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Theorem expgrowth 36052
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 36050 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 36050 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 9533 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  { RR ,  CC }
32a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  CC  e.  { RR ,  CC } )
4 expgrowth.k . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  K  e.  CC )
5 recnprss 22490 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
61, 5syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  S  C_  CC )
76sseld 3438 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( u  e.  S  ->  u  e.  CC ) )
8 mulcl 9524 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( K  e.  CC  /\  u  e.  CC )  ->  ( K  x.  u
)  e.  CC )
94, 7, 8syl6an 543 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( u  e.  S  ->  ( K  x.  u
)  e.  CC ) )
109imp 427 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  u  e.  S )  ->  ( K  x.  u )  e.  CC )
1110negcld 9872 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  u  e.  S )  ->  -u ( K  x.  u )  e.  CC )
124negcld 9872 . . . . . . . . . . . . . . . . . 18  |-  ( ph  -> 
-u K  e.  CC )
1312adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  u  e.  S )  ->  -u K  e.  CC )
14 efcl 13917 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
1514adantl 464 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  CC )  ->  ( exp `  y )  e.  CC )
164adantr 463 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  u  e.  S )  ->  K  e.  CC )
177imp 427 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  u  e.  S )  ->  u  e.  CC )
18 ax-1cn 9498 . . . . . . . . . . . . . . . . . . . . 21  |-  1  e.  CC
1918a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  u  e.  S )  ->  1  e.  CC )
201dvmptid 22542 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  u ) )  =  ( u  e.  S  |->  1 ) )
211, 17, 19, 20, 4dvmptcmul 22549 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  ( K  x.  u ) ) )  =  ( u  e.  S  |->  ( K  x.  1 ) ) )
224mulid1d 9561 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( K  x.  1 )  =  K )
2322mpteq2dv 4479 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( u  e.  S  |->  ( K  x.  1 ) )  =  ( u  e.  S  |->  K ) )
2421, 23eqtrd 2441 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  ( K  x.  u ) ) )  =  ( u  e.  S  |->  K ) )
251, 10, 16, 24dvmptneg 22551 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  -u ( K  x.  u
) ) )  =  ( u  e.  S  |-> 
-u K ) )
26 dvef 22563 . . . . . . . . . . . . . . . . . . 19  |-  ( CC 
_D  exp )  =  exp
27 eff 13916 . . . . . . . . . . . . . . . . . . . . . 22  |-  exp : CC
--> CC
28 ffn 5668 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( exp
: CC --> CC  ->  exp 
Fn  CC )
2927, 28ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21  |-  exp  Fn  CC
30 dffn5 5848 . . . . . . . . . . . . . . . . . . . . 21  |-  ( exp 
Fn  CC  <->  exp  =  ( y  e.  CC  |->  ( exp `  y ) ) )
3129, 30mpbi 208 . . . . . . . . . . . . . . . . . . . 20  |-  exp  =  ( y  e.  CC  |->  ( exp `  y ) )
3231oveq2i 6243 . . . . . . . . . . . . . . . . . . 19  |-  ( CC 
_D  exp )  =  ( CC  _D  ( y  e.  CC  |->  ( exp `  y ) ) )
3326, 32, 313eqtr3i 2437 . . . . . . . . . . . . . . . . . 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 5803 . . . . . . . . . . . . . . . . 17  |-  ( y  =  -u ( K  x.  u )  ->  ( exp `  y )  =  ( exp `  -u ( K  x.  u )
) )
361, 3, 11, 13, 15, 15, 25, 34, 35, 35dvmptco 22557 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  ( u  e.  S  |->  ( ( exp `  -u ( K  x.  u )
)  x.  -u K
) ) )
3736oveq2d 6248 . . . . . . . . . . . . . . 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 13917 . . . . . . . . . . . . . . . . . . . 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 9564 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  u  e.  S )  ->  (
( exp `  -u ( K  x.  u )
)  x.  -u K
)  e.  CC )
42 eqid 2400 . . . . . . . . . . . . . . . . . 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 5987 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( u  e.  S  |->  ( ( exp `  -u ( K  x.  u )
)  x.  -u K
) ) : S --> CC )
4436feq1d 5654 . . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( S  _D  (
u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) ) : S --> CC )
46 mulcom 9526 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
