Theorem expgrowth 38534

Description: Exponential growth and decay model. The derivative of a function y of variable t equals a constant k times y itself, iff y equals some constant C times the exponential of kt. This theorem and expgrowthi 38532 illustrate one of the simplest and most crucial classes of differential equations, equations that relate functions to their derivatives.

Section 6.3 of [Strang] p. 242 calls y' = ky "the most important differential equation in applied mathematics". In the field of population ecology it is known as the Malthusian growth model or exponential law, and C, k, and t correspond to initial population size, growth rate, and time respectively (https://en.wikipedia.org/wiki/Malthusian_growth_model); and in finance, the model appears in a similar role in continuous compounding with C as the initial amount of money. In exponential decay models, k is often expressed as the negative of a positive constant λ.

Here y' is given as ( S _D Y ), C as c, and ky as ( ( S X. { K } ) oF x. Y ). ( S X. { K } ) is the constant function that maps any real or complex input to k and oF x. is multiplication as a function operation.

The leftward direction of the biconditional is as given in http://www.saylor.org/site/wp-content/uploads/2011/06/MA221-2.1.1.pdf pp. 1-2, which also notes the reverse direction ("While we will not prove this here, it turns out that these are the only functions that satisfy this equation."). The rightward direction is Theorem 5.1 of [LarsonHostetlerEdwards] p. 375 (which notes " C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0."); its proof here closely follows the proof of y' = y in https://proofwiki.org/wiki/Exponential_Growth_Equation/Special_Case.

Statements for this and expgrowthi 38532 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)

Hypotheses

Ref	Expression
expgrowth.s	|- ( ph -> S e. { RR , CC } )
expgrowth.k	|- ( ph -> K e. CC )
expgrowth.y	|- ( ph -> Y : S --> CC )
expgrowth.dy	|- ( ph -> dom ( S _D Y ) = S )

Assertion

Ref	Expression
expgrowth	|- ( ph -> ( ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) <-> E. c e. CC Y = ( t e. S |-> ( c x. ( exp ` ( K x. t ) ) ) ) ) )

Distinct variable groups:   t, c, K   S, c, t   Y, c

Allowed substitution hints:   ph( t, c)   Y( t)

Proof of Theorem expgrowth

Dummy variables u x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		expgrowth.s	. . . . . . . . . . . . . . . . 17 |- ( ph -> S e. { RR , CC } )
2		cnelprrecn 10029	. . . . . . . . . . . . . . . . . 18 |- CC e. { RR , CC }
3	2	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> CC e. { RR , CC } )
4		expgrowth.k	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> K e. CC )
5		recnprss 23668	. . . . . . . . . . . . . . . . . . . . . 22 |- ( S e. { RR , CC } -> S C_ CC )
6	1, 5	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> S C_ CC )
7	6	sseld 3602	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( u e. S -> u e. CC ) )
8		mulcl 10020	. . . . . . . . . . . . . . . . . . . 20 |- ( ( K e. CC /\ u e. CC ) -> ( K x. u ) e. CC )
9	4, 7, 8	syl6an 568	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( u e. S -> ( K x. u ) e. CC ) )
10	9	imp 445	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ u e. S ) -> ( K x. u ) e. CC )
11	10	negcld 10379	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ u e. S ) -> -u ( K x. u ) e. CC )
12	4	negcld 10379	. . . . . . . . . . . . . . . . . 18 |- ( ph -> -u K e. CC )
13	12	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ u e. S ) -> -u K e. CC )
14		efcl 14813	. . . . . . . . . . . . . . . . . 18 |- ( y e. CC -> ( exp ` y ) e. CC )
15	14	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. CC ) -> ( exp ` y ) e. CC )
16	4	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ u e. S ) -> K e. CC )
17	7	imp 445	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ u e. S ) -> u e. CC )
18		ax-1cn 9994	. . . . . . . . . . . . . . . . . . . . 21 |- 1 e. CC
19	18	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ u e. S ) -> 1 e. CC )
20	1	dvmptid 23720	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( S _D ( u e. S |-> u ) ) = ( u e. S |-> 1 ) )
