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Theorem pellqrexplicit 30709
Description: Condition for a calculated real to be a Pell solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
Assertion
Ref Expression
pellqrexplicit  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e. 
NN0  /\  B  e.  NN0 )  /\  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 )  ->  ( A  +  ( ( sqr `  D )  x.  B
) )  e.  (Pell1QR `  D ) )

Proof of Theorem pellqrexplicit
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0re 10814 . . . . 5  |-  ( A  e.  NN0  ->  A  e.  RR )
213ad2ant2 1018 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  A  e.  RR )
3 eldifi 3631 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
433ad2ant1 1017 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  D  e.  NN )
54nnrpd 11265 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  D  e.  RR+ )
65rpsqrtcld 13218 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  ( sqr `  D
)  e.  RR+ )
76rpred 11266 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  ( sqr `  D
)  e.  RR )
8 nn0re 10814 . . . . . 6  |-  ( B  e.  NN0  ->  B  e.  RR )
983ad2ant3 1019 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  B  e.  RR )
107, 9remulcld 9634 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  ( ( sqr `  D
)  x.  B )  e.  RR )
112, 10readdcld 9633 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  +  ( ( sqr `  D
)  x.  B ) )  e.  RR )
1211adantr 465 . 2  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e. 
NN0  /\  B  e.  NN0 )  /\  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 )  ->  ( A  +  ( ( sqr `  D )  x.  B
) )  e.  RR )
13 simpl2 1000 . . 3  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e. 
NN0  /\  B  e.  NN0 )  /\  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 )  ->  A  e.  NN0 )
14 simpl3 1001 . . 3  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e. 
NN0  /\  B  e.  NN0 )  /\  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 )  ->  B  e.  NN0 )
15 eqidd 2468 . . 3  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e. 
NN0  /\  B  e.  NN0 )  /\  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 )  ->  ( A  +  ( ( sqr `  D )  x.  B
) )  =  ( A  +  ( ( sqr `  D )  x.  B ) ) )
16 simpr 461 . . 3  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e. 
NN0  /\  B  e.  NN0 )  /\  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 )  ->  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 )
17 oveq1 6301 . . . . . 6  |-  ( a  =  A  ->  (
a  +  ( ( sqr `  D )  x.  b ) )  =  ( A  +  ( ( sqr `  D
)  x.  b ) ) )
1817eqeq2d 2481 . . . . 5  |-  ( a  =  A  ->  (
( A  +  ( ( sqr `  D
)  x.  B ) )  =  ( a  +  ( ( sqr `  D )  x.  b
) )  <->  ( A  +  ( ( sqr `  D )  x.  B
) )  =  ( A  +  ( ( sqr `  D )  x.  b ) ) ) )
19 oveq1 6301 . . . . . . 7  |-  ( a  =  A  ->  (
a ^ 2 )  =  ( A ^
2 ) )
2019oveq1d 6309 . . . . . 6  |-  ( a  =  A  ->  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( A ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) ) )
2120eqeq1d 2469 . . . . 5  |-  ( a  =  A  ->  (
( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( A ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
2218, 21anbi12d 710 . . . 4  |-  ( a  =  A  ->  (
( ( A  +  ( ( sqr `  D
)  x.  B ) )  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 )  <->  ( ( A  +  ( ( sqr `  D )  x.  B ) )  =  ( A  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( A ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) )
23 oveq2 6302 . . . . . . 7  |-  ( b  =  B  ->  (
( sqr `  D
)  x.  b )  =  ( ( sqr `  D )  x.  B
) )