4746adantl 464 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  =  ( y  x.  x ) )
481, 38, 45, 47caofcom 6508 . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . 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 6248 . . . . . . . . . . . . 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 5711 . . . . . . . . . . . . . . . . . 18  |-  ( -u K  e.  CC  ->  ( S  X.  { -u K } ) : S --> CC )
5212, 51syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( S  X.  { -u K } ) : S --> CC )
53 eqid 2400 . . . . . . . . . . . . . . . . . 18  |-  ( u  e.  S  |->  ( exp `  -u ( K  x.  u ) ) )  =  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) )
5440, 53fmptd 5987 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) : S --> CC )
551, 52, 54, 47caofcom 6508 . . . . . . . . . . . . . . . 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 2401 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) )  =  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )
57 fconstmpt 4984 . . . . . . . . . . . . . . . . . 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 6492 . . . . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . 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 6248 . . . . . . . . . . . . . 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 6248 . . . . . . . . . . . . 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 5145 . . . . . . . . . . . . . . 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 5648 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( u  e.  S  |->  ( ( exp `  -u ( K  x.  u ) )  x.  -u K ) )  =  S )
6664, 65eqtrd 2441 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( S  _D  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  S )
671, 38, 54, 63, 66dvmulf 22528 . . . . . . . . . . . . 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 2452 . . . . . . . . . . . 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 36042 . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . 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 6248 . . . . . . . . . . . 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 2441 . . . . . . . . . . 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 6239 . . . . . . . . . . . 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 6247 . . . . . . . . . . 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 2461 . . . . . . . . . 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 9528 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
7776adantl 464 . . . . . . . . . . . . . 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 6510 . . . . . . . . . . . . 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 6248 . . . . . . . . . . . 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 2414 . . . . . . . . . . 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 463 . . . . . . . . . 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 232 . . . . . . . . 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 9524 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
8483adantl 464 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
85 fconst6g 5711 . . . . . . . . . . . . . 14  |-  ( K  e.  CC  ->  ( S  X.  { K }
) : S --> CC )
864, 85syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S  X.  { K } ) : S --> CC )
87 inidm 3645 . . . . . . . . . . . . 13  |-  ( S  i^i  S )  =  S
8884, 86, 38, 1, 1, 87off 6490 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( S  X.  { K } )  oF  x.  Y ) : S --> CC )
8984, 52, 38, 1, 1, 87off 6490 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( S  X.  { -u K } )  oF  x.  Y
) : S --> CC )
90 adddir 9535 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z
) ) )
9190adantl 464 . . . . . . . . . . . 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 6513 . . . . . . . . . . 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 2414 . . . . . . . . . 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 463 . . . . . . . . 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 232 . . . . . . . 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 10492 . . . . . . . . . . . . 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 1228 . . . . . . . . . . . 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 10598 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  CC
9998fconst6 5712 . . . . . . . . . . . . . . . 16  |-  ( S  X.  { -u 1 } ) : S --> CC
10099a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( S  X.  { -u 1 } ) : S --> CC )