21	1, 17, 19, 20, 4	dvmptcmul 23727	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( S _D ( u e. S |-> ( K x. u ) ) ) = ( u e. S |-> ( K x. 1 ) ) )
22	4	mulid1d 10057	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( K x. 1 ) = K )
23	22	mpteq2dv 4745	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( u e. S |-> ( K x. 1 ) ) = ( u e. S |-> K ) )
24	21, 23	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( S _D ( u e. S |-> ( K x. u ) ) ) = ( u e. S |-> K ) )
25	1, 10, 16, 24	dvmptneg 23729	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( S _D ( u e. S |-> -u ( K x. u ) ) ) = ( u e. S |-> -u K ) )
26		dvef 23743	. . . . . . . . . . . . . . . . . . 19 |- ( CC _D exp ) = exp
27		eff 14812	. . . . . . . . . . . . . . . . . . . . . 22 |- exp : CC --> CC
28		ffn 6045	. . . . . . . . . . . . . . . . . . . . . 22 |- ( exp : CC --> CC -> exp Fn CC )
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 |- exp Fn CC
30		dffn5 6241	. . . . . . . . . . . . . . . . . . . . 21 |- ( exp Fn CC <-> exp = ( y e. CC |-> ( exp ` y ) ) )
31	29, 30	mpbi 220	. . . . . . . . . . . . . . . . . . . 20 |- exp = ( y e. CC |-> ( exp ` y ) )
32	31	oveq2i 6661	. . . . . . . . . . . . . . . . . . 19 |- ( CC _D exp ) = ( CC _D ( y e. CC |-> ( exp ` y ) ) )
33	26, 32, 31	3eqtr3i 2652	. . . . . . . . . . . . . . . . . 18 |- ( CC _D ( y e. CC |-> ( exp ` y ) ) ) = ( y e. CC |-> ( exp ` y ) )
34	33	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( CC _D ( y e. CC |-> ( exp ` y ) ) ) = ( y e. CC |-> ( exp ` y ) ) )
35		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( y = -u ( K x. u ) -> ( exp ` y ) = ( exp ` -u ( K x. u ) ) )
36	1, 3, 11, 13, 15, 15, 25, 34, 35, 35	dvmptco 23735	. . . . . . . . . . . . . . . 16 |- ( ph -> ( S _D ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) )
37	36	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( ph -> ( Y oF x. ( S _D ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( Y oF x. ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) ) )
38		expgrowth.y	. . . . . . . . . . . . . . . 16 |- ( ph -> Y : S --> CC )
39		efcl 14813	. . . . . . . . . . . . . . . . . . . 20 |- ( -u ( K x. u ) e. CC -> ( exp ` -u ( K x. u ) ) e. CC )
40	11, 39	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ u e. S ) -> ( exp ` -u ( K x. u ) ) e. CC )
41	40, 13	mulcld 10060	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ u e. S ) -> ( ( exp ` -u ( K x. u ) ) x. -u K ) e. CC )
42		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) = ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) )
43	41, 42	fmptd 6385	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) : S --> CC )
44	36	feq1d 6030	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( S _D ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) : S --> CC <-> ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) : S --> CC ) )
45	43, 44	mpbird 247	. . . . . . . . . . . . . . . 16 |- ( ph -> ( S _D ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) : S --> CC )
46		mulcom 10022	. . . . . . . . . . . . . . . . 17 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) = ( y x. x ) )
47	46	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) = ( y x. x ) )
48	1, 38, 45, 47	caofcom 6929	. . . . . . . . . . . . . . 15 |- ( ph -> ( Y oF x. ( S _D ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( S _D ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF x. Y ) )
49	37, 48	eqtr3d 2658	. . . . . . . . . . . . . 14 |- ( ph -> ( Y oF x. ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) ) = ( ( S _D ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF x. Y ) )
50	49	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ph -> ( ( ( S _D Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( Y oF x. ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) ) ) = ( ( ( S _D Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( S _D ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF x. Y ) ) )
51		fconst6g 6094	. . . . . . . . . . . . . . . . . 18 |- ( -u K e. CC -> ( S X. { -u K } ) : S --> CC )
