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Theorem pellfund14 29239
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 29201 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR+ )
2 pellfundrp 29229 . . . . 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 29230 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =/=  1 )
54adantr 465 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  =/=  1 )
6 reglogcl 29231 . . . 4  |-  ( ( A  e.  RR+  /\  (PellFund `  D )  e.  RR+  /\  (PellFund `  D )  =/=  1 )  ->  (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR )
71, 3, 5, 6syl3anc 1218 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  RR )
87flcld 11648 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ )
9 pell14qrre 29198 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
109recnd 9412 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
113, 8rpexpcld 12031 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  RR+ )
1211rpcnd 11029 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  CC )
138znegcld 10749 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  ZZ )
143, 13rpexpcld 12031 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  RR+ )
1514rpcnd 11029 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  CC )
1614rpne0d 11032 . . 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 29209 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
19 pellfundex 29227 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
)
2018, 19sseldd 3357 . . . . . . . 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 29208 . . . . . . 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 1218 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ -u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  e.  (Pell14QR `  D
) )
24 pell14qrmulcl 29204 . . . . . 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 1315 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  (Pell14QR `  D
) )
26 1rp 10995 . . . . . . . . . 10  |-  1  e.  RR+
2726a1i 11 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  e.  RR+ )
28 modge0 11717 . . . . . . . . 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 9412 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( log `  A
)  /  ( log `  (PellFund `  D )
) )  e.  CC )
318zcnd 10748 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  e.  CC )
3230, 31negsubd 9725 . . . . . . . . 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 11721 . . . . . . . . . 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 2478 . . . . . . . 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 4318 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <_  ( (
( log `  A
)  /  ( log `  (PellFund `  D )
) )  +  -u ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D ) ) ) ) ) )
37 reglog1 29237 . . . . . . . 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 29234 . . . . . . . . 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 1222 . . . . . . . 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 29238 . . . . . . . . . 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 1216 . . . . . . . . 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 6107 . . . . . . . 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 2475 . . . . . . 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 4322 . . . . . 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 11043 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  e.  RR+ )
47 pellfundgt1 29224 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
1  <  (PellFund `  D
) )
4847adantr 465 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  <  (PellFund `  D
) )
49 reglogleb 29233 . . . . . . 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 1219 . . . . . 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 11718 . . . . . . . . 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 4312 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( ( log `  A )  /  ( log `  (PellFund `  D
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  <  1 )
55 reglogbas 29236 . . . . . . . 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 4322 . . . . . 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 29232 . . . . . . 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 1219 . . . . . 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 29228 . . . . 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 1222 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
(PellFund `  D ) ^ -u ( |_ `  (
( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) )  =  1 )
6331negidd 9709 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  + 
-u ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) )  =  0 )
6463oveq2d 6107 . . . . 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 11029 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  e.  CC )
663rpne0d 11032 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
(PellFund `  D )  =/=  0 )
67 expaddz 11908 . . . . . 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 1219 . . . . 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 12002 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( (PellFund `  D ) ^ 0 )  =  1 )
7064, 68, 693eqtr3rd 2484 . . . 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 2475 . . 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 9972 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =  ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )
73 oveq2 6099 . . . 4  |-  ( x  =  ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  -> 
( (PellFund `  D ) ^ x )  =  ( (PellFund `  D
) ^ ( |_
`  ( ( log `  A )  /  ( log `  (PellFund `  D
) ) ) ) ) )
7473eqeq2d 2454 . . 3  |-  ( x  =  ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) )  -> 
( A  =  ( (PellFund `  D ) ^ x )  <->  A  =  ( (PellFund `  D ) ^ ( |_ `  ( ( log `  A
)  /  ( log `  (PellFund `  D )
) ) ) ) ) )
7574rspcev 3073 . 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 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716    \ cdif 3325   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    < clt 9418    <_ cle 9419    - cmin 9595   -ucneg 9596    / cdiv 9993   NNcn 10322   ZZcz 10646   RR+crp 10991   |_cfl 11640    mod cmo 11708   ^cexp 11865   logclog 22006  ◻NNcsquarenn 29177  Pell1QRcpell1qr 29178  Pell14QRcpell14qr 29180  PellFundcpellfund 29181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-omul 6925  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-acn 8112  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-dvds 13536  df-gcd 13691  df-numer 13813  df-denom 13814  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008  df-squarenn 29182  df-pell1qr 29183  df-pell14qr 29184  df-pell1234qr 29185  df-pellfund 29186
This theorem is referenced by:  pellfund14b  29240
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