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Theorem expgrowth 36754
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 36752 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 36752 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 9650 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  { RR ,  CC }
32a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  CC  e.  { RR ,  CC } )
4 expgrowth.k . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  K  e.  CC )
5 recnprss 22938 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
61, 5syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  S  C_  CC )
76sseld 3417 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( u  e.  S  ->  u  e.  CC ) )
8 mulcl 9641 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( K  e.  CC  /\  u  e.  CC )  ->  ( K  x.  u
)  e.  CC )
94, 7, 8syl6an 554 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( u  e.  S  ->  ( K  x.  u
)  e.  CC ) )
109imp 436 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  u  e.  S )  ->  ( K  x.  u )  e.  CC )
1110negcld 9992 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  u  e.  S )  ->  -u ( K  x.  u )  e.  CC )
124negcld 9992 . . . . . . . . . . . . . . . . . 18  |-  ( ph  -> 
-u K  e.  CC )
1312adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  u  e.  S )  ->  -u K  e.  CC )
14 efcl 14214 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
1514adantl 473 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  CC )  ->  ( exp `  y )  e.  CC )
164adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  u  e.  S )  ->  K  e.  CC )
177imp 436 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  u  e.  S )  ->  u  e.  CC )
18 ax-1cn 9615 . . . . . . . . . . . . . . . . . . . . 21  |-  1  e.  CC
1918a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  u  e.  S )  ->  1  e.  CC )
201dvmptid 22990 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  u ) )  =  ( u  e.  S  |->  1 ) )
211, 17, 19, 20, 4dvmptcmul 22997 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  ( K  x.  u ) ) )  =  ( u  e.  S  |->  ( K  x.  1 ) ) )
224mulid1d 9678 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( K  x.  1 )  =  K )
2322mpteq2dv 4483 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( u  e.  S  |->  ( K  x.  1 ) )  =  ( u  e.  S  |->  K ) )
2421, 23eqtrd 2505 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  ( K  x.  u ) ) )  =  ( u  e.  S  |->  K ) )
251, 10, 16, 24dvmptneg 22999 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  -u ( K  x.  u
) ) )  =  ( u  e.  S  |-> 
-u K ) )
26 dvef 23011 . . . . . . . . . . . . . . . . . . 19  |-  ( CC 
_D  exp )  =  exp
27 eff 14213 . . . . . . . . . . . . . . . . . . . . . 22  |-  exp : CC
--> CC
28 ffn 5739 . . . . . . . . . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . . . . . . . . 20  |-  exp  =  ( y  e.  CC  |->  ( exp `  y ) )
3231oveq2i 6319 . . . . . . . . . . . . . . . . . . 19  |-  ( CC 
_D  exp )  =  ( CC  _D  ( y  e.  CC  |->  ( exp `  y ) ) )
3326, 32, 313eqtr3i 2501 . . . . . . . . . . . . . . . . . 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 23005 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  ( u  e.  S  |->  ( ( exp `  -u ( K  x.  u )
)  x.  -u K
) ) )
3736oveq2d 6324 . . . . . . . . . . . . . . 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 14214 . . . . . . . . . . . . . . . . . . . 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 9681 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  u  e.  S )  ->  (
( exp `  -u ( K  x.  u )
)  x.  -u K
)  e.  CC )
42 eqid 2471 . . . . . . . . . . . . . . . . . 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 6061 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( u  e.  S  |->  ( ( exp `  -u ( K  x.  u )
)  x.  -u K
) ) : S --> CC )
4436feq1d 5724 . . . . . . . . . . . . . . . . 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 240 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) ) : S --> CC )
46 mulcom 9643 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
4746adantl 473 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  =  ( y  x.  x ) )
481, 38, 45, 47caofcom 6582 . . . . . . . . . . . . . . 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 2507 . . . . . . . . . . . . . 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 6324 . . . . . . . . . . . . 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 5785 . . . . . . . . . . . . . . . . . 18  |-  ( -u K  e.  CC  ->  ( S  X.  { -u K } ) : S --> CC )
5212, 51syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( S  X.  { -u K } ) : S --> CC )
53 eqid 2471 . . . . . . . . . . . . . . . . . 18  |-  ( u  e.  S  |->  ( exp `  -u ( K  x.  u ) ) )  =  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) )
