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Theorem pellfund14 26999
Description: Every positive Pell solution is a power of the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
Assertion
Ref Expression
pellfund14  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  E. x  e.  ZZ  A  =  ( (PellFund `  D ) ^ x
) )
Distinct variable groups:    x, D    x, A

Proof of Theorem pellfund14
StepHypRef Expression
1 pell14qrrp 26961 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR+ )
2 pellfundrp 26989 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR+ )
32adantr 453 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  RR+ )
4 pellfundne1 26990 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =/=  1 )
54adantr 453 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  =/=  1 )
6 reglogcl 26991 . . . 4  |-  ( ( A  e.  RR+  /\  (PellFund `  D )  e.  RR+  /\  (PellFund `  D )  =/=  1 )  ->  (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR )
71, 3, 5, 6syl3anc 1185 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR )
87flcld 11238 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ )
9 pell14qrre 26958 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
109recnd 9145 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
113, 8rpexpcld 11577 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  RR+ )
1211rpcnd 10681 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  CC )
138znegcld 10408 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ )
143, 13rpexpcld 11577 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  RR+ )
1514rpcnd 10681 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  CC )
1614rpne0d 10684 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  =/=  0 )
17 simpl 445 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  D  e.  ( NN  \NN )
)
18 pell1qrss14 26969 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
19 pellfundex 26987 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
)
2018, 19sseldd 3335 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell14QR `  D )
)
2120adantr 453 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  (Pell14QR `  D )
)
22 pell14qrexpcl 26968 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  (PellFund `  D
)  e.  (Pell14QR `  D
)  /\  -u ( |_
`  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) )  e.  ZZ )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  (Pell14QR `  D
) )
2317, 21, 13, 22syl3anc 1185 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  (Pell14QR `  D
) )
24 pell14qrmulcl 26964 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  (Pell14QR `  D
) )  ->  ( A  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )  e.  (Pell14QR `  D ) )
2523, 24mpd3an3 1281 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  (Pell14QR `  D
) )
26 1rp 10647 . . . . . . . . . 10  |-  1  e.  RR+
2726a1i 11 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  e.  RR+ )
28 modge0 11288 . . . . . . . . 9  |-  ( ( ( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR  /\  1  e.  RR+ )  ->  0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  mod  1
) )
297, 27, 28syl2anc 644 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  mod  1
) )
307recnd 9145 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  CC )
318zcnd 10407 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  CC )
3230, 31negsubd 9448 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  =  ( ( ( log `  A )  /  ( log `  (PellFund `  D ) ) )  -  ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )
33 modfrac 11292 . . . . . . . . . 10  |-  ( ( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR  ->  ( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  mod  1 )  =  ( ( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  -  ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )
347, 33syl 16 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  mod  1 )  =  ( ( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  -  ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )
3532, 34eqtr4d 2477 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  =  ( ( ( log `  A )  /  ( log `  (PellFund `  D ) ) )  mod  1 ) )
3629, 35breqtrrd 4263 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  +  -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D ) ) ) ) ) )
37 reglog1 26997 . . . . . . . 8  |-  ( ( (PellFund `  D )  e.  RR+  /\  (PellFund `  D
)  =/=  1 )  ->  ( ( log `  1 )  / 
( log `  (PellFund `  D ) ) )  =  0 )
383, 5, 37syl2anc 644 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  1
)  /  ( log `  (PellFund `  D )
) )  =  0 )
39 reglogmul 26994 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  RR+  /\  (
(PellFund `  D )  e.  RR+  /\  (PellFund `  D
)  =/=  1 ) )  ->  ( ( log `  ( A  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) ) )  /  ( log `  (PellFund `  D
) ) )  =  ( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  +  ( ( log `  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  /  ( log `  (PellFund `  D )
) ) ) )
401, 14, 3, 5, 39syl112anc 1189 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  ( A  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) ) )  / 
