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Theorem expgrowthi 27212
Description: Exponential growth and decay model. See expgrowth 27214 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 } )  o F  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 6021 . . . . . . . 8  |-  ( t  =  y  ->  ( K  x.  t )  =  ( K  x.  y ) )
32fveq2d 5665 . . . . . . 7  |-  ( t  =  y  ->  ( exp `  ( K  x.  t ) )  =  ( exp `  ( K  x.  y )
) )
43oveq2d 6029 . . . . . 6  |-  ( t  =  y  ->  ( C  x.  ( exp `  ( K  x.  t
) ) )  =  ( C  x.  ( exp `  ( K  x.  y ) ) ) )
54cbvmptv 4234 . . . . 5  |-  ( t  e.  S  |->  ( C  x.  ( exp `  ( K  x.  t )
) ) )  =  ( y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y ) ) ) )
61, 5eqtri 2400 . . . 4  |-  Y  =  ( y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y ) ) ) )
76oveq2i 6024 . . 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 3770 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
10 eleq2 2441 . . . . . . . . . 10  |-  ( S  =  RR  ->  (
y  e.  S  <->  y  e.  RR ) )
11 recn 9006 . . . . . . . . . 10  |-  ( y  e.  RR  ->  y  e.  CC )
1210, 11syl6bi 220 . . . . . . . . 9  |-  ( S  =  RR  ->  (
y  e.  S  -> 
y  e.  CC ) )
13 eleq2 2441 . . . . . . . . . 10  |-  ( S  =  CC  ->  (
y  e.  S  <->  y  e.  CC ) )
1413biimpd 199 . . . . . . . . 9  |-  ( S  =  CC  ->  (
y  e.  S  -> 
y  e.  CC ) )
1512, 14jaoi 369 . . . . . . . 8  |-  ( ( S  =  RR  \/  S  =  CC )  ->  ( y  e.  S  ->  y  e.  CC ) )
168, 9, 153syl 19 . . . . . . 7  |-  ( ph  ->  ( y  e.  S  ->  y  e.  CC ) )
1716imp 419 . . . . . 6  |-  ( (
ph  /\  y  e.  S )  ->  y  e.  CC )
18 expgrowthi.k . . . . . . . 8  |-  ( ph  ->  K  e.  CC )
19 mulcl 9000 . . . . . . . 8  |-  ( ( K  e.  CC  /\  y  e.  CC )  ->  ( K  x.  y
)  e.  CC )
2018, 19sylan 458 . . . . . . 7  |-  ( (
ph  /\  y  e.  CC )  ->  ( K  x.  y )  e.  CC )
21 efcl 12605 . . . . . . 7  |-  ( ( K  x.  y )  e.  CC  ->  ( exp `  ( K  x.  y ) )  e.  CC )
2220, 21syl 16 . . . . . 6  |-  ( (
ph  /\  y  e.  CC )  ->  ( exp `  ( K  x.  y
) )  e.  CC )
2317, 22syldan 457 . . . . 5  |-  ( (
ph  /\  y  e.  S )  ->  ( exp `  ( K  x.  y ) )  e.  CC )
24 ovex 6038 . . . . . 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 cnex 8997 . . . . . . . . 9  |-  CC  e.  _V
2726prid2 3849 . . . . . . . 8  |-  CC  e.  { RR ,  CC }
2827a1i 11 . . . . . . 7  |-  ( ph  ->  CC  e.  { RR ,  CC } )
2917, 20syldan 457 . . . . . . 7  |-  ( (
ph  /\  y  e.  S )  ->  ( K  x.  y )  e.  CC )
3018adantr 452 . . . . . . 7  |-  ( (
ph  /\  y  e.  S )  ->  K  e.  CC )
31 efcl 12605 . . . . . . . 8  |-  ( x  e.  CC  ->  ( exp `  x )  e.  CC )
3231adantl 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  ( exp `  x )  e.  CC )
33 ax-1cn 8974 . . . . . . . . . 10  |-  1  e.  CC
3433a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  S )  ->  1  e.  CC )
358dvmptid 19703 . . . . . . . . 9  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  y ) )  =  ( y  e.  S  |->  1 ) )
368, 17, 34, 35, 18dvmptcmul 19710 . . . . . . . 8  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( K  x.  y ) ) )  =  ( y  e.  S  |->  ( K  x.  1 ) ) )
