |-  ( S  =  CC  ->  (
y  e.  S  -> 
y  e.  CC ) )
1512, 14jaoi 386 . . . . . . . 8  |-  ( ( S  =  RR  \/  S  =  CC )  ->  ( y  e.  S  ->  y  e.  CC ) )
168, 9, 153syl 18 . . . . . . 7  |-  ( ph  ->  ( y  e.  S  ->  y  e.  CC ) )
1716imp 436 . . . . . 6  |-  ( (
ph  /\  y  e.  S )  ->  y  e.  CC )
18 expgrowthi.k . . . . . . . 8  |-  ( ph  ->  K  e.  CC )
19 mulcl 9641 . . . . . . . 8  |-  ( ( K  e.  CC  /\  y  e.  CC )  ->  ( K  x.  y
)  e.  CC )
2018, 19sylan 479 . . . . . . 7  |-  ( (
ph  /\  y  e.  CC )  ->  ( K  x.  y )  e.  CC )
21 efcl 14214 . . . . . . 7  |-  ( ( K  x.  y )  e.  CC  ->  ( exp `  ( K  x.  y ) )  e.  CC )
2220, 21syl 17 . . . . . 6  |-  ( (
ph  /\  y  e.  CC )  ->  ( exp `  ( K  x.  y
) )  e.  CC )
2317, 22syldan 478 . . . . 5  |-  ( (
ph  /\  y  e.  S )  ->  ( exp `  ( K  x.  y ) )  e.  CC )
24 ovex 6336 . . . . . 6  |-  ( K  x.  ( exp `  ( K  x.  y )
) )  e.  _V
2524a1i 11 . . . . 5  |-  ( (
ph  /\  y  e.  S )  ->  ( K  x.  ( exp `  ( K  x.  y
) ) )  e. 
_V )
26 cnelprrecn 9650 . . . . . . . 8  |-  CC  e.  { RR ,  CC }
2726a1i 11 . . . . . . 7  |-  ( ph  ->  CC  e.  { RR ,  CC } )
2817, 20syldan 478 . . . . . . 7  |-  ( (
ph  /\  y  e.  S )  ->  ( K  x.  y )  e.  CC )
2918adantr 472 . . . . . . 7  |-  ( (
ph  /\  y  e.  S )  ->  K  e.  CC )
30 efcl 14214 . . . . . . . 8  |-  ( x  e.  CC  ->  ( exp `  x )  e.  CC )
3130adantl 473 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  ( exp `  x )  e.  CC )
32 1cnd 9677 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  S )  ->  1  e.  CC )
338dvmptid 22990 . . . . . . . . 9  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  y ) )  =  ( y  e.  S  |->  1 ) )
348, 17, 32, 33, 18dvmptcmul 22997 . . . . . . . 8  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( K  x.  y ) ) )  =  ( y  e.  S  |->  ( K  x.  1 ) ) )
3518mulid1d 9678 . . . . . . . . 9  |-  ( ph  ->  ( K  x.  1 )  =  K )
3635mpteq2dv 4483 . . . . . . . 8  |-  ( ph  ->  ( y  e.  S  |->  ( K  x.  1 ) )  =  ( y  e.  S  |->  K ) )
3734, 36eqtrd 2505 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( K  x.  y ) ) )  =  ( y  e.  S  |->  K ) )
38 dvef 23011 . . . . . . . . 9  |-  ( CC 
_D  exp )  =  exp
39 eff 14213 . . . . . . . . . . . 12  |-  exp : CC
--> CC
40 ffn 5739 . . . . . . . . . . . 12  |-  ( exp
: CC --> CC  ->  exp 
Fn  CC )
4139, 40ax-mp 5 . . . . . . . . . . 11  |-  exp  Fn  CC
42 dffn5 5924 . . . . . . . . . . 11  |-  ( exp 
Fn  CC  <->  exp  =  ( x  e.  CC  |->  ( exp `  x ) ) )
4341, 42mpbi 213 . . . . . . . . . 10  |-  exp  =  ( x  e.  CC  |->  ( exp `  x ) )
4443oveq2i 6319 . . . . . . . . 9  |-  ( CC 
_D  exp )  =  ( CC  _D  ( x  e.  CC  |->  ( exp `  x ) ) )
4538, 44, 433eqtr3i 2501 . . . . . . . 8  |-  ( CC 
_D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( x  e.  CC  |->  ( exp `  x ) )
4645a1i 11 . . . . . . 7  |-  ( ph  ->  ( CC  _D  (
x  e.  CC  |->  ( exp `  x ) ) )  =  ( x  e.  CC  |->  ( exp `  x ) ) )
47 fveq2 5879 . . . . . . 7  |-  ( x  =  ( K  x.  y )  ->  ( exp `  x )  =  ( exp `  ( K  x.  y )
) )
488, 27, 28, 29, 31, 31, 37, 46, 47, 47dvmptco 23005 . . . . . 6  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( exp `  ( K  x.  y ) ) ) )  =  ( y  e.  S  |->  ( ( exp `  ( K  x.  y )
)  x.  K ) ) )
49 mulcom 9643 . . . . . . . . 9  |-  ( ( ( exp `  ( K  x.  y )
)  e.  CC  /\  K  e.  CC )  ->  ( ( exp `  ( K  x.  y )
)  x.  K )  =  ( K  x.  ( exp `  ( K  x.  y ) ) ) )
5023, 18, 49syl2anr 486 . . . . . . . 8  |-  ( (
ph  /\  ( ph  /\  y  e.  S ) )  ->  ( ( exp `  ( K  x.  y ) )  x.  K )  =  ( K  x.  ( exp `  ( K  x.  y