52	12, 51	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( S X. { -u K } ) : S --> CC )
53		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( u e. S |-> ( exp ` -u ( K x. u ) ) ) = ( u e. S |-> ( exp ` -u ( K x. u ) ) )
54	40, 53	fmptd 6385	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( u e. S |-> ( exp ` -u ( K x. u ) ) ) : S --> CC )
55	1, 52, 54, 47	caofcom 6929	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( S X. { -u K } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( ( u e. S |-> ( exp ` -u ( K x. u ) ) ) oF x. ( S X. { -u K } ) ) )
56		eqidd 2623	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( u e. S |-> ( exp ` -u ( K x. u ) ) ) = ( u e. S |-> ( exp ` -u ( K x. u ) ) ) )
57		fconstmpt 5163	. . . . . . . . . . . . . . . . . 18 |- ( S X. { -u K } ) = ( u e. S |-> -u K )
58	57	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( S X. { -u K } ) = ( u e. S |-> -u K ) )
59	1, 40, 13, 56, 58	offval2 6914	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( u e. S |-> ( exp ` -u ( K x. u ) ) ) oF x. ( S X. { -u K } ) ) = ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) )
60	55, 59	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( S X. { -u K } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) )
61	60	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ph -> ( Y oF x. ( ( S X. { -u K } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( Y oF x. ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) ) )
62	61	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ph -> ( ( ( S _D Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( Y oF x. ( ( S X. { -u K } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) = ( ( ( S _D Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( Y oF x. ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) ) ) )
63		expgrowth.dy	. . . . . . . . . . . . . 14 |- ( ph -> dom ( S _D Y ) = S )
64	36	dmeqd 5326	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( S _D ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = dom ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) )
65	42, 41	dmmptd 6024	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( u e. S |-> ( ( exp ` -u ( K x. u ) ) x. -u K ) ) = S )
66	64, 65	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ph -> dom ( S _D ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = S )
67	1, 38, 54, 63, 66	dvmulf 23706	. . . . . . . . . . . . 13 |- ( ph -> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( S _D Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( S _D ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF x. Y ) ) )
68	50, 62, 67	3eqtr4rd 2667	. . . . . . . . . . . 12 |- ( ph -> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( S _D Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( Y oF x. ( ( S X. { -u K } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) )
69		ofmul12 38524	. . . . . . . . . . . . . 14 |- ( ( ( S e. { RR , CC } /\ Y : S --> CC ) /\ ( ( S X. { -u K } ) : S --> CC /\ ( u e. S |-> ( exp ` -u ( K x. u ) ) ) : S --> CC ) ) -> ( Y oF x. ( ( S X. { -u K } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( S X. { -u K } ) oF x. ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) )
70	1, 38, 52, 54, 69	syl22anc 1327	. . . . . . . . . . . . 13 |- ( ph -> ( Y oF x. ( ( S X. { -u K } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( S X. { -u K } ) oF x. ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) )
71	70	oveq2d 6666	. . . . . . . . . . . 12 |- ( ph -> ( ( ( S _D Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( Y oF x. ( ( S X. { -u K } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) = ( ( ( S _D Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( S X. { -u K } ) oF x. ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) )
72	68, 71	eqtrd 2656	. . . . . . . . . . 11 |- ( ph -> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( S _D Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( S X. { -u K } ) oF x. ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) )