3718mulid1d 9031 . . . . . . . . 9  |-  ( ph  ->  ( K  x.  1 )  =  K )
3837mpteq2dv 4230 . . . . . . . 8  |-  ( ph  ->  ( y  e.  S  |->  ( K  x.  1 ) )  =  ( y  e.  S  |->  K ) )
3936, 38eqtrd 2412 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( K  x.  y ) ) )  =  ( y  e.  S  |->  K ) )
40 dvef 19724 . . . . . . . . 9  |-  ( CC 
_D  exp )  =  exp
41 eff 12604 . . . . . . . . . . . 12  |-  exp : CC
--> CC
42 ffn 5524 . . . . . . . . . . . 12  |-  ( exp
: CC --> CC  ->  exp 
Fn  CC )
4341, 42ax-mp 8 . . . . . . . . . . 11  |-  exp  Fn  CC
44 dffn5 5704 . . . . . . . . . . 11  |-  ( exp 
Fn  CC  <->  exp  =  ( x  e.  CC  |->  ( exp `  x ) ) )
4543, 44mpbi 200 . . . . . . . . . 10  |-  exp  =  ( x  e.  CC  |->  ( exp `  x ) )
4645oveq2i 6024 . . . . . . . . 9  |-  ( CC 
_D  exp )  =  ( CC  _D  ( x  e.  CC  |->  ( exp `  x ) ) )
4740, 46, 453eqtr3i 2408 . . . . . . . 8  |-  ( CC 
_D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( x  e.  CC  |->  ( exp `  x ) )
4847a1i 11 . . . . . . 7  |-  ( ph  ->  ( CC  _D  (
x  e.  CC  |->  ( exp `  x ) ) )  =  ( x  e.  CC  |->  ( exp `  x ) ) )
49 fveq2 5661 . . . . . . 7  |-  ( x  =  ( K  x.  y )  ->  ( exp `  x )  =  ( exp `  ( K  x.  y )
) )
508, 28, 29, 30, 32, 32, 39, 48, 49, 49dvmptco 19718 . . . . . 6  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( exp `  ( K  x.  y ) ) ) )  =  ( y  e.  S  |->  ( ( exp `  ( K  x.  y )
)  x.  K ) ) )
51 mulcom 9002 . . . . . . . . 9  |-  ( ( ( exp `  ( K  x.  y )
)  e.  CC  /\  K  e.  CC )  ->  ( ( exp `  ( K  x.  y )
)  x.  K )  =  ( K  x.  ( exp `  ( K  x.  y ) ) ) )
5223, 18, 51syl2anr 465 . . . . . . . 8  |-  ( (
ph  /\  ( ph  /\  y  e.  S ) )  ->  ( ( exp `  ( K  x.  y ) )  x.  K )  =  ( K  x.  ( exp `  ( K  x.  y
) ) ) )
5352anabss5 790 . . . . . . 7  |-  ( (
ph  /\  y  e.  S )  ->  (
( exp `  ( K  x.  y )
)  x.  K )  =  ( K  x.  ( exp `  ( K  x.  y ) ) ) )
5453mpteq2dva 4229 . . . . . 6  |-  ( ph  ->  ( y  e.  S  |->  ( ( exp `  ( K  x.  y )
)  x.  K ) )  =  ( y  e.  S  |->  ( K  x.  ( exp `  ( K  x.  y )
) ) ) )
5550, 54eqtrd 2412 . . . . 5  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( exp `  ( K  x.  y ) ) ) )  =  ( y  e.  S  |->  ( K  x.  ( exp `  ( K  x.  y
) ) ) ) )
56 expgrowthi.y0 . . . . 5  |-  ( ph  ->  C  e.  CC )
578, 23, 25, 55, 56dvmptcmul 19710 . . . 4  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y
) ) ) ) )  =  ( y  e.  S  |->  ( C  x.  ( K  x.  ( exp `  ( K  x.  y ) ) ) ) ) )
5856, 18, 233anim123i 1139 . . . . . . . 8  |-  ( (
ph  /\  ph  /\  ( ph  /\  y  e.  S
) )  ->  ( C  e.  CC  /\  K  e.  CC  /\  ( exp `  ( K  x.  y
) )  e.  CC ) )
59583anidm12 1241 . . . . . . 7  |-  ( (
ph  /\  ( ph  /\  y  e.  S ) )  ->  ( C  e.  CC  /\  K  e.  CC  /\  ( exp `  ( K  x.  y
) )  e.  CC ) )
6059anabss5 790 . . . . . 6  |-  ( (
ph  /\  y  e.  S )  ->  ( C  e.  CC  /\  K  e.  CC  /\  ( exp `  ( K  x.  y
) )  e.  CC ) )
61 mul12 9157 . . . . . 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
) ) ) ) )
6260, 61syl 16 . . . . 5  |-  ( (
