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Theorem expgrowth 36548
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 36546 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 36546 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.
StepHypRef Expression
1 expgrowth.s . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  S  e.  { RR ,  CC } )
2 cnelprrecn 9634 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  { RR ,  CC }
32a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  CC  e.  { RR ,  CC } )
4 expgrowth.k . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  K  e.  CC )
5 recnprss 22851 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
61, 5syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  S  C_  CC )
76sseld 3464 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( u  e.  S  ->  u  e.  CC ) )
8 mulcl 9625 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( K  e.  CC  /\  u  e.  CC )  ->  ( K  x.  u
)  e.  CC )
94, 7, 8syl6an 548 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( u  e.  S  ->  ( K  x.  u
)  e.  CC ) )
109imp 431 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  u  e.  S )  ->  ( K  x.  u )  e.  CC )
1110negcld 9975 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  u  e.  S )  ->  -u ( K  x.  u )  e.  CC )
124negcld 9975 . . . . . . . . . . . . . . . . . 18  |-  ( ph  -> 
-u K  e.  CC )
1312adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  u  e.  S )  ->  -u K  e.  CC )
14 efcl 14130 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
1514adantl 468 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  CC )  ->  ( exp `  y )  e.  CC )
164adantr 467 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  u  e.  S )  ->  K  e.  CC )
177imp 431 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  u  e.  S )  ->  u  e.  CC )
18 ax-1cn 9599 . . . . . . . . . . . . . . . . . . . . 21  |-  1  e.  CC
1918a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  u  e.  S )  ->  1  e.  CC )
201dvmptid 22903 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  u ) )  =  ( u  e.  S  |->  1 ) )
211, 17, 19, 20, 4dvmptcmul 22910 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  ( K  x.  u ) ) )  =  ( u  e.  S  |->  ( K  x.  1 ) ) )
224mulid1d 9662 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( K  x.  1 )  =  K )
2322mpteq2dv 4509 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( u  e.  S  |->  ( K  x.  1 ) )  =  ( u  e.  S  |->  K ) )
2421, 23eqtrd 2464 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  ( K  x.  u ) ) )  =  ( u  e.  S  |->  K ) )
251, 10, 16, 24dvmptneg 22912 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  -u ( K  x.  u
) ) )  =  ( u  e.  S  |-> 
-u K ) )
26 dvef 22924 . . . . . . . . . . . . . . . . . . 19  |-  ( CC 
_D  exp )  =  exp
27 eff 14129 . . . . . . . . . . . . . . . . . . . . . 22  |-  exp : CC
--> CC
28 ffn 5744 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( exp
: CC --> CC  ->  exp 
Fn  CC )
2927, 28ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21  |-  exp  Fn  CC
30 dffn5 5924 . . . . . . . . . . . . . . . . . . . . 21  |-  ( exp 
Fn  CC  <->  exp  =  ( y  e.  CC  |->  ( exp `  y ) ) )
3129, 30mpbi 212 . . . . . . . . . . . . . . . . . . . 20  |-  exp  =  ( y  e.  CC  |->  ( exp `  y ) )
3231oveq2i 6314 . . . . . . . . . . . . . . . . . . 19  |-  ( CC 
_D  exp )  =  ( CC  _D  ( y  e.  CC  |->  ( exp `  y ) ) )
3326, 32, 313eqtr3i 2460 . . . . . . . . . . . . . . . . . 18  |-  ( CC 
_D  ( y  e.  CC  |->  ( exp `  y
) ) )  =  ( y  e.  CC  |->  ( exp `  y ) )
3433a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( CC  _D  (
y  e.  CC  |->  ( exp `  y ) ) )  =  ( y  e.  CC  |->  ( exp `  y ) ) )
35 fveq2 5879 . . . . . . . . . . . . . . . . 17  |-  ( y  =  -u ( K  x.  u )  ->  ( exp `  y )  =  ( exp `  -u ( K  x.  u )
) )
361, 3, 11, 13, 15, 15, 25, 34, 35, 35dvmptco 22918 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  ( u  e.  S  |->  ( ( exp `  -u ( K  x.  u )