73		oveq1 6657	. . . . . . . . . . . 12 |- ( ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) -> ( ( S _D Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) )
74	73	oveq1d 6665	. . . . . . . . . . 11 |- ( ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) -> ( ( ( S _D Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( S X. { -u K } ) oF x. ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( S X. { -u K } ) oF x. ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) )
75	72, 74	sylan9eq 2676	. . . . . . . . . 10 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( S X. { -u K } ) oF x. ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) )
76		mulass 10024	. . . . . . . . . . . . . . 15 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x x. y ) x. z ) = ( x x. ( y x. z ) ) )
77	76	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. CC /\ y e. CC /\ z e. CC ) ) -> ( ( x x. y ) x. z ) = ( x x. ( y x. z ) ) )
78	1, 52, 38, 54, 77	caofass 6931	. . . . . . . . . . . . 13 |- ( ph -> ( ( ( S X. { -u K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( ( S X. { -u K } ) oF x. ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) )
79	78	oveq2d 6666	. . . . . . . . . . . 12 |- ( ph -> ( ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( ( S X. { -u K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( S X. { -u K } ) oF x. ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) )
80	79	eqeq2d 2632	. . . . . . . . . . 11 |- ( ph -> ( ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( ( S X. { -u K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) <-> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( S X. { -u K } ) oF x. ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) ) )
81	80	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> ( ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( ( S X. { -u K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) <-> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( S X. { -u K } ) oF x. ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) ) )
82	75, 81	mpbird 247	. . . . . . . . 9 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( ( S X. { -u K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) )
83		mulcl 10020	. . . . . . . . . . . . . 14 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
84	83	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
85		fconst6g 6094	. . . . . . . . . . . . . 14 |- ( K e. CC -> ( S X. { K } ) : S --> CC )
86	4, 85	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( S X. { K } ) : S --> CC )
87		inidm 3822	. . . . . . . . . . . . 13 |- ( S i^i S ) = S
88	84, 86, 38, 1, 1, 87	off 6912	. . . . . . . . . . . 12 |- ( ph -> ( ( S X. { K } ) oF x. Y ) : S --> CC )
89	84, 52, 38, 1, 1, 87	off 6912	. . . . . . . . . . . 12 |- ( ph -> ( ( S X. { -u K } ) oF x. Y ) : S --> CC )
90		adddir 10031	. . . . . . . . . . . . 13 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x + y ) x. z ) = ( ( x x. z ) + ( y x. z ) ) )
91	90	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. CC /\ y e. CC /\ z e. CC ) ) -> ( ( x + y ) x. z ) = ( ( x x. z ) + ( y x. z ) ) )
92	1, 54, 88, 89, 91	caofdir 6934	. . . . . . . . . . 11 |- ( ph -> ( ( ( ( S X. { K } ) oF x. Y ) oF + ( ( S X. { -u K } ) oF x. Y ) ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( ( S X. { -u K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) )
93	92	eqeq2d 2632	. . . . . . . . . 10 |- ( ph -> ( ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF + ( ( S X. { -u K } ) oF x. Y ) ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) <-> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( ( S X. { -u K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) )
94	93	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> ( ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF + ( ( S X. { -u K } ) oF x. Y ) ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) <-> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF + ( ( ( S X. { -u K } ) oF x. Y ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) ) )
95	82, 94	mpbird 247	. . . . . . . 8 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF + ( ( S X. { -u K } ) oF x. Y ) ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) )