1011, 100, 86, 38, 77caofass 6510 . . . . . . . . . . . . . 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 6501 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( S  X.  { -u 1 } )  oF  x.  ( S  X.  { K }
) )  =  ( S  X.  { (
-u 1  x.  K
) } ) )
1044mulm1d 9967 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( -u 1  x.  K )  =  -u K )
105104sneqd 3981 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  { ( -u 1  x.  K ) }  =  { -u K } )
106105xpeq2d 4964 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( S  X.  {
( -u 1  x.  K
) } )  =  ( S  X.  { -u K } ) )
107103, 106eqtrd 2441 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( S  X.  { -u 1 } )  oF  x.  ( S  X.  { K }
) )  =  ( S  X.  { -u K } ) )
108107oveq1d 6247 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( S  X.  { -u 1 } )  oF  x.  ( S  X.  { K } ) )  oF  x.  Y
)  =  ( ( S  X.  { -u K } )  oF  x.  Y ) )
109101, 108eqtr3d 2443 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( S  X.  { -u 1 } )  oF  x.  (
( S  X.  { K } )  oF  x.  Y ) )  =  ( ( S  X.  { -u K } )  oF  x.  Y ) )
110109oveq2d 6248 . . . . . . . . . . . 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 36041 . . . . . . . . . . . . 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 659 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( S  X.  { K }
)  oF  x.  Y )  oF  -  ( ( S  X.  { K }
)  oF  x.  Y ) )  =  ( S  X.  {
0 } ) )
11397, 110, 1123eqtr3d 2449 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( S  X.  { K }
)  oF  x.  Y )  oF  +  ( ( S  X.  { -u K } )  oF  x.  Y ) )  =  ( S  X.  { 0 } ) )
114113oveq1d 6247 . . . . . . . . . 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 2414 . . . . . . . . 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 463 . . . . . . . 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 210 . . . . . . 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 9537 . . . . . . . . 9  |-  ( ph  ->  0  e.  CC )
119 mul02 9710 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
120119adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
1211, 54, 118, 118, 120caofid2 6507 . . . . . . . 8  |-  ( ph  ->  ( ( S  X.  { 0 } )  oF  x.  (
u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  ( S  X.  {
0 } ) )
122121adantr 463 . . . . . . 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 2441 . . . . . 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 463 . . . . . . 7  |-  ( (
ph  /\  ( S  _D  Y )  =  ( ( S  X.  { K } )  oF  x.  Y ) )  ->  S  e.  { RR ,  CC } )
12584, 38, 54, 1, 1, 87off 6490 . . . . . . . 8  |-  ( ph  ->  ( Y  oF  x.  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) ) : S --> CC )
126125adantr 463 . . . . . . 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 5145 . . . . . . . 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 9536 . . . . . . . . . 10  |-  0  e.  CC
129128fconst6 5712 . . . . . . . . 9  |-  ( S  X.  { 0 } ) : S --> CC
130129fdmi 5673 . . . . . . . 8  |-  dom  ( S  X.  { 0 } )  =  S
131127, 130syl6eq 2457 . . . . . . 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 36051 . . . . . 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 210 . . . . 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 6239 . . . . . . . . . 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 13931 . . . . . . . . . . . . . . 15  |-  ( -u ( K  x.  u
)  e.  CC  ->  ( exp `  -u ( K  x.  u )
)  =/=  0 )
136 eldifsn 4094 . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . 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 5987 . . . . . . . . . . . 12  |-  ( ph  ->  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) : S --> ( CC  \  { 0 } ) )
140 ofdivcan4 36044 . . . . . . . . . . . 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 1228 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Y  oF  x.  ( u  e.  S  |->  ( exp `  -u ( K  x.  u ) ) ) )  oF  / 
( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  Y )
142141eqeq1d 2402 . . . . . . . . . 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 219 . . . . . . . . 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 463 . . . . . . . 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 3059 . . . . . . . . . . . . 13  |-  x  e. 
_V
146145a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  S )  ->  x  e.  _V )
147 ovex 6260 . . . . . . . . . . . . 13  |-  ( 1  /  ( exp `  ( K  x.  u )
) )  e.  _V
148147a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  S )  ->  (
1  /  ( exp `  ( K  x.  u
) ) )  e. 