)  x.  -u K
) ) )
3736oveq2d 6319 . . . . . . . . . . . . . . 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 14130 . . . . . . . . . . . . . . . . . . . 20  |-  ( -u ( K  x.  u
)  e.  CC  ->  ( exp `  -u ( K  x.  u )
)  e.  CC )
4011, 39syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  u  e.  S )  ->  ( exp `  -u ( K  x.  u ) )  e.  CC )
4140, 13mulcld 9665 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  u  e.  S )  ->  (
( exp `  -u ( K  x.  u )
)  x.  -u K
)  e.  CC )
42 eqid 2423 . . . . . . . . . . . . . . . . . 18  |-  ( u  e.  S  |->  ( ( exp `  -u ( K  x.  u )
)  x.  -u K
) )  =  ( u  e.  S  |->  ( ( exp `  -u ( K  x.  u )
)  x.  -u K
) )
4341, 42fmptd 6059 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( u  e.  S  |->  ( ( exp `  -u ( K  x.  u )
)  x.  -u K
) ) : S --> CC )
4436feq1d 5730 . . . . . . . . . . . . . . . . 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 ) )
4543, 44mpbird 236 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) ) : S --> CC )
46 mulcom 9627 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
4746adantl 468 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  =  ( y  x.  x ) )
481, 38, 45, 47caofcom 6575 . . . . . . . . . . . . . . 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 ) )
4937, 48eqtr3d 2466 . . . . . . . . . . . . . 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 ) )
5049oveq2d 6319 . . . . . . . . . . . . 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 5787 . . . . . . . . . . . . . . . . . 18  |-  ( -u K  e.  CC  ->  ( S  X.  { -u K } ) : S --> CC )
5212, 51syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( S  X.  { -u K } ) : S --> CC )
53 eqid 2423 . . . . . . . . . . . . . . . . . 18  |-  ( u  e.  S  |->  ( exp `  -u ( K  x.  u ) ) )  =  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) )
5440, 53fmptd 6059 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) : S --> CC )
551, 52, 54, 47caofcom 6575 . . . . . . . . . . . . . . . 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 2424 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) )  =  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )
57 fconstmpt 4895 . . . . . . . . . . . . . . . . . 18  |-  ( S  X.  { -u K } )  =  ( u  e.  S  |->  -u K )
5857a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( S  X.  { -u K } )  =  ( u  e.  S  |-> 
-u K ) )
591, 40, 13, 56, 58offval2 6560 . . . . . . . . . . . . . . . 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
) ) )
6055, 59eqtrd 2464 . . . . . . . . . . . . . . 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
) ) )
6160oveq2d 6319 . . . . . . . . . . . . . 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
) ) ) )
6261oveq2d 6319 . . . . . . . . . . . . 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 )
6436dmeqd 5054 . . . . . . . . . . . . . . 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 ) ) )
6542, 41dmmptd 5724 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( u  e.  S  |->  ( ( exp `  -u ( K  x.  u ) )  x.  -u K ) )  =  S )
6664, 65eqtrd 2464 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( S  _D  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  S )
671, 38, 54, 63, 66dvmulf 22889 . . . . . . . . . . . . 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 ) ) )
6850, 62, 673eqtr4rd 2475 . . . . . . . . . . . 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 36538 . . . . . . . . . . . . . 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 )
) ) ) ) )
701, 38, 52, 54, 69syl22anc 1266 . . . . . . . . . . . . 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 ) ) ) ) ) )
7170oveq2d 6319 . . . . . . . . . . . 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 ) ) ) ) ) ) )
7268, 71eqtrd 2464 . . . . . . . . . . 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 6310 . . . . . . . . . . . 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 )
) ) ) )