96		ofnegsub 11018	. . . . . . . . . . . . 13 |- ( ( S e. { RR , CC } /\ ( ( S X. { K } ) oF x. Y ) : S --> CC /\ ( ( S X. { K } ) oF x. Y ) : S --> CC ) -> ( ( ( S X. { K } ) oF x. Y ) oF + ( ( S X. { -u 1 } ) oF x. ( ( S X. { K } ) oF x. Y ) ) ) = ( ( ( S X. { K } ) oF x. Y ) oF - ( ( S X. { K } ) oF x. Y ) ) )
97	1, 88, 88, 96	syl3anc 1326	. . . . . . . . . . . 12 |- ( ph -> ( ( ( S X. { K } ) oF x. Y ) oF + ( ( S X. { -u 1 } ) oF x. ( ( S X. { K } ) oF x. Y ) ) ) = ( ( ( S X. { K } ) oF x. Y ) oF - ( ( S X. { K } ) oF x. Y ) ) )
98		neg1cn 11124	. . . . . . . . . . . . . . . . 17 |- -u 1 e. CC
99	98	fconst6 6095	. . . . . . . . . . . . . . . 16 |- ( S X. { -u 1 } ) : S --> CC
100	99	a1i 11	. . . . . . . . . . . . . . 15 |- ( ph -> ( S X. { -u 1 } ) : S --> CC )
101	1, 100, 86, 38, 77	caofass 6931	. . . . . . . . . . . . . 14 |- ( ph -> ( ( ( S X. { -u 1 } ) oF x. ( S X. { K } ) ) oF x. Y ) = ( ( S X. { -u 1 } ) oF x. ( ( S X. { K } ) oF x. Y ) ) )
102	98	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> -u 1 e. CC )
103	1, 102, 4	ofc12 6922	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( S X. { -u 1 } ) oF x. ( S X. { K } ) ) = ( S X. { ( -u 1 x. K ) } ) )
104	4	mulm1d 10482	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( -u 1 x. K ) = -u K )
105	104	sneqd 4189	. . . . . . . . . . . . . . . . 17 |- ( ph -> { ( -u 1 x. K ) } = { -u K } )
106	105	xpeq2d 5139	. . . . . . . . . . . . . . . 16 |- ( ph -> ( S X. { ( -u 1 x. K ) } ) = ( S X. { -u K } ) )
107	103, 106	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( S X. { -u 1 } ) oF x. ( S X. { K } ) ) = ( S X. { -u K } ) )
108	107	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( ph -> ( ( ( S X. { -u 1 } ) oF x. ( S X. { K } ) ) oF x. Y ) = ( ( S X. { -u K } ) oF x. Y ) )
109	101, 108	eqtr3d 2658	. . . . . . . . . . . . 13 |- ( ph -> ( ( S X. { -u 1 } ) oF x. ( ( S X. { K } ) oF x. Y ) ) = ( ( S X. { -u K } ) oF x. Y ) )
110	109	oveq2d 6666	. . . . . . . . . . . 12 |- ( ph -> ( ( ( S X. { K } ) oF x. Y ) oF + ( ( S X. { -u 1 } ) oF x. ( ( S X. { K } ) oF x. Y ) ) ) = ( ( ( S X. { K } ) oF x. Y ) oF + ( ( S X. { -u K } ) oF x. Y ) ) )
111		ofsubid 38523	. . . . . . . . . . . . 13 |- ( ( S e. { RR , CC } /\ ( ( S X. { K } ) oF x. Y ) : S --> CC ) -> ( ( ( S X. { K } ) oF x. Y ) oF - ( ( S X. { K } ) oF x. Y ) ) = ( S X. { 0 } ) )
112	1, 88, 111	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( ( ( S X. { K } ) oF x. Y ) oF - ( ( S X. { K } ) oF x. Y ) ) = ( S X. { 0 } ) )
113	97, 110, 112	3eqtr3d 2664	. . . . . . . . . . 11 |- ( ph -> ( ( ( S X. { K } ) oF x. Y ) oF + ( ( S X. { -u K } ) oF x. Y ) ) = ( S X. { 0 } ) )
114	113	oveq1d 6665	. . . . . . . . . 10 |- ( ph -> ( ( ( ( S X. { K } ) oF x. Y ) oF + ( ( S X. { -u K } ) oF x. Y ) ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( ( S X. { 0 } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) )
115	114	eqeq2d 2632	. . . . . . . . 9 |- ( ph -> ( ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF + ( ( S X. { -u K } ) oF x. Y ) ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) <-> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( S X. { 0 } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) )
116	115	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> ( ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( ( ( S X. { K } ) oF x. Y ) oF + ( ( S X. { -u K } ) oF x. Y ) ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) <-> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( S X. { 0 } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) )
117	95, 116	mpbid 222	. . . . . . 7 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( ( S X. { 0 } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) )
118		0cnd 10033	. . . . . . . . 9 |- ( ph -> 0 e. CC )
119		mul02 10214	. . . . . . . . . 10 |- ( x e. CC -> ( 0 x. x ) = 0 )
120	119	adantl 482	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> ( 0 x. x ) = 0 )
121	1, 54, 118, 118, 120	caofid2 6928	. . . . . . . 8 |- ( ph -> ( ( S X. { 0 } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( S X. { 0 } ) )
