Theorem expgrowthi 38532

Description: Exponential growth and decay model. See expgrowth 38534 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.

Step	Hyp	Ref	Expression
1		expgrowthi.yt	. . . . 5 |- Y = ( t e. S |-> ( C x. ( exp ` ( K x. t ) ) ) )
2		oveq2 6658	. . . . . . . 8 |- ( t = y -> ( K x. t ) = ( K x. y ) )
3	2	fveq2d 6195	. . . . . . 7 |- ( t = y -> ( exp ` ( K x. t ) ) = ( exp ` ( K x. y ) ) )
4	3	oveq2d 6666	. . . . . 6 |- ( t = y -> ( C x. ( exp ` ( K x. t ) ) ) = ( C x. ( exp ` ( K x. y ) ) ) )
5	4	cbvmptv 4750	. . . . 5 |- ( t e. S |-> ( C x. ( exp ` ( K x. t ) ) ) ) = ( y e. S |-> ( C x. ( exp ` ( K x. y ) ) ) )
6	1, 5	eqtri 2644	. . . 4 |- Y = ( y e. S |-> ( C x. ( exp ` ( K x. y ) ) ) )
7	6	oveq2i 6661	. . 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 4197	. . . . . . . 8 |- ( S e. { RR , CC } -> ( S = RR \/ S = CC ) )
10		eleq2 2690	. . . . . . . . . 10 |- ( S = RR -> ( y e. S <-> y e. RR ) )
11		recn 10026	. . . . . . . . . 10 |- ( y e. RR -> y e. CC )
12	10, 11	syl6bi 243	. . . . . . . . 9 |- ( S = RR -> ( y e. S -> y e. CC ) )
13		eleq2 2690	. . . . . . . . . 10 |- ( S = CC -> ( y e. S <-> y e. CC ) )
14	13	biimpd 219	. . . . . . . . 9 |- ( S = CC -> ( y e. S -> y e. CC ) )
15	12, 14	jaoi 394	. . . . . . . 8 |- ( ( S = RR \/ S = CC ) -> ( y e. S -> y e. CC ) )
16	8, 9, 15	3syl 18	. . . . . . 7 |- ( ph -> ( y e. S -> y e. CC ) )
17	16	imp 445	. . . . . 6 |- ( ( ph /\ y e. S ) -> y e. CC )
18		expgrowthi.k	. . . . . . . 8 |- ( ph -> K e. CC )
19		mulcl 10020	. . . . . . . 8 |- ( ( K e. CC /\ y e. CC ) -> ( K x. y ) e. CC )
20	18, 19	sylan 488	. . . . . . 7 |- ( ( ph /\ y e. CC ) -> ( K x. y ) e. CC )
21		efcl 14813	. . . . . . 7 |- ( ( K x. y ) e. CC -> ( exp ` ( K x. y ) ) e. CC )
22	20, 21	syl 17	. . . . . 6 |- ( ( ph /\ y e. CC ) -> ( exp ` ( K x. y ) ) e. CC )
23	17, 22	syldan 487	. . . . 5 |- ( ( ph /\ y e. S ) -> ( exp ` ( K x. y ) ) e. CC )
24		ovexd 6680	. . . . 5 |- ( ( ph /\ y e. S ) -> ( K x. ( exp ` ( K x. y ) ) ) e. _V )
25		cnelprrecn 10029	. . . . . . . 8 |- CC e. { RR , CC }
26	25	a1i 11	. . . . . . 7 |- ( ph -> CC e. { RR , CC } )
27	17, 20	syldan 487	. . . . . . 7 |- ( ( ph /\ y e. S ) -> ( K x. y ) e. CC )
28	18	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. S ) -> K e. CC )
29		efcl 14813	. . . . . . . 8 |- ( x e. CC -> ( exp ` x ) e. CC )
30	29	adantl 482	. . . . . . 7 |- ( ( ph /\ x e. CC ) -> ( exp ` x ) e. CC )
31		1cnd 10056	. . . . . . . . 9 |- ( ( ph /\ y e. S ) -> 1 e. CC )
32	8	dvmptid 23720	. . . . . . . . 9 |- ( ph -> ( S _D ( y e. S |-> y ) ) = ( y e. S |-> 1 ) )
33	8, 17, 31, 32, 18	dvmptcmul 23727	. . . . . . . 8 |- ( ph -> ( S _D ( y e. S |-> ( K x. y ) ) ) = ( y e. S |-> ( K x. 1 ) ) )
34	18	mulid1d 10057	. . . . . . . . 9 |- ( ph -> ( K x. 1 ) = K )
35	34	mpteq2dv 4745	. . . . . . . 8 |- ( ph -> ( y e. S |-> ( K x. 1 ) ) = ( y e. S |-> K ) )
36	33, 35	eqtrd 2656	. . . . . . 7 |- ( ph -> ( S _D ( y e. S |-> ( K x. y ) ) ) = ( y e. S |-> K ) )
37		dvef 23743	. . . . . . . . 9 |- ( CC _D exp ) = exp
38		eff 14812	. . . . . . . . . . . 12 |- exp : CC --> CC
39		ffn 6045	. . . . . . . . . . . 12 |- ( exp : CC --> CC -> exp Fn CC )
40	38, 39	ax-mp 5	. . . . . . . . . . 11 |- exp Fn CC
41		dffn5 6241	. . . . . . . . . . 11 |- ( exp Fn CC <-> exp = ( x e. CC |-> ( exp ` x ) ) )