7473oveq1d 6318 . . . . . . . . . . 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 )
) ) ) ) ) )
7572, 74sylan9eq 2484 . . . . . . . . . 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 9629 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
7776adantl 468 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  x.  y )  x.  z
)  =  ( x  x.  ( y  x.  z ) ) )
781, 52, 38, 54, 77caofass 6577 . . . . . . . . . . . . 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 ) ) ) ) ) )
7978oveq2d 6319 . . . . . . . . . . . 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 ) ) ) ) ) ) )
8079eqeq2d 2437 . . . . . . . . . . 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 )
) ) ) ) ) ) )
8180adantr 467 . . . . . . . . . 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 )
) ) ) ) ) ) )
8275, 81mpbird 236 . . . . . . . . 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 9625 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
8483adantl 468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
85 fconst6g 5787 . . . . . . . . . . . . . 14  |-  ( K  e.  CC  ->  ( S  X.  { K }
) : S --> CC )
864, 85syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S  X.  { K } ) : S --> CC )
87 inidm 3672 . . . . . . . . . . . . 13  |-  ( S  i^i  S )  =  S
8884, 86, 38, 1, 1, 87off 6558 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( S  X.  { K } )  oF  x.  Y ) : S --> CC )
8984, 52, 38, 1, 1, 87off 6558 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( S  X.  { -u K } )  oF  x.  Y
) : S --> CC )
90 adddir 9636 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z
) ) )
9190adantl 468 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  +  y )  x.  z
)  =  ( ( x  x.  z )  +  ( y  x.  z ) ) )
921, 54, 88, 89, 91caofdir 6580 . . . . . . . . . . 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 ) ) ) ) ) )
9392eqeq2d 2437 . . . . . . . . . 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 ) ) ) ) ) ) )
9493adantr 467 . . . . . . . . 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 ) ) ) ) ) ) )
9582, 94mpbird 236 . . . . . . . 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 10609 . . . . . . . . . . . . 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 ) ) )
971, 88, 88, 96syl3anc 1265 . . . . . . . . . . . 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 10715 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  CC
9998fconst6 5788 . . . . . . . . . . . . . . . 16  |-  ( S  X.  { -u 1 } ) : S --> CC
10099a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( S  X.  { -u 1 } ) : S --> CC )
1011, 100, 86, 38, 77caofass 6577 . . . . . . . . . . . . . 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 ) ) )
10298a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  -> 
-u 1  e.  CC )
1031, 102, 4ofc12 6568 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( S  X.  { -u 1 } )  oF  x.  ( S  X.  { K }
) )  =  ( S  X.  { (
-u 1  x.  K
) } ) )
1044mulm1d 10072 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( -u 1  x.  K )  =  -u K )
105104sneqd 4009 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  { ( -u 1  x.  K ) }  =  { -u K } )
106105xpeq2d 4875 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( S  X.  {
( -u 1  x.  K
) } )  =  ( S  X.  { -u K } ) )
107103, 106eqtrd 2464 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( S  X.  { -u 1 } )  oF  x.  ( S  X.  { K }
) )  =  ( S  X.  { -u K } ) )
108107oveq1d 6318 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( S  X.  { -u 1 } )  oF  x.  ( S  X.  { K } ) )  oF  x.  Y
)  =  ( ( S  X.  { -u K } )  oF  x.  Y ) )
109101, 108eqtr3d 2466 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( S  X.  { -u 1 } )  oF  x.  (
( S  X.  { K } )  oF  x.  Y ) )  =  ( ( S  X.  { -u K } )  oF  x.  Y ) )
110109oveq2d 6319 . . . . . . . . . . . 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 36537 . . . . . . . . . . . . 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 } ) )
