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Theorem pellfund14 26652
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 26614 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR+ )
2 pellfundrp 26642 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR+ )
32adantr 452 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  RR+ )
4 pellfundne1 26643 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =/=  1 )
54adantr 452 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  =/=  1 )
6 reglogcl 26644 . . . 4  |-  ( ( A  e.  RR+  /\  (PellFund `  D )  e.  RR+  /\  (PellFund `  D )  =/=  1 )  ->  (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR )
71, 3, 5, 6syl3anc 1184 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR )
87flcld 11134 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ )
9 pell14qrre 26611 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
109recnd 9047 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
113, 8rpexpcld 11473 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  RR+ )
1211rpcnd 10582 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  CC )
138znegcld 10309 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ )
143, 13rpexpcld 11473 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  RR+ )
1514rpcnd 10582 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  CC )
1614rpne0d 10585 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  =/=  0 )
17 simpl 444 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  D  e.  ( NN  \NN )
)
18 pell1qrss14 26622 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
19 pellfundex 26640 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
)
2018, 19sseldd 3292 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell14QR `  D )
)
2120adantr 452 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  (Pell14QR `  D )
)
22 pell14qrexpcl 26621 . . . . . . 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 1184 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  (Pell14QR `  D
) )
24 pell14qrmulcl 26617 . . . . . 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 1280 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  (Pell14QR `  D
) )
26 1rp 10548 . . . . . . . . . 10  |-  1  e.  RR+
2726a1i 11 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  e.  RR+ )
28 modge0 11184 . . . . . . . . 9  |-  ( ( ( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR  /\  1  e.  RR+ )  ->  0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  mod  1
) )
297, 27, 28syl2anc 643 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  mod  1
) )
307recnd 9047 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  CC )
318zcnd 10308 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  CC )
3230, 31negsubd 9349 . . . . . . . . 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 11188 . . . . . . . . . 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 2422 . . . . . . . 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 4179 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  +  -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D ) ) ) ) ) )
37 reglog1 26650 . . . . . . . 8  |-  ( ( (PellFund `  D )  e.  RR+  /\  (PellFund `  D
)  =/=  1 )  ->  ( ( log `  1 )  / 
( log `  (PellFund `  D ) ) )  =  0 )
383, 5, 37syl2anc 643 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  1
)  /  ( log `  (PellFund `  D )
) )  =  0 )
39 reglogmul 26647 . . . . . . . . 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 1188 . . . . . . . 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 26651 . . . . . . . . . 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 1182 . . . . . . . . 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 6036 . . . . . . . 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 2419 . . . . . . 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 4183 . . . . . 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 10596 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  RR+ )
47 pellfundgt1 26637 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
1  <  (PellFund `  D
) )
4847adantr 452 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  <  (PellFund `  D
) )
49 reglogleb 26646 . . . . . . 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 1185 . . . . . 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 224 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  <_  ( A  x.  ( (PellFund `  D
) ^ -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) ) )
52 modlt 11185 . . . . . . . . 9  |-  ( ( ( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR  /\  1  e.  RR+ )  ->  ( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  mod  1 )  <  1
)
537, 27, 52syl2anc 643 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  mod  1 )  <  1
)
5435, 53eqbrtrd 4173 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  <  1 )
55 reglogbas 26649 . . . . . . . 8  |-  ( ( (PellFund `  D )  e.  RR+  /\  (PellFund `  D
)  =/=  1 )  ->  ( ( log `  (PellFund `  D )
)  /  ( log `  (PellFund `  D )
) )  =  1 )
563, 5, 55syl2anc 643 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  (PellFund `  D ) )  / 
( log `  (PellFund `  D ) ) )  =  1 )
5754, 44, 563brtr4d 4183 . . . . . 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 26645 . . . . . . 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 1185 . . . . . 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 224 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  <  (PellFund `  D
) )
61 pellfund14gap 26641 . . . . 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 1188 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  =  1 )
6331negidd 9333 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  =  0 )
6463oveq2d 6036 . . . . 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 10582 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  CC )
663rpne0d 10585 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  =/=  0 )
67 expaddz 11351 . . . . . 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 1185 . . . . 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 11444 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ 0 )  =  1 )
7064, 68, 693eqtr3rd 2428 . . . 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 2419 . . 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 9590 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =  ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )
73 oveq2 6028 . . . 4  |-  ( x  =  ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  -> 
( (PellFund `  D ) ^ x )  =  ( (PellFund `  D
) ^ ( |_
`  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )
7473eqeq2d 2398 . . 3  |-  ( x  =  ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  -> 
( A  =  ( (PellFund `  D ) ^ x )  <->  A  =  ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) ) )
7574rspcev 2995 . 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 643 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 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650    \ cdif 3260   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    < clt 9053    <_ cle 9054    - cmin 9223   -ucneg 9224    / cdiv 9609   NNcn 9932   ZZcz 10214   RR+crp 10544   |_cfl 11128    mod cmo 11177   ^cexp 11309   logclog 20319  ◻NNcsquarenn 26590  Pell1QRcpell1qr 26591  Pell14QRcpell14qr 26593  PellFundcpellfund 26594
This theorem is referenced by:  pellfund14b  26653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-acn 7762  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ioc 10853  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-fac 11494  df-bc 11521  df-hash 11546  df-shft 11809  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-limsup 12192  df-clim 12209  df-rlim 12210  df-sum 12407  df-ef 12597  df-sin 12599  df-cos 12600  df-pi 12602  df-dvds 12780  df-gcd 12934  df-numer 13054  df-denom 13055  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-rest 13577  df-topn 13578  df-topgen 13594  df-pt 13595  df-prds 13598  df-xrs 13653  df-0g 13654  df-gsum 13655  df-qtop 13660  df-imas 13661  df-xps 13663  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-mulg 14742  df-cntz 15043  df-cmn 15341  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-fbas 16623  df-fg 16624  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-ntr 17007  df-cls 17008  df-nei 17085  df-lp 17123  df-perf 17124  df-cn 17213  df-cnp 17214  df-haus 17301  df-tx 17515  df-hmeo 17708  df-fil 17799  df-fm 17891  df-flim 17892  df-flf 17893  df-xms 18259  df-ms 18260  df-tms 18261  df-cncf 18779  df-limc 19620  df-dv 19621  df-log 20321  df-squarenn 26595  df-pell1qr 26596  df-pell14qr 26597  df-pell1234qr 26598  df-pellfund 26599
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