) ) ) )
5150anabss5 832 . . . . . . 7  |-  ( (
ph  /\  y  e.  S )  ->  (
( exp `  ( K  x.  y )
)  x.  K )  =  ( K  x.  ( exp `  ( K  x.  y ) ) ) )
5251mpteq2dva 4482 . . . . . 6  |-  ( ph  ->  ( y  e.  S  |->  ( ( exp `  ( K  x.  y )
)  x.  K ) )  =  ( y  e.  S  |->  ( K  x.  ( exp `  ( K  x.  y )
) ) ) )
5348, 52eqtrd 2505 . . . . 5  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( exp `  ( K  x.  y ) ) ) )  =  ( y  e.  S  |->  ( K  x.  ( exp `  ( K  x.  y
) ) ) ) )
54 expgrowthi.y0 . . . . 5  |-  ( ph  ->  C  e.  CC )
558, 23, 25, 53, 54dvmptcmul 22997 . . . 4  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y
) ) ) ) )  =  ( y  e.  S  |->  ( C  x.  ( K  x.  ( exp `  ( K  x.  y ) ) ) ) ) )
5654, 18, 233anim123i 1215 . . . . . . . 8  |-  ( (
ph  /\  ph  /\  ( ph  /\  y  e.  S
) )  ->  ( C  e.  CC  /\  K  e.  CC  /\  ( exp `  ( K  x.  y
) )  e.  CC ) )
57563anidm12 1349 . . . . . . 7  |-  ( (
ph  /\  ( ph  /\  y  e.  S ) )  ->  ( C  e.  CC  /\  K  e.  CC  /\  ( exp `  ( K  x.  y
) )  e.  CC ) )
5857anabss5 832 . . . . . 6  |-  ( (
ph  /\  y  e.  S )  ->  ( C  e.  CC  /\  K  e.  CC  /\  ( exp `  ( K  x.  y
) )  e.  CC ) )
59 mul12 9817 . . . . . 6  |-  ( ( C  e.  CC  /\  K  e.  CC  /\  ( exp `  ( K  x.  y ) )  e.  CC )  ->  ( C  x.  ( K  x.  ( exp `  ( K  x.  y )
) ) )  =  ( K  x.  ( C  x.  ( exp `  ( K  x.  y
) ) ) ) )
6058, 59syl 17 . . . . 5  |-  ( (
ph  /\  y  e.  S )  ->  ( C  x.  ( K  x.  ( exp `  ( K  x.  y )
) ) )  =  ( K  x.  ( C  x.  ( exp `  ( K  x.  y
) ) ) ) )
6160mpteq2dva 4482 . . . 4  |-  ( ph  ->  ( y  e.  S  |->  ( C  x.  ( K  x.  ( exp `  ( K  x.  y
) ) ) ) )  =  ( y  e.  S  |->  ( K  x.  ( C  x.  ( exp `  ( K  x.  y ) ) ) ) ) )
6255, 61eqtrd 2505 . . 3  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y
) ) ) ) )  =  ( y  e.  S  |->  ( K  x.  ( C  x.  ( exp `  ( K  x.  y ) ) ) ) ) )
637, 62syl5eq 2517 . 2  |-  ( ph  ->  ( S  _D  Y
)  =  ( y  e.  S  |->  ( K  x.  ( C  x.  ( exp `  ( K  x.  y ) ) ) ) ) )
64 ovex 6336 . . . 4  |-  ( C  x.  ( exp `  ( K  x.  y )
) )  e.  _V
6564a1i 11 . . 3  |-  ( (
ph  /\  y  e.  S )  ->  ( C  x.  ( exp `  ( K  x.  y
) ) )  e. 
_V )
66 fconstmpt 4883 . . . 4  |-  ( S  X.  { K }
)  =  ( y  e.  S  |->  K )
6766a1i 11 . . 3  |-  ( ph  ->  ( S  X.  { K } )  =  ( y  e.  S  |->  K ) )
686a1i 11 . . 3  |-  ( ph  ->  Y  =  ( y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y )
) ) ) )
698, 29, 65, 67, 68offval2 6567 . 2  |-  ( ph  ->  ( ( S  X.  { K } )  oF  x.  Y )  =  ( y  e.  S  |->  ( K  x.  ( C  x.  ( exp `  ( K  x.  y ) ) ) ) ) )
7063, 69eqtr4d 2508 1  |-  ( ph  ->  ( S  _D  Y
)  =  ( ( S  X.  { K } )  oF  x.  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031   {csn 3959   {cpr 3961    |-> cmpt 4454    X. cxp 4837    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    oFcof 6548   CCcc 9555   RRcr 9556   1c1 9558    x. cmul 9562   expce 14191    _D cdv 22897
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-inf2 8164  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  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  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-int 4227  df-iun 4271  df-iin 4272  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-se 4799  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-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  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-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901
This theorem is referenced by:  expgrowth  36754
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