5440, 53fmptd 6061 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) : S --> CC )
551, 52, 54, 47caofcom 6582 . . . . . . . . . . . . . . . 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 2472 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) )  =  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )
57 fconstmpt 4883 . . . . . . . . . . . . . . . . . 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 6567 . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . . 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 6324 . . . . . . . . . . . . . 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 6324 . . . . . . . . . . . . 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 5042 . . . . . . . . . . . . . . 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 5718 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( u  e.  S  |->  ( ( exp `  -u ( K  x.  u ) )  x.  -u K ) )  =  S )
6664, 65eqtrd 2505 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( S  _D  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  S )
671, 38, 54, 63, 66dvmulf 22976 . . . . . . . . . . . . 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 2516 . . . . . . . . . . . 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 36744 . . . . . . . . . . . . . 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 1293 . . . . . . . . . . . . 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 6324 . . . . . . . . . . . 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 2505 . . . . . . . . . . 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 6315 . . . . . . . . . . . 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 6323 . . . . . . . . . . 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 2525 . . . . . . . . . 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 9645 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
7776adantl 473 . . . . . . . . . . . . . 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 6584 . . . . . . . . . . . . 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 6324 . . . . . . . . . . . 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 2481 . . . . . . . . . . 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 472 . . . . . . . . . 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 240 . . . . . . . . 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 9641 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
8483adantl 473 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
85 fconst6g 5785 . . . . . . . . . . . . . 14  |-  ( K  e.  CC  ->  ( S  X.  { K }
) : S --> CC )
864, 85syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S  X.  { K } ) : S --> CC )
87 inidm 3632 . . . . . . . . . . . . 13  |-  ( S  i^i  S )  =  S
8884, 86, 38, 1, 1, 87off 6565 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( S  X.  { K } )  oF  x.  Y ) : S --> CC )
8984, 52, 38, 1, 1, 87off 6565 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( S  X.  { -u K } )  oF  x.  Y
) : S --> CC )
90 adddir 9652 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z
) ) )
9190adantl 473 . . . . . . . . . . . 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 6587 . . . . . . . . . . 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 2481 . . . . . . . . . 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 472 . . . . . . . . 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 240 . . . . . . . 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 10629 . . . . . . . . . . . . 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 1292 . . . . . . . . . . . 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 10735 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  CC
9998fconst6 5786 . . . . . . . . . . . . . . . 16  |-  ( S  X.  { -u 1 } ) : S --> CC
10099a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( S  X.  { -u 1 } ) : S --> CC )
1011, 100, 86, 38, 77caofass 6584 . . . . . . . . . . . . . 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 6575 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( S  X.  { -u 1 } )  oF  x.  ( S  X.  { K }
) )  =  ( S  X.  { (
-u 1  x.  K
) } ) )
1044mulm1d 10091 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( -u 1  x.  K )  =  -u K )
105104sneqd 3971 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  { ( -u 1  x.  K ) }  =  { -u K } )
106105xpeq2d 4863 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( S  X.  {
( -u 1  x.  K
) } )  =  ( S  X.  { -u K } ) )
107103, 106eqtrd 2505 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( S  X.  { -u 1 } )  oF  x.  ( S  X.  { K }
) )  =  ( S  X.  { -u K } ) )
108107oveq1d 6323 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( S  X.  { -u 1 } )  oF  x.  ( S  X.  { K } ) )  oF  x.  Y
