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Theorem expgrowthi 27550
Description: Exponential growth and decay model. See expgrowth 27552 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 5866 . . . . . . . 8  |-  ( t  =  y  ->  ( K  x.  t )  =  ( K  x.  y ) )
32fveq2d 5529 . . . . . . 7  |-  ( t  =  y  ->  ( exp `  ( K  x.  t ) )  =  ( exp `  ( K  x.  y )
) )
43oveq2d 5874 . . . . . 6  |-  ( t  =  y  ->  ( C  x.  ( exp `  ( K  x.  t
) ) )  =  ( C  x.  ( exp `  ( K  x.  y ) ) ) )
54cbvmptv 4111 . . . . 5  |-  ( t  e.  S  |->  ( C  x.  ( exp `  ( K  x.  t )
) ) )  =  ( y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y ) ) ) )
61, 5eqtri 2303 . . . 4  |-  Y  =  ( y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y ) ) ) )
76oveq2i 5869 . . 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 3660 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
10 eleq2 2344 . . . . . . . . . 10  |-  ( S  =  RR  ->  (
y  e.  S  <->  y  e.  RR ) )
11 recn 8827 . . . . . . . . . 10  |-  ( y  e.  RR  ->  y  e.  CC )
1210, 11syl6bi 219 . . . . . . . . 9  |-  ( S  =  RR  ->  (
y  e.  S  -> 
y  e.  CC ) )
13 eleq2 2344 . . . . . . . . . 10  |-  ( S  =  CC  ->  (
y  e.  S  <->  y  e.  CC ) )
1413biimpd 198 . . . . . . . . 9  |-  ( S  =  CC  ->  (
y  e.  S  -> 
y  e.  CC ) )
1512, 14jaoi 368 . . . . . . . 8  |-  ( ( S  =  RR  \/  S  =  CC )  ->  ( y  e.  S  ->  y  e.  CC ) )
168, 9, 153syl 18 . . . . . . 7  |-  ( ph  ->  ( y  e.  S  ->  y  e.  CC ) )
1716imp 418 . . . . . 6  |-  ( (
ph  /\  y  e.  S )  ->  y  e.  CC )
18 expgrowthi.k . . . . . . . 8  |-  ( ph  ->  K  e.  CC )
19 mulcl 8821 . . . . . . . 8  |-  ( ( K  e.  CC  /\  y  e.  CC )  ->  ( K  x.  y
)  e.  CC )
2018, 19sylan 457 . . . . . . 7  |-  ( (
ph  /\  y  e.  CC )  ->  ( K  x.  y )  e.  CC )
21 efcl 12364 . . . . . . 7  |-  ( ( K  x.  y )  e.  CC  ->  ( exp `  ( K  x.  y ) )  e.  CC )
2220, 21syl 15 . . . . . 6  |-  ( (
ph  /\  y  e.  CC )  ->  ( exp `  ( K  x.  y
) )  e.  CC )
2317, 22syldan 456 . . . . 5  |-  ( (
ph  /\  y  e.  S )  ->  ( exp `  ( K  x.  y ) )  e.  CC )
24 ovex 5883 . . . . . 6  |-  ( K  x.  ( exp `  ( K  x.  y )
) )  e.  _V
2524a1i 10 . . . . 5  |-  ( (
ph  /\  y  e.  S )  ->  ( K  x.  ( exp `  ( K  x.  y
) ) )  e. 
_V )
26 cnex 8818 . . . . . . . . 9  |-  CC  e.  _V
2726prid2 3735 . . . . . . . 8  |-  CC  e.  { RR ,  CC }
2827a1i 10 . . . . . . 7  |-  ( ph  ->  CC  e.  { RR ,  CC } )
2917, 20syldan 456 . . . . . . 7  |-  ( (
ph  /\  y  e.  S )  ->  ( K  x.  y )  e.  CC )
3018adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  S )  ->  K  e.  CC )
31 efcl 12364 . . . . . . . 8  |-  ( x  e.  CC  ->  ( exp `  x )  e.  CC )
3231adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  ( exp `  x )  e.  CC )
33 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
3433a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  S )  ->  1  e.  CC )
358dvmptid 19306 . . . . . . . . 9  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  y ) )  =  ( y  e.  S  |->  1 ) )
368, 17, 34, 35, 18dvmptcmul 19313 . . . . . . . 8  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( K  x.  y ) ) )  =  ( y  e.  S  |->  ( K  x.  1 ) ) )
3718mulid1d 8852 . . . . . . . . 9  |-  ( ph  ->  ( K  x.  1 )  =  K )
3837mpteq2dv 4107 . . . . . . . 8  |-  ( ph  ->  ( y  e.  S  |->  ( K  x.  1 ) )  =  ( y  e.  S  |->  K ) )
3936, 38eqtrd 2315 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( K  x.  y ) ) )  =  ( y  e.  S  |->  K ) )
40 dvef 19327 . . . . . . . . 9  |-  ( CC 
_D  exp )  =  exp
41 eff 12363 . . . . . . . . . . . 12  |-  exp : CC