_V )
149 fconstmpt 4984 . . . . . . . . . . . . 13  |-  ( S  X.  { x }
)  =  ( u  e.  S  |->  x )
150149a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( S  X.  {
x } )  =  ( u  e.  S  |->  x ) )
151 efneg 13932 . . . . . . . . . . . . . 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 4478 . . . . . . . . . . . 12  |-  ( ph  ->  ( u  e.  S  |->  ( exp `  -u ( K  x.  u )
) )  =  ( u  e.  S  |->  ( 1  /  ( exp `  ( K  x.  u
) ) ) ) )
1541, 146, 148, 150, 153offval2 6492 . . . . . . . . . . 11  |-  ( ph  ->  ( ( S  X.  { x } )  oF  /  (
u  e.  S  |->  ( exp `  -u ( K  x.  u )
) ) )  =  ( u  e.  S  |->  ( x  /  (
1  /  ( exp `  ( K  x.  u
) ) ) ) ) )
155154adantr 463 . . . . . . . . . 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 13917 . . . . . . . . . . . . . . . . 17  |-  ( ( K  x.  u )  e.  CC  ->  ( exp `  ( K  x.  u ) )  e.  CC )
157 efne0 13931 . . . . . . . . . . . . . . . . 17  |-  ( ( K  x.  u )  e.  CC  ->  ( exp `  ( K  x.  u ) )  =/=  0 )
158156, 157jca 530 . . . . . . . . . . . . . . . 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 9509 . . . . . . . . . . . . . . . . 17  |-  1  =/=  0
16118, 160pm3.2i 453 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  CC  /\  1  =/=  0 )
162 divdiv2 10215 . . . . . . . . . . . . . . . 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 1312 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 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 9524 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CC  /\  ( exp `  ( K  x.  u ) )  e.  CC )  -> 
( x  x.  ( exp `  ( K  x.  u ) ) )  e.  CC )
167165, 166sylan2 472 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CC  /\  ( ph  /\  u  e.  S ) )  -> 
( x  x.  ( exp `  ( K  x.  u ) ) )  e.  CC )
168167div1d 10271 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  ( ph  /\  u  e.  S ) )  -> 
( ( x  x.  ( exp `  ( K  x.  u )
) )  /  1
)  =  ( x  x.  ( exp `  ( K  x.  u )
) ) )
169164, 168eqtrd 2441 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  ( ph  /\  u  e.  S ) )  -> 
( x  /  (
1  /  ( exp `  ( K  x.  u
) ) ) )  =  ( x  x.  ( exp `  ( K  x.  u )
) ) )
170169ancoms 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  u  e.  S )  /\  x  e.  CC )  ->  (
x  /  ( 1  /  ( exp `  ( K  x.  u )
) ) )  =  ( x  x.  ( exp `  ( K  x.  u ) ) ) )
171170an32s 803 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  CC )  /\  u  e.  S )  ->  (
x  /  ( 1  /  ( exp `  ( K  x.  u )
) ) )  =  ( x  x.  ( exp `  ( K  x.  u ) ) ) )
172171mpteq2dva 4478 . . . . . . . . . 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 2441 . . . . . . . . 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 2414 . . . . . . . 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 214 . . . . . . 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 2876 . . . . . 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 463 . . . . 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 432 . . 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 463 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u ) ) ) ) ) )  ->  S  e.  { RR ,  CC } )
1814adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u ) ) ) ) ) )  ->  K  e.  CC )
182 simprl 755 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u ) ) ) ) ) )  ->  x  e.  CC )
183 eqid 2400 . . . . . . 7  |-  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u )
) ) )  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u ) ) ) )