1121, 88, 111syl2anc 666 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( S  X.  { K }
)  oF  x.  Y )  oF  -  ( ( S  X.  { K }
)  oF  x.  Y ) )  =  ( S  X.  {
0 } ) )
11397, 110, 1123eqtr3d 2472 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( S  X.  { K }
)  oF  x.  Y )  oF  +  ( ( S  X.  { -u K } )  oF  x.  Y ) )  =  ( S  X.  { 0 } ) )
114113oveq1d 6318 . . . . . . . . . 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 )
) ) ) )
115114eqeq2d 2437 . . . . . . . . 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 )
) ) ) ) )
116115adantr 467 . . . . . . . 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 )
) ) ) ) )
11795, 116mpbid 214 . . . . . . 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 9638 . . . . . . . . 9  |-  ( ph  ->  0  e.  CC )
119 mul02 9813 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
120119adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
1211, 54, 118, 118, 120caofid2 6574 . . . . . . . 8  |-  ( ph  ->  ( ( S  X.  { 0 } )  oF  x.  (
u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  ( S  X.  {
0 } ) )
122121adantr 467 . . . . . . 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 } ) )
123117, 122eqtrd 2464 . . . . . 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 } ) )
1241adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( S  _D  Y )  =  ( ( S  X.  { K } )  oF  x.  Y ) )  ->  S  e.  { RR ,  CC } )
12584, 38, 54, 1, 1, 87off 6558 . . . . . . . 8  |-  ( ph  ->  ( Y  oF  x.  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) ) : S --> CC )
126125adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( S  _D  Y )  =  ( ( S  X.  { K } )  oF  x.  Y ) )  ->  ( Y  oF  x.  ( u  e.  S  |->  ( exp `  -u ( K  x.  u ) ) ) ) : S --> CC )
127123dmeqd 5054 . . . . . . . 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 9637 . . . . . . . . . 10  |-  0  e.  CC
129128fconst6 5788 . . . . . . . . 9  |-  ( S  X.  { 0 } ) : S --> CC
130129fdmi 5749 . . . . . . . 8  |-  dom  ( S  X.  { 0 } )  =  S
131127, 130syl6eq 2480 . . . . . . 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 )
132124, 126, 131dvconstbi 36547 . . . . . 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 }
) ) )
133123, 132mpbid 214 . . . . 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 6310 . . . . . . . . . 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 14144 . . . . . . . . . . . . . . 15  |-  ( -u ( K  x.  u
)  e.  CC  ->  ( exp `  -u ( K  x.  u )
)  =/=  0 )
136 eldifsn 4123 . . . . . . . . . . . . . . 15  |-  ( ( exp `  -u ( K  x.  u )
)  e.  ( CC 
\  { 0 } )  <->  ( ( exp `  -u ( K  x.  u ) )  e.  CC  /\  ( exp `  -u ( K  x.  u ) )  =/=  0 ) )
13739, 135, 136sylanbrc 669 . . . . . . . . . . . . . 14  |-  ( -u ( K  x.  u
)  e.  CC  ->  ( exp `  -u ( K  x.  u )
)  e.  ( CC 
\  { 0 } ) )
13811, 137syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  u  e.  S )  ->  ( exp `  -u ( K  x.  u ) )  e.  ( CC  \  {
0 } ) )
139138, 53fmptd 6059 . . . . . . . . . . . 12  |-  ( ph  ->  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) : S --> ( CC  \  { 0 } ) )
140 ofdivcan4 36540 . . . . . . . . . . . 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 )
1411, 38, 139, 140syl3anc 1265 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Y  oF  x.  ( u  e.  S  |->  ( exp `  -u ( K  x.  u ) ) ) )  oF  / 
( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  Y )
142141eqeq1d 2425 . . . . . . . . . 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 )
) ) ) ) )
143134, 142syl5ib 223 . . . . . . . . 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 )
) ) ) ) )
144143adantr 467 . . . . . . . 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 3085 . . . . . . . . . . . . 13  |-  x  e. 
_V
146145a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  S )  ->  x  e.  _V )
147 ovex 6331 . . . . . . . . . . . . 13  |-  ( 1  /  ( exp `  ( K  x.  u )
) )  e.  _V
148147a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  S )  ->  (
1  /  ( exp `  ( K  x.  u
) ) )  e. 