)  =  ( ( S  X.  { -u K } )  oF  x.  Y ) )
109101, 108eqtr3d 2507 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( S  X.  { -u 1 } )  oF  x.  (
( S  X.  { K } )  oF  x.  Y ) )  =  ( ( S  X.  { -u K } )  oF  x.  Y ) )
110109oveq2d 6324 . . . . . . . . . . . 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 36743 . . . . . . . . . . . . 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 673 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( S  X.  { K }
)  oF  x.  Y )  oF  -  ( ( S  X.  { K }
)  oF  x.  Y ) )  =  ( S  X.  {
0 } ) )
11397, 110, 1123eqtr3d 2513 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( S  X.  { K }
)  oF  x.  Y )  oF  +  ( ( S  X.  { -u K } )  oF  x.  Y ) )  =  ( S  X.  { 0 } ) )
114113oveq1d 6323 . . . . . . . . . 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 2481 . . . . . . . . 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 472 . . . . . . . 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 215 . . . . . . 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 9654 . . . . . . . . 9  |-  ( ph  ->  0  e.  CC )
119 mul02 9829 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
120119adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
1211, 54, 118, 118, 120caofid2 6581 . . . . . . . 8  |-  ( ph  ->  ( ( S  X.  { 0 } )  oF  x.  (
u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  ( S  X.  {
0 } ) )
122121adantr 472 . . . . . . 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 2505 . . . . . 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 472 . . . . . . 7  |-  ( (
ph  /\  ( S  _D  Y )  =  ( ( S  X.  { K } )  oF  x.  Y ) )  ->  S  e.  { RR ,  CC } )
12584, 38, 54, 1, 1, 87off 6565 . . . . . . . 8  |-  ( ph  ->  ( Y  oF  x.  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) ) : S --> CC )
126125adantr 472 . . . . . . 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 5042 . . . . . . . 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 9653 . . . . . . . . . 10  |-  0  e.  CC
129128fconst6 5786 . . . . . . . . 9  |-  ( S  X.  { 0 } ) : S --> CC
130129fdmi 5746 . . . . . . . 8  |-  dom  ( S  X.  { 0 } )  =  S
131127, 130syl6eq 2521 . . . . . . 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 36753 . . . . . 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 215 . . . . 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 6315 . . . . . . . . . 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 14228 . . . . . . . . . . . . . . 15  |-  ( -u ( K  x.  u
)  e.  CC  ->  ( exp `  -u ( K  x.  u )
)  =/=  0 )
136 eldifsn 4088 . . . . . . . . . . . . . . 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 677 . . . . . . . . . . . . . 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 6061 . . . . . . . . . . . 12  |-  ( ph  ->  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) : S --> ( CC  \  { 0 } ) )
140 ofdivcan4 36746 . . . . . . . . . . . 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 1292 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Y  oF  x.  ( u  e.  S  |->  ( exp `  -u ( K  x.  u ) ) ) )  oF  / 
( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  Y )
142141eqeq1d 2473 . . . . . . . . . 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 227 . . . . . . . . 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 472 . . . . . . . 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 3034 . . . . . . . . . . . . 13  |-  x  e. 
_V
146145a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  S )  ->  x  e.  _V )
147 ovex 6336 . . . . . . . . . . . . 13  |-  ( 1  /  ( exp `  ( K  x.  u )
) )  e.  _V
148147a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  S )  ->  (
1  /  ( exp `  ( K  x.  u
) ) )  e. 
_V )
149 fconstmpt 4883 . . . . . . . . . . . . 13  |-  ( S  X.  { x }
)  =  ( u  e.  S  |->  x )
150149a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( S  X.  {
x } )  =  ( u  e.  S  |->  x ) )
151 efneg 14229 . . . . . . . . . . . . . 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 4482 . . . . . . . . . . . 12  |-  ( ph  ->  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) )  =  ( u  e.  S  |->  ( 1  /  ( exp `  ( K  x.  u
) ) ) ) )
1541, 146, 148, 150, 153offval2 6567 . . . . . . . . . . 11  |-  ( ph  ->  ( ( S  X.  { x } )  oF  /  (
u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  ( u  e.  S  |->  ( x  /  (
1  /  ( exp `  ( K  x.  u
) ) ) ) ) )
155154adantr 472 . . . . . . . . . 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 14214 . . . . . . . . . . . . . . . . 17  |-  ( ( K  x.  u )  e.  CC  ->  ( exp `  ( K  x.  u ) )  e.  CC )
157 efne0 14228 . . . . . . . . . . . . . . . . 17  |-  ( ( K  x.  u )  e.  CC  ->  ( exp `  ( K  x.  u ) )  =/=  0 )