--> CC
42 ffn 5389 . . . . . . . . . . . 12  |-  ( exp
: CC --> CC  ->  exp 
Fn  CC )
4341, 42ax-mp 8 . . . . . . . . . . 11  |-  exp  Fn  CC
44 dffn5 5568 . . . . . . . . . . 11  |-  ( exp 
Fn  CC  <->  exp  =  ( x  e.  CC  |->  ( exp `  x ) ) )
4543, 44mpbi 199 . . . . . . . . . 10  |-  exp  =  ( x  e.  CC  |->  ( exp `  x ) )
4645oveq2i 5869 . . . . . . . . 9  |-  ( CC 
_D  exp )  =  ( CC  _D  ( x  e.  CC  |->  ( exp `  x ) ) )
4740, 46, 453eqtr3i 2311 . . . . . . . 8  |-  ( CC 
_D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( x  e.  CC  |->  ( exp `  x ) )
4847a1i 10 . . . . . . 7  |-  ( ph  ->  ( CC  _D  (
x  e.  CC  |->  ( exp `  x ) ) )  =  ( x  e.  CC  |->  ( exp `  x ) ) )
49 fveq2 5525 . . . . . . 7  |-  ( x  =  ( K  x.  y )  ->  ( exp `  x )  =  ( exp `  ( K  x.  y )
) )
508, 28, 29, 30, 32, 32, 39, 48, 49, 49dvmptco 19321 . . . . . 6  |-  ( ph  ->  ( S  _D  (
y  e.  S  |->  ( exp `  ( K  x.  y ) ) ) )  =  ( y  e.  S  |->  ( ( exp `  ( K  x.  y )
)  x.  K ) ) )
51 mulcom 8823 . . . . . . . . 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 464 . . . . . . . 8  |-  ( (
ph  /\  ( ph  /\  y  e.  S ) )  ->  ( ( exp `  ( K  x.  y ) )  x.  K )  =  ( K  x.  ( exp `  ( K  x.  y
) ) ) )
5352anabss5 789 . . . . . . 7  |-  ( (
ph  /\  y  e.  S )  ->  (
( exp `  ( K  x.  y )
)  x.  K )  =  ( K  x.  ( exp `  ( K  x.  y ) ) ) )
5453mpteq2dva 4106 . . . . . 6  |-  ( ph  ->  ( y  e.  S  |->  ( ( exp `  ( K  x.  y )
)  x.  K ) )  =  ( y  e.  S  |->  ( K  x.  ( exp `  ( K  x.  y )
) ) ) )
5550, 54eqtrd 2315 . . . . 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 19313 . . . 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 1137 . . . . . . . 8  |-  ( (
ph  /\  ph  /\  ( ph  /\  y  e.  S
) )  ->  ( C  e.  CC  /\  K  e.  CC  /\  ( exp `  ( K  x.  y
) )  e.  CC ) )
59583anidm12 1239 . . . . . . 7  |-  ( (
ph  /\  ( ph  /\  y  e.  S ) )  ->  ( C  e.  CC  /\  K  e.  CC  /\  ( exp `  ( K  x.  y
) )  e.  CC ) )
6059anabss5 789 . . . . . 6  |-  ( (
ph  /\  y  e.  S )  ->  ( C  e.  CC  /\  K  e.  CC  /\  ( exp `  ( K  x.  y
) )  e.  CC ) )
61 mul12 8978 . . . . . 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 15 . . . . 5  |-  ( (
ph  /\  y  e.  S )  ->  ( C  x.  ( K  x.  ( exp `  ( K  x.  y )
) ) )  =  ( K  x.  ( C  x.  ( exp `  ( K  x.  y
) ) ) ) )
6362mpteq2dva 4106 . . . 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 2315 . . 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 2327 . 2  |-  ( ph  ->  ( S  _D  Y
)  =  ( y  e.  S  |->  ( K  x.  ( C  x.  ( exp `  ( K  x.  y ) ) ) ) ) )
66 ovex 5883 . . . 4  |-  ( C  x.  ( exp `  ( K  x.  y )
) )  e.  _V
6766a1i 10 . . 3  |-  ( (
ph  /\  y  e.  S )  ->  ( C  x.  ( exp `  ( K  x.  y
) ) )  e. 
_V )
68 fconstmpt 4732 . . . 4  |-  ( S  X.  { K }
)  =  ( y  e.  S  |->  K )
6968a1i 10 . . 3  |-  ( ph  ->  ( S  X.  { K } )  =  ( y  e.  S  |->  K ) )
706a1i 10 . . 3  |-  ( ph  ->  Y  =  ( y  e.  S  |->  ( C  x.  ( exp `  ( K  x.  y )
) ) ) )
718, 30, 67, 69, 70offval2 6095 . 2  |-  ( ph  ->  ( ( S  X.  { K } )  o F  x.  Y )  =  ( y  e.  S  |->  ( K  x.  ( C  x.  ( exp `  ( K  x.  y ) ) ) ) ) )
7265, 71eqtr4d 2318 1  |-  ( ph  ->  ( S  _D  Y
)  =  ( ( S  X.  { K } )  o F  x.  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   {cpr 3641    e. cmpt 4077    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   RRcr 8736   1c1 8738    x. cmul 8742   expce 12343    _D cdv 19213
This theorem is referenced by:  expgrowth  27552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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