_V )
149 fconstmpt 4895 . . . . . . . . . . . . 13  |-  ( S  X.  { x }
)  =  ( u  e.  S  |->  x )
150149a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( S  X.  {
x } )  =  ( u  e.  S  |->  x ) )
151 efneg 14145 . . . . . . . . . . . . . 14  |-  ( ( K  x.  u )  e.  CC  ->  ( exp `  -u ( K  x.  u ) )  =  ( 1  /  ( exp `  ( K  x.  u ) ) ) )
15210, 151syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  u  e.  S )  ->  ( exp `  -u ( K  x.  u ) )  =  ( 1  /  ( exp `  ( K  x.  u ) ) ) )
153152mpteq2dva 4508 . . . . . . . . . . . 12  |-  ( ph  ->  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) )  =  ( u  e.  S  |->  ( 1  /  ( exp `  ( K  x.  u
) ) ) ) )
1541, 146, 148, 150, 153offval2 6560 . . . . . . . . . . 11  |-  ( ph  ->  ( ( S  X.  { x } )  oF  /  (
u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  ( u  e.  S  |->  ( x  /  (
1  /  ( exp `  ( K  x.  u
) ) ) ) ) )
155154adantr 467 . . . . . . . . . 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
) ) ) ) ) )
156 efcl 14130 . . . . . . . . . . . . . . . . 17  |-  ( ( K  x.  u )  e.  CC  ->  ( exp `  ( K  x.  u ) )  e.  CC )
157 efne0 14144 . . . . . . . . . . . . . . . . 17  |-  ( ( K  x.  u )  e.  CC  ->  ( exp `  ( K  x.  u ) )  =/=  0 )
158156, 157jca 535 . . . . . . . . . . . . . . . 16  |-  ( ( K  x.  u )  e.  CC  ->  (
( exp `  ( K  x.  u )
)  e.  CC  /\  ( exp `  ( K  x.  u ) )  =/=  0 ) )
15910, 158syl 17 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  u  e.  S )  ->  (
( exp `  ( K  x.  u )
)  e.  CC  /\  ( exp `  ( K  x.  u ) )  =/=  0 ) )
160 ax-1ne0 9610 . . . . . . . . . . . . . . . . 17  |-  1  =/=  0
16118, 160pm3.2i 457 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  CC  /\  1  =/=  0 )
162 divdiv2 10321 . . . . . . . . . . . . . . . 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
) )
163161, 162mp3an2 1349 . . . . . . . . . . . . . . 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 ) )
164159, 163sylan2 477 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  ( ph  /\  u  e.  S ) )  -> 
( x  /  (
1  /  ( exp `  ( K  x.  u
) ) ) )  =  ( ( x  x.  ( exp `  ( K  x.  u )
) )  /  1
) )
16510, 156syl 17 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  u  e.  S )  ->  ( exp `  ( K  x.  u ) )  e.  CC )
166 mulcl 9625 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CC  /\  ( exp `  ( K  x.  u ) )  e.  CC )  -> 
( x  x.  ( exp `  ( K  x.  u ) ) )  e.  CC )
167165, 166sylan2 477 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CC  /\  ( ph  /\  u  e.  S ) )  -> 
( x  x.  ( exp `  ( K  x.  u ) ) )  e.  CC )
168167div1d 10377 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  ( ph  /\  u  e.  S ) )  -> 
( ( x  x.  ( exp `  ( K  x.  u )
) )  /  1
)  =  ( x  x.  ( exp `  ( K  x.  u )
) ) )
169164, 168eqtrd 2464 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  ( ph  /\  u  e.  S ) )  -> 
( x  /  (
1  /  ( exp `  ( K  x.  u
) ) ) )  =  ( x  x.  ( exp `  ( K  x.  u )
) ) )
170169ancoms 455 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  u  e.  S )  /\  x  e.  CC )  ->  (
x  /  ( 1  /  ( exp `  ( K  x.  u )
) ) )  =  ( x  x.  ( exp `  ( K  x.  u ) ) ) )
171170an32s 812 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  CC )  /\  u  e.  S )  ->  (
x  /  ( 1  /  ( exp `  ( K  x.  u )
) ) )  =  ( x  x.  ( exp `  ( K  x.  u ) ) ) )
172171mpteq2dva 4508 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  ( u  e.  S  |->  ( x  /  ( 1  / 
( exp `  ( K  x.  u )
) ) ) )  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u )
) ) ) )
173155, 172eqtrd 2464 . . . . . . . . 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 ) ) ) ) )
174173eqeq2d 2437 . . . . . . . 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 ) ) ) ) ) )
175144, 174sylibd 218 . . . . . . 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 )
) ) ) ) )
176175reximdva 2901 . . . . . 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
) ) ) ) ) )
177176adantr 467 . . . . 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