158156, 157jca 541 . . . . . . . . . . . . . . . 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 9626 . . . . . . . . . . . . . . . . 17  |-  1  =/=  0
16118, 160pm3.2i 462 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  CC  /\  1  =/=  0 )
162 divdiv2 10341 . . . . . . . . . . . . . . . 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 1378 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 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 9641 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CC  /\  ( exp `  ( K  x.  u ) )  e.  CC )  -> 
( x  x.  ( exp `  ( K  x.  u ) ) )  e.  CC )
167165, 166sylan2 482 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CC  /\  ( ph  /\  u  e.  S ) )  -> 
( x  x.  ( exp `  ( K  x.  u ) ) )  e.  CC )
168167div1d 10397 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  ( ph  /\  u  e.  S ) )  -> 
( ( x  x.  ( exp `  ( K  x.  u )
) )  /  1
)  =  ( x  x.  ( exp `  ( K  x.  u )
) ) )
169164, 168eqtrd 2505 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  ( ph  /\  u  e.  S ) )  -> 
( x  /  (
1  /  ( exp `  ( K  x.  u
) ) ) )  =  ( x  x.  ( exp `  ( K  x.  u )
) ) )
170169ancoms 460 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  u  e.  S )  /\  x  e.  CC )  ->  (
x  /  ( 1  /  ( exp `  ( K  x.  u )
) ) )  =  ( x  x.  ( exp `  ( K  x.  u ) ) ) )
171170an32s 821 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  CC )  /\  u  e.  S )  ->  (
x  /  ( 1  /  ( exp `  ( K  x.  u )
) ) )  =  ( x  x.  ( exp `  ( K  x.  u ) ) ) )
172171mpteq2dva 4482 . . . . . . . . . 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 2505 . . . . . . . . 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 2481 . . . . . . . 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 222 . . . . . . 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 2858 . . . . . 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 472 . . . . 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 441 . . 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 472 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u ) ) ) ) ) )  ->  S  e.  { RR ,  CC } )
1814adantr 472 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u ) ) ) ) ) )  ->  K  e.  CC )
182 simprl 772 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u ) ) ) ) ) )  ->  x  e.  CC )
183 eqid 2471 . . . . . . 7  |-  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u )
) ) )  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u ) ) ) )
184180, 181, 182, 183expgrowthi 36752 . . . . . 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 1227 . . . . 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 6316 . . . . . . 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 6316 . . . . . . 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 2486 . . . . . 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 1053 . . . . 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 240 . . . 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 2873 . . 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 195 . 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 6316 . . . . . . . 8  |-  ( u  =  t  ->  ( K  x.  u )  =  ( K  x.  t ) )
194193fveq2d 5883 . . . . . . 7  |-  ( u  =  t  ->  ( exp `  ( K  x.  u ) )  =  ( exp `  ( K  x.  t )
) )
195194oveq2d 6324 . . . . . 6  |-  ( u  =  t  ->  (
x  x.  ( exp `  ( K  x.  u
) ) )  =  ( x  x.  ( exp `  ( K  x.  t ) ) ) )
196195cbvmptv 4488 . . . . 5  |-  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u )
) ) )  =  ( t  e.  S  |->  ( x  x.  ( exp `  ( K  x.  t ) ) ) )
197 oveq1 6315 . . . . . 6  |-  ( x  =  c  ->  (
x  x.  ( exp `  ( K  x.  t
) ) )  =  ( c  x.  ( exp `  ( K  x.  t ) ) ) )
198197mpteq2dv 4483 . . . . 5  |-  ( x  =  c  ->  (
t  e.  S  |->  ( x  x.  ( exp `  ( K  x.  t
) ) ) )  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t )
) ) ) )
199196, 198syl5eq 2517 . . . 4  |-  ( x  =  c  ->  (
u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u
) ) ) )  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t )
) ) ) )
200199eqeq2d 2481 . . 3  |-  ( x  =  c  ->  ( Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u )
) ) )  <->  Y  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t ) ) ) ) ) )
201200cbvrexv 3006 . 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 269 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   _Vcvv 3031    \ cdif 3387    C_ wss 3390   {csn 3959   {cpr 3961    |-> cmpt 4454    X. cxp 4837   dom cdm 4839    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    oFcof 6548   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    - cmin 9880   -ucneg 9881    / cdiv 10291   expce 14191    _D cdv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901
This theorem is referenced by: (None)
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