2423oveq2d 6310 . . . . . 6  |-  ( b  =  B  ->  ( A  +  ( ( sqr `  D )  x.  b ) )  =  ( A  +  ( ( sqr `  D
)  x.  B ) ) )
2524eqeq2d 2481 . . . . 5  |-  ( b  =  B  ->  (
( A  +  ( ( sqr `  D
)  x.  B ) )  =  ( A  +  ( ( sqr `  D )  x.  b
) )  <->  ( A  +  ( ( sqr `  D )  x.  B
) )  =  ( A  +  ( ( sqr `  D )  x.  B ) ) ) )
26 oveq1 6301 . . . . . . . 8  |-  ( b  =  B  ->  (
b ^ 2 )  =  ( B ^
2 ) )
2726oveq2d 6310 . . . . . . 7  |-  ( b  =  B  ->  ( D  x.  ( b ^ 2 ) )  =  ( D  x.  ( B ^ 2 ) ) )
2827oveq2d 6310 . . . . . 6  |-  ( b  =  B  ->  (
( A ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) ) )
2928eqeq1d 2469 . . . . 5  |-  ( b  =  B  ->  (
( ( A ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( A ^ 2 )  -  ( D  x.  ( B ^
2 ) ) )  =  1 ) )
3025, 29anbi12d 710 . . . 4  |-  ( b  =  B  ->  (
( ( A  +  ( ( sqr `  D
)  x.  B ) )  =  ( A  +  ( ( sqr `  D )  x.  b
) )  /\  (
( A ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 )  <->  ( ( A  +  ( ( sqr `  D )  x.  B ) )  =  ( A  +  ( ( sqr `  D
)  x.  B ) )  /\  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 ) ) )
3122, 30rspc2ev 3230 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN0  /\  (
( A  +  ( ( sqr `  D
)  x.  B ) )  =  ( A  +  ( ( sqr `  D )  x.  B
) )  /\  (
( A ^ 2 )  -  ( D  x.  ( B ^
2 ) ) )  =  1 ) )  ->  E. a  e.  NN0  E. b  e.  NN0  (
( A  +  ( ( sqr `  D
)  x.  B ) )  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
3213, 14, 15, 16, 31syl112anc 1232 . 2  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e. 
NN0  /\  B  e.  NN0 )  /\  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 )  ->  E. a  e.  NN0  E. b  e. 
NN0  ( ( A  +  ( ( sqr `  D )  x.  B
) )  =  ( a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )
33 elpell1qr 30679 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( ( A  +  ( ( sqr `  D
)  x.  B ) )  e.  (Pell1QR `  D
)  <->  ( ( A  +  ( ( sqr `  D )  x.  B
) )  e.  RR  /\ 
E. a  e.  NN0  E. b  e.  NN0  (
( A  +  ( ( sqr `  D
)  x.  B ) )  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) ) ) )
34333ad2ant1 1017 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  ( ( A  +  ( ( sqr `  D
)  x.  B ) )  e.  (Pell1QR `  D
)  <->  ( ( A  +  ( ( sqr `  D )  x.  B
) )  e.  RR  /\ 
E. a  e.  NN0  E. b  e.  NN0  (
( A  +  ( ( sqr `  D
)  x.  B ) )  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) ) ) )
3534adantr 465 . 2  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e. 
NN0  /\  B  e.  NN0 )  /\  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 )  ->  ( ( A  +  ( ( sqr `  D )  x.  B ) )  e.  (Pell1QR `  D )  <->  ( ( A  +  ( ( sqr `  D
)  x.  B ) )  e.  RR  /\  E. a  e.  NN0  E. b  e.  NN0  ( ( A  +  ( ( sqr `  D )  x.  B
) )  =  ( a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) ) ) )
3612, 32, 35mpbir2and 920 1  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e. 
NN0  /\  B  e.  NN0 )  /\  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 )  ->  ( A  +  ( ( sqr `  D )  x.  B
) )  e.  (Pell1QR `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818    \ cdif 3478   ` cfv 5593  (class class class)co 6294   RRcr 9501   1c1 9503    + caddc 9505    x. cmul 9507    - cmin 9815   NNcn 10546   2c2 10595   NN0cn0 10805   ^cexp 12144   sqrcsqrt 13041  ◻NNcsquarenn 30668  Pell1QRcpell1qr 30669
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-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-sup 7911  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-seq 12086  df-exp 12145  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-pell1qr 30674
This theorem is referenced by:  pellqrex  30711  rmspecfund  30741
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