) ) ) ) ) )
178133, 177mpd 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 )
) ) ) )
179178ex 436 . . 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 )
) ) ) ) )
1801adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u ) ) ) ) ) )  ->  S  e.  { RR ,  CC } )
1814adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u ) ) ) ) ) )  ->  K  e.  CC )
182 simprl 763 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u ) ) ) ) ) )  ->  x  e.  CC )
183 eqid 2423 . . . . . . 7  |-  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u )
) ) )  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u ) ) ) )
184180, 181, 182, 183expgrowthi 36546 . . . . . 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 )
) ) ) ) )
1851843impb 1202 . . . . 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 )
) ) ) ) )
186 oveq2 6311 . . . . . . 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 )
) ) ) ) )
187 oveq2 6311 . . . . . . 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 ) ) ) ) ) )
188186, 187eqeq12d 2445 . . . . . 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 )
) ) ) ) ) )
1891883ad2ant3 1029 . . . . 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 )
) ) ) ) ) )
190185, 189mpbird 236 . . . 4  |-  ( (
ph  /\  x  e.  CC  /\  Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u
) ) ) ) )  ->  ( S  _D  Y )  =  ( ( S  X.  { K } )  oF  x.  Y ) )
191190rexlimdv3a 2920 . . 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 ) ) )
192179, 191impbid 194 . 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
) ) ) ) ) )
193 oveq2 6311 . . . . . . . 8  |-  ( u  =  t  ->  ( K  x.  u )  =  ( K  x.  t ) )
194193fveq2d 5883 . . . . . . 7  |-  ( u  =  t  ->  ( exp `  ( K  x.  u ) )  =  ( exp `  ( K  x.  t )
) )
195194oveq2d 6319 . . . . . 6  |-  ( u  =  t  ->  (
x  x.  ( exp `  ( K  x.  u
) ) )  =  ( x  x.  ( exp `  ( K  x.  t ) ) ) )
196195cbvmptv 4514 . . . . 5  |-  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u )
) ) )  =  ( t  e.  S  |->  ( x  x.  ( exp `  ( K  x.  t ) ) ) )
197 oveq1 6310 . . . . . 6  |-  ( x  =  c  ->  (
x  x.  ( exp `  ( K  x.  t
) ) )  =  ( c  x.  ( exp `  ( K  x.  t ) ) ) )
198197mpteq2dv 4509 . . . . 5  |-  ( x  =  c  ->  (
t  e.  S  |->  ( x  x.  ( exp `  ( K  x.  t
) ) ) )  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t )
) ) ) )
199196, 198syl5eq 2476 . . . 4  |-  ( x  =  c  ->  (
u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u
) ) ) )  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t )
) ) ) )
200199eqeq2d 2437 . . 3  |-  ( x  =  c  ->  ( Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u )
) ) )  <->  Y  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t ) ) ) ) ) )
201200cbvrexv 3057 . 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
) ) ) ) )
202192, 201syl6bb 265 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
) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   E.wrex 2777   _Vcvv 3082    \ cdif 3434    C_ wss 3437   {csn 3997   {cpr 3999    |-> cmpt 4480    X. cxp 4849   dom cdm 4851    Fn wfn 5594   -->wf 5595   ` cfv 5599  (class class class)co 6303    oFcof 6541   CCcc 9539   RRcr 9540   0cc0 9541   1c1 9542    + caddc 9544    x. cmul 9546    - cmin 9862   -ucneg 9863    / cdiv 10271   expce 14107    _D cdv 22810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620  ax-mulf 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ioo 11641  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-fl 12029  df-seq 12215  df-exp 12274  df-fac 12461  df-bc 12489  df-hash 12517  df-shft 13124  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cld 20026  df-ntr 20027  df-cls 20028  df-nei 20106  df-lp 20144  df-perf 20145  df-cn 20235  df-cnp 20236  df-haus 20323  df-cmp 20394  df-tx 20569  df-hmeo 20762  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-xms 21327  df-ms 21328  df-tms 21329  df-cncf 21902  df-limc 22813  df-dv 22814
This theorem is referenced by: (None)
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