184180, 181, 182, 183expgrowthi 36050 . . . . . 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 1191 . . . . 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 6240 . . . . . . 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 6240 . . . . . . 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 2422 . . . . . 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 1018 . . . . 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 232 . . . 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 2895 . . 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 191 . 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 6240 . . . . . . . 8  |-  ( u  =  t  ->  ( K  x.  u )  =  ( K  x.  t ) )
194193fveq2d 5807 . . . . . . 7  |-  ( u  =  t  ->  ( exp `  ( K  x.  u ) )  =  ( exp `  ( K  x.  t )
) )
195194oveq2d 6248 . . . . . 6  |-  ( u  =  t  ->  (
x  x.  ( exp `  ( K  x.  u
) ) )  =  ( x  x.  ( exp `  ( K  x.  t ) ) ) )
196195cbvmptv 4484 . . . . 5  |-  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u )
) ) )  =  ( t  e.  S  |->  ( x  x.  ( exp `  ( K  x.  t ) ) ) )
197 oveq1 6239 . . . . . 6  |-  ( x  =  c  ->  (
x  x.  ( exp `  ( K  x.  t
) ) )  =  ( c  x.  ( exp `  ( K  x.  t ) ) ) )
198197mpteq2dv 4479 . . . . 5  |-  ( x  =  c  ->  (
t  e.  S  |->  ( x  x.  ( exp `  ( K  x.  t
) ) ) )  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t )
) ) ) )
199196, 198syl5eq 2453 . . . 4  |-  ( x  =  c  ->  (
u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u
) ) ) )  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t )
) ) ) )
200199eqeq2d 2414 . . 3  |-  ( x  =  c  ->  ( Y  =  ( u  e.  S  |->  ( x  x.  ( exp `  ( K  x.  u )
) ) )  <->  Y  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t ) ) ) ) ) )
201200cbvrexv 3032 . 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 261 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 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   E.wrex 2752   _Vcvv 3056    \ cdif 3408    C_ wss 3411   {csn 3969   {cpr 3971    |-> cmpt 4450    X. cxp 4938   dom cdm 4940    Fn wfn 5518   -->wf 5519   ` cfv 5523  (class class class)co 6232    oFcof 6473   CCcc 9438   RRcr 9439   0cc0 9440   1c1 9441    + caddc 9443    x. cmul 9445    - cmin 9759   -ucneg 9760    / cdiv 10165   expce 13896    _D cdv 22449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518  ax-addf 9519  ax-mulf 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-fi 7823  df-sup 7853  df-oi 7887  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-q 11144  df-rp 11182  df-xneg 11287  df-xadd 11288  df-xmul 11289  df-ioo 11502  df-ico 11504  df-icc 11505  df-fz 11642  df-fzo 11766  df-fl 11877  df-seq 12060  df-exp 12119  df-fac 12306  df-bc 12333  df-hash 12358  df-shft 12954  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-limsup 13348  df-clim 13365  df-rlim 13366  df-sum 13563  df-ef 13902  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-starv 14814  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-unif 14822  df-hom 14823  df-cco 14824  df-rest 14927  df-topn 14928  df-0g 14946  df-gsum 14947  df-topgen 14948  df-pt 14949  df-prds 14952  df-xrs 15006  df-qtop 15011  df-imas 15012  df-xps 15014  df-mre 15090  df-mrc 15091  df-acs 15093  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-submnd 16181  df-mulg 16274  df-cntz 16569  df-cmn 17014  df-psmet 18621  df-xmet 18622  df-met 18623  df-bl 18624  df-mopn 18625  df-fbas 18626  df-fg 18627  df-cnfld 18631  df-top 19581  df-bases 19583  df-topon 19584  df-topsp 19585  df-cld 19702  df-ntr 19703  df-cls 19704  df-nei 19782  df-lp 19820  df-perf 19821  df-cn 19911  df-cnp 19912  df-haus 19999  df-cmp 20070  df-tx 20245  df-hmeo 20438  df-fil 20529  df-fm 20621  df-flim 20622  df-flf 20623  df-xms 21005  df-ms 21006  df-tms 21007  df-cncf 21564  df-limc 22452  df-dv 22453
This theorem is referenced by: (None)
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