122	121	adantr 481	. . . . . . 7 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> ( ( S X. { 0 } ) oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( S X. { 0 } ) )
123	117, 122	eqtrd 2656	. . . . . 6 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( S X. { 0 } ) )
124	1	adantr 481	. . . . . . 7 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> S e. { RR , CC } )
125	84, 38, 54, 1, 1, 87	off 6912	. . . . . . . 8 |- ( ph -> ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) : S --> CC )
126	125	adantr 481	. . . . . . 7 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) : S --> CC )
127	123	dmeqd 5326	. . . . . . . 8 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> dom ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = dom ( S X. { 0 } ) )
128		0cn 10032	. . . . . . . . . 10 |- 0 e. CC
129	128	fconst6 6095	. . . . . . . . 9 |- ( S X. { 0 } ) : S --> CC
130	129	fdmi 6052	. . . . . . . 8 |- dom ( S X. { 0 } ) = S
131	127, 130	syl6eq 2672	. . . . . . 7 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> dom ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = S )
132	124, 126, 131	dvconstbi 38533	. . . . . 6 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> ( ( S _D ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) = ( S X. { 0 } ) <-> E. x e. CC ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( S X. { x } ) ) )
133	123, 132	mpbid 222	. . . . 5 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> E. x e. CC ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( S X. { x } ) )
134		oveq1 6657	. . . . . . . . . 10 |- ( ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( S X. { x } ) -> ( ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( ( S X. { x } ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) )
135		efne0 14827	. . . . . . . . . . . . . . 15 |- ( -u ( K x. u ) e. CC -> ( exp ` -u ( K x. u ) ) =/= 0 )
136		eldifsn 4317	. . . . . . . . . . . . . . 15 |- ( ( exp ` -u ( K x. u ) ) e. ( CC \ { 0 } ) <-> ( ( exp ` -u ( K x. u ) ) e. CC /\ ( exp ` -u ( K x. u ) ) =/= 0 ) )
137	39, 135, 136	sylanbrc 698	. . . . . . . . . . . . . 14 |- ( -u ( K x. u ) e. CC -> ( exp ` -u ( K x. u ) ) e. ( CC \ { 0 } ) )
138	11, 137	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ u e. S ) -> ( exp ` -u ( K x. u ) ) e. ( CC \ { 0 } ) )
139	138, 53	fmptd 6385	. . . . . . . . . . . 12 |- ( ph -> ( u e. S |-> ( exp ` -u ( K x. u ) ) ) : S --> ( CC \ { 0 } ) )
140		ofdivcan4 38526	. . . . . . . . . . . 12 |- ( ( S e. { RR , CC } /\ Y : S --> CC /\ ( u e. S |-> ( exp ` -u ( K x. u ) ) ) : S --> ( CC \ { 0 } ) ) -> ( ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = Y )
141	1, 38, 139, 140	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> ( ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = Y )
142	141	eqeq1d 2624	. . . . . . . . . 10 |- ( ph -> ( ( ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( ( S X. { x } ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) <-> Y = ( ( S X. { x } ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) )
143	134, 142	syl5ib 234	. . . . . . . . 9 |- ( ph -> ( ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( S X. { x } ) -> Y = ( ( S X. { x } ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) )
144	143	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> ( ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( S X. { x } ) -> Y = ( ( S X. { x } ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) ) )
145		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
146	145	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ u e. S ) -> x e. _V )
147		ovexd 6680	. . . . . . . . . . . 12 |- ( ( ph /\ u e. S ) -> ( 1 / ( exp ` ( K x. u ) ) ) e. _V )
148		fconstmpt 5163	. . . . . . . . . . . . 13 |- ( S X. { x } ) = ( u e. S |-> x )
149	148	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( S X. { x } ) = ( u e. S |-> x ) )
150		efneg 14828	. . . . . . . . . . . . . 14 |- ( ( K x. u ) e. CC -> ( exp ` -u ( K x. u ) ) = ( 1 / ( exp ` ( K x. u ) ) ) )