( log `  (PellFund `  D ) ) )  =  ( ( ( log `  A )  /  ( log `  (PellFund `  D ) ) )  +  ( ( log `  ( (PellFund `  D
) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )  /  ( log `  (PellFund `  D
) ) ) ) )
41 reglogexpbas 26998 . . . . . . . . . 10  |-  ( (
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ  /\  ( (PellFund `  D )  e.  RR+  /\  (PellFund `  D )  =/=  1 ) )  -> 
( ( log `  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  /  ( log `  (PellFund `  D )
) )  =  -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D ) ) ) ) )
4213, 3, 5, 41syl12anc 1183 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  /  ( log `  (PellFund `  D )
) )  =  -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D ) ) ) ) )
4342oveq2d 6126 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  +  ( ( log `  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  /  ( log `  (PellFund `  D )
) ) )  =  ( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )
4440, 43eqtrd 2474 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  ( A  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) ) )  / 
( log `  (PellFund `  D ) ) )  =  ( ( ( log `  A )  /  ( log `  (PellFund `  D ) ) )  +  -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )
4536, 38, 443brtr4d 4267 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  1
)  /  ( log `  (PellFund `  D )
) )  <_  (
( log `  ( A  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) ) )  / 
( log `  (PellFund `  D ) ) ) )
461, 14rpmulcld 10695 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  RR+ )
47 pellfundgt1 26984 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
1  <  (PellFund `  D
) )
4847adantr 453 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  <  (PellFund `  D
) )
49 reglogleb 26993 . . . . . . 7  |-  ( ( ( 1  e.  RR+  /\  ( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  RR+ )  /\  ( (PellFund `  D
)  e.  RR+  /\  1  <  (PellFund `  D )
) )  ->  (
1  <_  ( A  x.  ( (PellFund `  D
) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )  <->  ( ( log `  1 )  / 
( log `  (PellFund `  D ) ) )  <_  ( ( log `  ( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) ) )  /  ( log `  (PellFund `  D
) ) ) ) )
5027, 46, 3, 48, 49syl22anc 1186 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <_  ( A  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )  <->  ( ( log `  1 )  / 
( log `  (PellFund `  D ) ) )  <_  ( ( log `  ( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) ) )  /  ( log `  (PellFund `  D
) ) ) ) )
5145, 50mpbird 225 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  <_  ( A  x.  ( (PellFund `  D
) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) ) )
52 modlt 11289 . . . . . . . . 9  |-  ( ( ( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR  /\  1  e.  RR+ )  ->  ( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  mod  1 )  <  1
)
537, 27, 52syl2anc 644 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  mod  1 )  <  1
)
5435, 53eqbrtrd 4257 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  <  1 )
55 reglogbas 26996 . . . . . . . 8  |-  ( ( (PellFund `  D )  e.  RR+  /\  (PellFund `  D
)  =/=  1 )  ->  ( ( log `  (PellFund `  D )
)  /  ( log `  (PellFund `  D )
) )  =  1 )
563, 5, 55syl2anc 644 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  (PellFund `  D ) )  / 
( log `  (PellFund `  D ) ) )  =  1 )
5754, 44, 563brtr4d 4267 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  ( A  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) ) )  / 
( log `  (PellFund `  D ) ) )  <  ( ( log `  (PellFund `  D )
)  /  ( log `  (PellFund `  D )
) ) )
58 reglogltb 26992 . . . . . . 7  |-  ( ( ( ( A  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  RR+  /\  (PellFund `  D )  e.  RR+ )  /\  ( (PellFund `  D
)  e.  RR+  /\  1  <  (PellFund `  D )
) )  ->  (
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  <  (PellFund `  D
)  <->  ( ( log `  ( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) ) )  /  ( log `  (PellFund `  D
) ) )  < 
( ( log `  (PellFund `  D ) )  / 
( log `  (PellFund `  D ) ) ) ) )
5946, 3, 3, 48, 58syl22anc 1186 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( A  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  <  (PellFund `  D
)  <->  ( ( log `  ( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) ) )  /  ( log `  (PellFund `  D
) ) )  < 
( ( log `  (PellFund `  D ) )  / 
( log `  (PellFund `  D ) ) ) ) )
6057, 59mpbird 225 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  <  (PellFund `  D
) )
61 pellfund14gap 26988 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  (Pell14QR `  D
)  /\  ( 1  <_  ( A  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  /\  ( A  x.  ( (PellFund `  D
) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )  <  (PellFund `  D ) ) )  ->  ( A  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  =  1 )
6217, 25, 51, 60, 61syl112anc 1189 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  =  1 )
6331negidd 9432 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  =  0 )
6463oveq2d 6126 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ ( ( |_