ph  /\  y  e.  S )  ->  ( C  x.  ( K  x.  ( exp `  ( K  x.  y )
) ) )  =  ( K  x.  ( C  x.  ( exp `  ( K  x.  y
) ) ) ) )
6362mpteq2dva 4229 . . . 4  |-  ( ph  ->  ( y  e.  S  |->  ( C  x.  ( K  x.  ( exp `  ( K  x.  y
) ) ) ) )  =  ( y  e.  S  |->  ( K  x.  ( C  x.  ( exp `  ( K  x.  y ) ) ) ) ) )
6457, 63eqtrd 2412 . . 3  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y
) ) ) ) )  =  ( y  e.  S  |->  ( K  x.  ( C  x.  ( exp `  ( K  x.  y ) ) ) ) ) )
657, 64syl5eq 2424 . 2  |-  ( ph  ->  ( S  _D  Y
)  =  ( y  e.  S  |->  ( K  x.  ( C  x.  ( exp `  ( K  x.  y ) ) ) ) ) )
66 ovex 6038 . . . 4  |-  ( C  x.  ( exp `  ( K  x.  y )
) )  e.  _V
6766a1i 11 . . 3  |-  ( (
ph  /\  y  e.  S )  ->  ( C  x.  ( exp `  ( K  x.  y
) ) )  e. 
_V )
68 fconstmpt 4854 . . . 4  |-  ( S  X.  { K }
)  =  ( y  e.  S  |->  K )
6968a1i 11 . . 3  |-  ( ph  ->  ( S  X.  { K } )  =  ( y  e.  S  |->  K ) )
706a1i 11 . . 3  |-  ( ph  ->  Y  =  ( y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y )
) ) ) )
718, 30, 67, 69, 70offval2 6254 . 2  |-  ( ph  ->  ( ( S  X.  { K } )  o F  x.  Y )  =  ( y  e.  S  |->  ( K  x.  ( C  x.  ( exp `  ( K  x.  y ) ) ) ) ) )
7265, 71eqtr4d 2415 1  |-  ( ph  ->  ( S  _D  Y
)  =  ( ( S  X.  { K } )  o F  x.  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2892   {csn 3750   {cpr 3751    e. cmpt 4200    X. cxp 4809    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013    o Fcof 6235   CCcc 8914   RRcr 8915   1c1 8917    x. cmul 8921   expce 12584    _D cdv 19610
This theorem is referenced by:  expgrowth  27214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995  ax-mulf 8996
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-fi 7344  df-sup 7374  df-oi 7405  df-card 7752  df-cda 7974  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-q 10500  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-ico 10847  df-icc 10848  df-fz 10969  df-fzo 11059  df-fl 11122  df-seq 11244  df-exp 11303  df-fac 11487  df-bc 11514  df-hash 11539  df-shft 11802  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-limsup 12185  df-clim 12202  df-rlim 12203  df-sum 12400  df-ef 12590  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-starv 13464  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-unif 13472  df-hom 13473  df-cco 13474  df-rest 13570  df-topn 13571  df-topgen 13587  df-pt 13588  df-prds 13591  df-xrs 13646  df-0g 13647  df-gsum 13648  df-qtop 13653  df-imas 13654  df-xps 13656  df-mre 13731  df-mrc 13732  df-acs 13734  df-mnd 14610  df-submnd 14659  df-mulg 14735  df-cntz 15036  df-cmn 15334  df-xmet 16612  df-met 16613  df-bl 16614  df-mopn 16615  df-fbas 16616  df-fg 16617  df-cnfld 16620  df-top 16879  df-bases 16881  df-topon 16882  df-topsp 16883  df-cld 16999  df-ntr 17000  df-cls 17001  df-nei 17078  df-lp 17116  df-perf 17117  df-cn 17206  df-cnp 17207  df-haus 17294  df-tx 17508  df-hmeo 17701  df-fil 17792  df-fm 17884  df-flim 17885  df-flf 17886  df-xms 18252  df-ms 18253  df-tms 18254  df-cncf 18772  df-limc 19613  df-dv 19614
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