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Theorem pellfund14 30466
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 30428 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR+ )
2 pellfundrp 30456 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR+ )
32adantr 465 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  RR+ )
4 pellfundne1 30457 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =/=  1 )
54adantr 465 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  =/=  1 )
6 reglogcl 30458 . . . 4  |-  ( ( A  e.  RR+  /\  (PellFund `  D )  e.  RR+  /\  (PellFund `  D )  =/=  1 )  ->  (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR )
71, 3, 5, 6syl3anc 1228 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR )
87flcld 11903 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ )
9 pell14qrre 30425 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
109recnd 9622 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
113, 8rpexpcld 12301 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  RR+ )
1211rpcnd 11258 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  CC )
138znegcld 10968 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ )
143, 13rpexpcld 12301 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  RR+ )
1514rpcnd 11258 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  CC )
1614rpne0d 11261 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  =/=  0 )
17 simpl 457 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  D  e.  ( NN  \NN )
)
18 pell1qrss14 30436 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
19 pellfundex 30454 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
)
2018, 19sseldd 3505 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell14QR `  D )
)
2120adantr 465 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  (Pell14QR `  D )
)
22 pell14qrexpcl 30435 . . . . . . 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 1228 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  (Pell14QR `  D
) )
24 pell14qrmulcl 30431 . . . . . 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 1325 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  (Pell14QR `  D
) )
26 1rp 11224 . . . . . . . . . 10  |-  1  e.  RR+
2726a1i 11 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  e.  RR+ )
28 modge0 11973 . . . . . . . . 9  |-  ( ( ( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR  /\  1  e.  RR+ )  ->  0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  mod  1
) )
297, 27, 28syl2anc 661 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  mod  1
) )
307recnd 9622 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  CC )
318zcnd 10967 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  CC )
3230, 31negsubd 9936 . . . . . . . . 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 11977 . . . . . . . . . 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 2511 . . . . . . . 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 4473 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  +  -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D ) ) ) ) ) )
37 reglog1 30464 . . . . . . . 8  |-  ( ( (PellFund `  D )  e.  RR+  /\  (PellFund `  D
)  =/=  1 )  ->  ( ( log `  1 )  / 
( log `  (PellFund `  D ) ) )  =  0 )
383, 5, 37syl2anc 661 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  1
)  /  ( log `  (PellFund `  D )
) )  =  0 )
39 reglogmul 30461 . . . . . . . . 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 1232 . . . . . . . 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 30465 . . . . . . . . . 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 1226 . . . . . . . . 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 6300 . . . . . . . 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 2508 . . . . . . 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 4477 . . . . . 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 11272 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  RR+ )
47 pellfundgt1 30451 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
1  <  (PellFund `  D
) )
4847adantr 465 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  <  (PellFund `  D
) )
49 reglogleb 30460 . . . . . . 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 1229 . . . . . 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 11974 . . . . . . . . 9  |-  ( ( ( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR  /\  1  e.  RR+ )  ->  ( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  mod  1 )  <  1
)
537, 27, 52syl2anc 661 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  mod  1 )  <  1
)
5435, 53eqbrtrd 4467 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  <  1 )
55 reglogbas 30463 . . . . . . . 8  |-  ( ( (PellFund `  D )  e.  RR+  /\  (PellFund `  D
)  =/=  1 )  ->  ( ( log `  (PellFund `  D )
)  /  ( log `  (PellFund `  D )
) )  =  1 )
563, 5, 55syl2anc 661 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  (PellFund `  D ) )  / 
( log `  (PellFund `  D ) ) )  =  1 )
5754, 44, 563brtr4d 4477 . . . . . 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 30459 . . . . . . 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 1229 . . . . . 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 30455 . . . . 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 1232 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  =  1 )
6331negidd 9920 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  =  0 )
6463oveq2d 6300 . . . . 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 11258 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  CC )
663rpne0d 11261 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  =/=  0 )
67 expaddz 12178 . . . . . 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 1229 . . . . 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 12272 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ 0 )  =  1 )
7064, 68, 693eqtr3rd 2517 . . . 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 2508 . . 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 10185 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =  ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )
73 oveq2 6292 . . . 4  |-  ( x  =  ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  -> 
( (PellFund `  D ) ^ x )  =  ( (PellFund `  D
) ^ ( |_
`  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )
7473eqeq2d 2481 . . 3  |-  ( x  =  ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  -> 
( A  =  ( (PellFund `  D ) ^ x )  <->  A  =  ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) ) )
7574rspcev 3214 . 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 661 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 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    \ cdif 3473   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    <_ cle 9629    - cmin 9805   -ucneg 9806    / cdiv 10206   NNcn 10536   ZZcz 10864   RR+crp 11220   |_cfl 11895    mod cmo 11964   ^cexp 12134   logclog 22698  ◻NNcsquarenn 30404  Pell1QRcpell1qr 30405  Pell14QRcpell14qr 30407  PellFundcpellfund 30408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-pi 13670  df-dvds 13848  df-gcd 14004  df-numer 14127  df-denom 14128  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-limc 22033  df-dv 22034  df-log 22700  df-squarenn 30409  df-pell1qr 30410  df-pell14qr 30411  df-pell1234qr 30412  df-pellfund 30413
This theorem is referenced by:  pellfund14b  30467
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