`  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) )  +  -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  =  ( (PellFund `  D ) ^ 0 ) )
653rpcnd 10681 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  CC )
663rpne0d 10684 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  =/=  0 )
67 expaddz 11455 . . . . . 6  |-  ( ( ( (PellFund `  D
)  e.  CC  /\  (PellFund `  D )  =/=  0 )  /\  (
( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ  /\  -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) )  e.  ZZ ) )  ->  ( (PellFund `  D
) ^ ( ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D ) ) ) )  +  -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )  =  ( ( (PellFund `  D
) ^ ( |_
`  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) )  x.  ( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) ) )
6865, 66, 8, 13, 67syl22anc 1186 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ ( ( |_
`  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) )  +  -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  =  ( ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  x.  ( (PellFund `  D
) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) ) )
6965exp0d 11548 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ 0 )  =  1 )
7064, 68, 693eqtr3rd 2483 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  =  ( ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  x.  ( (PellFund `  D
) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) ) )
7162, 70eqtrd 2474 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  =  ( ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  x.  ( (PellFund `  D
) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) ) )
7210, 12, 15, 16, 71mulcan2ad 9689 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =  ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )
73 oveq2 6118 . . . 4  |-  ( x  =  ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  -> 
( (PellFund `  D ) ^ x )  =  ( (PellFund `  D
) ^ ( |_
`  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )
7473eqeq2d 2453 . . 3  |-  ( x  =  ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  -> 
( A  =  ( (PellFund `  D ) ^ x )  <->  A  =  ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) ) )
7574rspcev 3058 . 2  |-  ( ( ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ  /\  A  =  ( (PellFund `  D
) ^ ( |_
`  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )  ->  E. x  e.  ZZ  A  =  ( (PellFund `  D ) ^ x ) )
768, 72, 75syl2anc 644 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  E. x  e.  ZZ  A  =  ( (PellFund `  D ) ^ x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   E.wrex 2712    \ cdif 3303   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   CCcc 9019   RRcr 9020   0cc0 9021   1c1 9022    + caddc 9024    x. cmul 9026    < clt 9151    <_ cle 9152    - cmin 9322   -ucneg 9323    / cdiv 9708   NNcn 10031   ZZcz 10313   RR+crp 10643   |_cfl 11232    mod cmo 11281   ^cexp 11413   logclog 20483  ◻NNcsquarenn 26937  Pell1QRcpell1qr 26938  Pell14QRcpell14qr 26940  PellFundcpellfund 26941
This theorem is referenced by:  pellfund14b  27000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099  ax-addf 9100  ax-mulf 9101
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-2o 6754  df-oadd 6757  df-omul 6758  df-er 6934  df-map 7049  df-pm 7050  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-fi 7445  df-sup 7475  df-oi 7508  df-card 7857  df-acn 7860  df-cda 8079  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-q 10606  df-rp 10644  df-xneg 10741  df-xadd 10742  df-xmul 10743  df-ioo 10951  df-ioc 10952  df-ico 10953  df-icc 10954  df-fz 11075  df-fzo 11167  df-fl 11233  df-mod 11282  df-seq 11355  df-exp 11414  df-fac 11598  df-bc 11625  df-hash 11650  df-shft 11913  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-limsup 12296  df-clim 12313  df-rlim 12314  df-sum 12511  df-ef 12701  df-sin 12703  df-cos 12704  df-pi 12706  df-dvds 12884  df-gcd 13038  df-numer 13158  df-denom 13159  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-mulr 13574  df-starv 13575  df-sca 13576  df-vsca 13577  df-tset 13579  df-ple 13580  df-ds 13582  df-unif 13583  df-hom 13584  df-cco 13585  df-rest 13681  df-topn 13682  df-topgen 13698  df-pt 13699  df-prds 13702  df-xrs 13757  df-0g 13758  df-gsum 13759  df-qtop 13764  df-imas 13765  df-xps 13767  df-mre 13842  df-mrc 13843  df-acs 13845  df-mnd 14721  df-submnd 14770  df-mulg 14846  df-cntz 15147  df-cmn 15445  df-psmet 16725  df-xmet 16726  df-met 16727  df-bl 16728  df-mopn 16729  df-fbas 16730  df-fg 16731  df-cnfld 16735  df-top 16994  df-bases 16996  df-topon 16997  df-topsp 16998  df-cld 17114  df-ntr 17115  df-cls 17116  df-nei 17193  df-lp 17231  df-perf 17232  df-cn 17322  df-cnp 17323  df-haus 17410  df-tx 17625  df-hmeo 17818  df-fil 17909  df-fm 18001  df-flim 18002  df-flf 18003  df-xms 18381  df-ms 18382  df-tms 18383  df-cncf 18939  df-limc 19784  df-dv 19785  df-log 20485  df-squarenn 26942  df-pell1qr 26943  df-pell14qr 26944  df-pell1234qr 26945  df-pellfund 26946
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