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Theorem pellfund14 29164
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 29126 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR+ )
2 pellfundrp 29154 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR+ )
32adantr 462 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  RR+ )
4 pellfundne1 29155 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =/=  1 )
54adantr 462 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  =/=  1 )
6 reglogcl 29156 . . . 4  |-  ( ( A  e.  RR+  /\  (PellFund `  D )  e.  RR+  /\  (PellFund `  D )  =/=  1 )  ->  (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR )
71, 3, 5, 6syl3anc 1213 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR )
87flcld 11644 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ )
9 pell14qrre 29123 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
109recnd 9408 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
113, 8rpexpcld 12027 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  RR+ )
1211rpcnd 11025 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  CC )
138znegcld 10745 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ )
143, 13rpexpcld 12027 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  RR+ )
1514rpcnd 11025 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  CC )
1614rpne0d 11028 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  =/=  0 )
17 simpl 454 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  D  e.  ( NN  \NN )
)
18 pell1qrss14 29134 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
19 pellfundex 29152 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
)
2018, 19sseldd 3354 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell14QR `  D )
)
2120adantr 462 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  (Pell14QR `  D )
)
22 pell14qrexpcl 29133 . . . . . . 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 1213 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  (Pell14QR `  D
) )
24 pell14qrmulcl 29129 . . . . . 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 1310 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  (Pell14QR `  D
) )
26 1rp 10991 . . . . . . . . . 10  |-  1  e.  RR+
2726a1i 11 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  e.  RR+ )
28 modge0 11713 . . . . . . . . 9  |-  ( ( ( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR  /\  1  e.  RR+ )  ->  0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  mod  1
) )
297, 27, 28syl2anc 656 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  mod  1
) )
307recnd 9408 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  CC )
318zcnd 10744 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  CC )
3230, 31negsubd 9721 . . . . . . . . 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 11717 . . . . . . . . . 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 2476 . . . . . . . 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 4315 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  +  -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D ) ) ) ) ) )
37 reglog1 29162 . . . . . . . 8  |-  ( ( (PellFund `  D )  e.  RR+  /\  (PellFund `  D
)  =/=  1 )  ->  ( ( log `  1 )  / 
( log `  (PellFund `  D ) ) )  =  0 )
383, 5, 37syl2anc 656 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  1
)  /  ( log `  (PellFund `  D )
) )  =  0 )
39 reglogmul 29159 . . . . . . . . 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 1217 . . . . . . . 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 29163 . . . . . . . . . 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 1211 . . . . . . . . 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 6106 . . . . . . . 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 2473 . . . . . . 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 4319 . . . . . 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 11039 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  RR+ )
47 pellfundgt1 29149 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
1  <  (PellFund `  D
) )
4847adantr 462 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  <  (PellFund `  D
) )
49 reglogleb 29158 . . . . . . 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 1214 . . . . . 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 232 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  <_  ( A  x.  ( (PellFund `  D
) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) ) )
52 modlt 11714 . . . . . . . . 9  |-  ( ( ( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR  /\  1  e.  RR+ )  ->  ( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  mod  1 )  <  1
)
537, 27, 52syl2anc 656 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  mod  1 )  <  1
)
5435, 53eqbrtrd 4309 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  <  1 )
55 reglogbas 29161 . . . . . . . 8  |-  ( ( (PellFund `  D )  e.  RR+  /\  (PellFund `  D
)  =/=  1 )  ->  ( ( log `  (PellFund `  D )
)  /  ( log `  (PellFund `  D )
) )  =  1 )
563, 5, 55syl2anc 656 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  (PellFund `  D ) )  / 
( log `  (PellFund `  D ) ) )  =  1 )
5754, 44, 563brtr4d 4319 . . . . . 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 29157 . . . . . . 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 1214 . . . . . 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 232 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  <  (PellFund `  D
) )
61 pellfund14gap 29153 . . . . 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 1217 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  =  1 )
6331negidd 9705 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  =  0 )
6463oveq2d 6106 . . . . 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 11025 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  CC )
663rpne0d 11028 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  =/=  0 )
67 expaddz 11904 . . . . . 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 1214 . . . . 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 11998 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ 0 )  =  1 )
7064, 68, 693eqtr3rd 2482 . . . 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 2473 . . 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 9968 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =  ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )
73 oveq2 6098 . . . 4  |-  ( x  =  ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  -> 
( (PellFund `  D ) ^ x )  =  ( (PellFund `  D
) ^ ( |_
`  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )
7473eqeq2d 2452 . . 3  |-  ( x  =  ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  -> 
( A  =  ( (PellFund `  D ) ^ x )  <->  A  =  ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) ) )
7574rspcev 3070 . 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 656 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  E. x  e.  ZZ  A  =  ( (PellFund `  D ) ^ x
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714    \ cdif 3322   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    < clt 9414    <_ cle 9415    - cmin 9591   -ucneg 9592    / cdiv 9989   NNcn 10318   ZZcz 10642   RR+crp 10987   |_cfl 11636    mod cmo 11704   ^cexp 11861   logclog 21965  ◻NNcsquarenn 29102  Pell1QRcpell1qr 29103  Pell14QRcpell14qr 29105  PellFundcpellfund 29106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-omul 6921  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-dvds 13532  df-gcd 13687  df-numer 13809  df-denom 13810  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967  df-squarenn 29107  df-pell1qr 29108  df-pell14qr 29109  df-pell1234qr 29110  df-pellfund 29111
This theorem is referenced by:  pellfund14b  29165
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