151	10, 150	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ u e. S ) -> ( exp ` -u ( K x. u ) ) = ( 1 / ( exp ` ( K x. u ) ) ) )
152	151	mpteq2dva 4744	. . . . . . . . . . . 12 |- ( ph -> ( u e. S |-> ( exp ` -u ( K x. u ) ) ) = ( u e. S |-> ( 1 / ( exp ` ( K x. u ) ) ) ) )
153	1, 146, 147, 149, 152	offval2 6914	. . . . . . . . . . 11 |- ( ph -> ( ( S X. { x } ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( u e. S |-> ( x / ( 1 / ( exp ` ( K x. u ) ) ) ) ) )
154	153	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> ( ( S X. { x } ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( u e. S |-> ( x / ( 1 / ( exp ` ( K x. u ) ) ) ) ) )
155		efcl 14813	. . . . . . . . . . . . . . . . 17 |- ( ( K x. u ) e. CC -> ( exp ` ( K x. u ) ) e. CC )
156		efne0 14827	. . . . . . . . . . . . . . . . 17 |- ( ( K x. u ) e. CC -> ( exp ` ( K x. u ) ) =/= 0 )
157	155, 156	jca 554	. . . . . . . . . . . . . . . 16 |- ( ( K x. u ) e. CC -> ( ( exp ` ( K x. u ) ) e. CC /\ ( exp ` ( K x. u ) ) =/= 0 ) )
158	10, 157	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ph /\ u e. S ) -> ( ( exp ` ( K x. u ) ) e. CC /\ ( exp ` ( K x. u ) ) =/= 0 ) )
159		ax-1ne0 10005	. . . . . . . . . . . . . . . . 17 |- 1 =/= 0
160	18, 159	pm3.2i 471	. . . . . . . . . . . . . . . 16 |- ( 1 e. CC /\ 1 =/= 0 )
161		divdiv2 10737	. . . . . . . . . . . . . . . 16 |- ( ( x e. CC /\ ( 1 e. CC /\ 1 =/= 0 ) /\ ( ( exp ` ( K x. u ) ) e. CC /\ ( exp ` ( K x. u ) ) =/= 0 ) ) -> ( x / ( 1 / ( exp ` ( K x. u ) ) ) ) = ( ( x x. ( exp ` ( K x. u ) ) ) / 1 ) )
162	160, 161	mp3an2 1412	. . . . . . . . . . . . . . 15 |- ( ( x e. CC /\ ( ( exp ` ( K x. u ) ) e. CC /\ ( exp ` ( K x. u ) ) =/= 0 ) ) -> ( x / ( 1 / ( exp ` ( K x. u ) ) ) ) = ( ( x x. ( exp ` ( K x. u ) ) ) / 1 ) )
163	158, 162	sylan2 491	. . . . . . . . . . . . . 14 |- ( ( x e. CC /\ ( ph /\ u e. S ) ) -> ( x / ( 1 / ( exp ` ( K x. u ) ) ) ) = ( ( x x. ( exp ` ( K x. u ) ) ) / 1 ) )
164	10, 155	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ u e. S ) -> ( exp ` ( K x. u ) ) e. CC )
165		mulcl 10020	. . . . . . . . . . . . . . . 16 |- ( ( x e. CC /\ ( exp ` ( K x. u ) ) e. CC ) -> ( x x. ( exp ` ( K x. u ) ) ) e. CC )
166	164, 165	sylan2 491	. . . . . . . . . . . . . . 15 |- ( ( x e. CC /\ ( ph /\ u e. S ) ) -> ( x x. ( exp ` ( K x. u ) ) ) e. CC )
167	166	div1d 10793	. . . . . . . . . . . . . 14 |- ( ( x e. CC /\ ( ph /\ u e. S ) ) -> ( ( x x. ( exp ` ( K x. u ) ) ) / 1 ) = ( x x. ( exp ` ( K x. u ) ) ) )
168	163, 167	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( x e. CC /\ ( ph /\ u e. S ) ) -> ( x / ( 1 / ( exp ` ( K x. u ) ) ) ) = ( x x. ( exp ` ( K x. u ) ) ) )
169	168	ancoms 469	. . . . . . . . . . . 12 |- ( ( ( ph /\ u e. S ) /\ x e. CC ) -> ( x / ( 1 / ( exp ` ( K x. u ) ) ) ) = ( x x. ( exp ` ( K x. u ) ) ) )
170	169	an32s 846	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. CC ) /\ u e. S ) -> ( x / ( 1 / ( exp ` ( K x. u ) ) ) ) = ( x x. ( exp ` ( K x. u ) ) ) )
171	170	mpteq2dva 4744	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> ( u e. S |-> ( x / ( 1 / ( exp ` ( K x. u ) ) ) ) ) = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) )
172	154, 171	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> ( ( S X. { x } ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) )
173	172	eqeq2d 2632	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> ( Y = ( ( S X. { x } ) oF / ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) <-> Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) )
174	144, 173	sylibd 229	. . . . . . 7 |- ( ( ph /\ x e. CC ) -> ( ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( S X. { x } ) -> Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) )
175	174	reximdva 3017	. . . . . 6 |- ( ph -> ( E. x e. CC ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( S X. { x } ) -> E. x e. CC Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) )
176	175	adantr 481	. . . . 5 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> ( E. x e. CC ( Y oF x. ( u e. S |-> ( exp ` -u ( K x. u ) ) ) ) = ( S X. { x } ) -> E. x e. CC Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) )