42	40, 41	mpbi 220	. . . . . . . . . 10 |- exp = ( x e. CC |-> ( exp ` x ) )
43	42	oveq2i 6661	. . . . . . . . 9 |- ( CC _D exp ) = ( CC _D ( x e. CC |-> ( exp ` x ) ) )
44	37, 43, 42	3eqtr3i 2652	. . . . . . . 8 |- ( CC _D ( x e. CC |-> ( exp ` x ) ) ) = ( x e. CC |-> ( exp ` x ) )
45	44	a1i 11	. . . . . . 7 |- ( ph -> ( CC _D ( x e. CC |-> ( exp ` x ) ) ) = ( x e. CC |-> ( exp ` x ) ) )
46		fveq2 6191	. . . . . . 7 |- ( x = ( K x. y ) -> ( exp ` x ) = ( exp ` ( K x. y ) ) )
47	8, 26, 27, 28, 30, 30, 36, 45, 46, 46	dvmptco 23735	. . . . . 6 |- ( ph -> ( S _D ( y e. S |-> ( exp ` ( K x. y ) ) ) ) = ( y e. S |-> ( ( exp ` ( K x. y ) ) x. K ) ) )
48		mulcom 10022	. . . . . . . . 9 |- ( ( ( exp ` ( K x. y ) ) e. CC /\ K e. CC ) -> ( ( exp ` ( K x. y ) ) x. K ) = ( K x. ( exp ` ( K x. y ) ) ) )
49	23, 18, 48	syl2anr 495	. . . . . . . 8 |- ( ( ph /\ ( ph /\ y e. S ) ) -> ( ( exp ` ( K x. y ) ) x. K ) = ( K x. ( exp ` ( K x. y ) ) ) )
50	49	anabss5 857	. . . . . . 7 |- ( ( ph /\ y e. S ) -> ( ( exp ` ( K x. y ) ) x. K ) = ( K x. ( exp ` ( K x. y ) ) ) )
51	50	mpteq2dva 4744	. . . . . 6 |- ( ph -> ( y e. S |-> ( ( exp ` ( K x. y ) ) x. K ) ) = ( y e. S |-> ( K x. ( exp ` ( K x. y ) ) ) ) )
52	47, 51	eqtrd 2656	. . . . 5 |- ( ph -> ( S _D ( y e. S |-> ( exp ` ( K x. y ) ) ) ) = ( y e. S |-> ( K x. ( exp ` ( K x. y ) ) ) ) )
53		expgrowthi.y0	. . . . 5 |- ( ph -> C e. CC )
54	8, 23, 24, 52, 53	dvmptcmul 23727	. . . 4 |- ( ph -> ( S _D ( y e. S |-> ( C x. ( exp ` ( K x. y ) ) ) ) ) = ( y e. S |-> ( C x. ( K x. ( exp ` ( K x. y ) ) ) ) ) )
55	53, 18, 23	3anim123i 1247	. . . . . . . 8 |- ( ( ph /\ ph /\ ( ph /\ y e. S ) ) -> ( C e. CC /\ K e. CC /\ ( exp ` ( K x. y ) ) e. CC ) )
56	55	3anidm12 1383	. . . . . . 7 |- ( ( ph /\ ( ph /\ y e. S ) ) -> ( C e. CC /\ K e. CC /\ ( exp ` ( K x. y ) ) e. CC ) )
57	56	anabss5 857	. . . . . 6 |- ( ( ph /\ y e. S ) -> ( C e. CC /\ K e. CC /\ ( exp ` ( K x. y ) ) e. CC ) )
58		mul12 10202	. . . . . 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 ) ) ) ) )
59	57, 58	syl 17	. . . . 5 |- ( ( ph /\ y e. S ) -> ( C x. ( K x. ( exp ` ( K x. y ) ) ) ) = ( K x. ( C x. ( exp ` ( K x. y ) ) ) ) )
60	59	mpteq2dva 4744	. . . 4 |- ( ph -> ( y e. S |-> ( C x. ( K x. ( exp ` ( K x. y ) ) ) ) ) = ( y e. S |-> ( K x. ( C x. ( exp ` ( K x. y ) ) ) ) ) )
61	54, 60	eqtrd 2656	. . 3 |- ( ph -> ( S _D ( y e. S |-> ( C x. ( exp ` ( K x. y ) ) ) ) ) = ( y e. S |-> ( K x. ( C x. ( exp ` ( K x. y ) ) ) ) ) )
62	7, 61	syl5eq 2668	. 2 |- ( ph -> ( S _D Y ) = ( y e. S |-> ( K x. ( C x. ( exp ` ( K x. y ) ) ) ) ) )
63		ovexd 6680	. . 3 |- ( ( ph /\ y e. S ) -> ( C x. ( exp ` ( K x. y ) ) ) e. _V )
64		fconstmpt 5163	. . . 4 |- ( S X. { K } ) = ( y e. S |-> K )
65	64	a1i 11	. . 3 |- ( ph -> ( S X. { K } ) = ( y e. S |-> K ) )
66	6	a1i 11	. . 3 |- ( ph -> Y = ( y e. S |-> ( C x. ( exp ` ( K x. y ) ) ) ) )
67	8, 28, 63, 65, 66	offval2 6914	. 2 |- ( ph -> ( ( S X. { K } ) oF x. Y ) = ( y e. S |-> ( K x. ( C x. ( exp ` ( K x. y ) ) ) ) ) )
68	62, 67	eqtr4d 2659	1 |- ( ph -> ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   {cpr 4179   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   1c1 9937   x. cmul 9941   expce 14792   _D cdv 23627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  expgrowth  38534