177	133, 176	mpd 15	. . . 4 |- ( ( ph /\ ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) -> E. x e. CC Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) )
178	177	ex 450	. . 3 |- ( ph -> ( ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) -> E. x e. CC Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) )
179	1	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) ) -> S e. { RR , CC } )
180	4	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) ) -> K e. CC )
181		simprl 794	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) ) -> x e. CC )
182		eqid 2622	. . . . . . 7 |- ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) )
183	179, 180, 181, 182	expgrowthi 38532	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) ) -> ( S _D ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) = ( ( S X. { K } ) oF x. ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) )
184	183	3impb 1260	. . . . 5 |- ( ( ph /\ x e. CC /\ Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) -> ( S _D ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) = ( ( S X. { K } ) oF x. ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) )
185		oveq2 6658	. . . . . . 7 |- ( Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) -> ( S _D Y ) = ( S _D ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) )
186		oveq2 6658	. . . . . . 7 |- ( Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) -> ( ( S X. { K } ) oF x. Y ) = ( ( S X. { K } ) oF x. ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) )
187	185, 186	eqeq12d 2637	. . . . . 6 |- ( Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) -> ( ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) <-> ( S _D ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) = ( ( S X. { K } ) oF x. ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) ) )
188	187	3ad2ant3 1084	. . . . 5 |- ( ( ph /\ x e. CC /\ Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) -> ( ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) <-> ( S _D ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) = ( ( S X. { K } ) oF x. ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) ) )
189	184, 188	mpbird 247	. . . 4 |- ( ( ph /\ x e. CC /\ Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) -> ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) )
190	189	rexlimdv3a 3033	. . 3 |- ( ph -> ( E. x e. CC Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) -> ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) ) )
191	178, 190	impbid 202	. 2 |- ( ph -> ( ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) <-> E. x e. CC Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) ) )
192		oveq2 6658	. . . . . . . 8 |- ( u = t -> ( K x. u ) = ( K x. t ) )
193	192	fveq2d 6195	. . . . . . 7 |- ( u = t -> ( exp ` ( K x. u ) ) = ( exp ` ( K x. t ) ) )
194	193	oveq2d 6666	. . . . . 6 |- ( u = t -> ( x x. ( exp ` ( K x. u ) ) ) = ( x x. ( exp ` ( K x. t ) ) ) )
195	194	cbvmptv 4750	. . . . 5 |- ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) = ( t e. S |-> ( x x. ( exp ` ( K x. t ) ) ) )
196		oveq1 6657	. . . . . 6 |- ( x = c -> ( x x. ( exp ` ( K x. t ) ) ) = ( c x. ( exp ` ( K x. t ) ) ) )
197	196	mpteq2dv 4745	. . . . 5 |- ( x = c -> ( t e. S |-> ( x x. ( exp ` ( K x. t ) ) ) ) = ( t e. S |-> ( c x. ( exp ` ( K x. t ) ) ) ) )
198	195, 197	syl5eq 2668	. . . 4 |- ( x = c -> ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) = ( t e. S |-> ( c x. ( exp ` ( K x. t ) ) ) ) )
199	198	eqeq2d 2632	. . 3 |- ( x = c -> ( Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) <-> Y = ( t e. S |-> ( c x. ( exp ` ( K x. t ) ) ) ) ) )
200	199	cbvrexv 3172	. 2 |- ( E. x e. CC Y = ( u e. S |-> ( x x. ( exp ` ( K x. u ) ) ) ) <-> E. c e. CC Y = ( t e. S |-> ( c x. ( exp ` ( K x. t ) ) ) ) )
201	191, 200	syl6bb 276	1 |- ( ph -> ( ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) <-> E. c e. CC Y = ( t e. S |-> ( c x. ( exp ` ( K x. t ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   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-ioo 12179  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-cmp 21190  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: (None)