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Theorem pellqrexplicit 26962
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 9974 . . . . 5  |-  ( A  e.  NN0  ->  A  e.  RR )
213ad2ant2 977 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  A  e.  RR )
3 eldifi 3298 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
433ad2ant1 976 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  D  e.  NN )
54nnrpd 10389 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  D  e.  RR+ )
65rpsqrcld 11894 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  ( sqr `  D
)  e.  RR+ )
76rpred 10390 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  ( sqr `  D
)  e.  RR )
8 nn0re 9974 . . . . . 6  |-  ( B  e.  NN0  ->  B  e.  RR )
983ad2ant3 978 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  B  e.  RR )
107, 9remulcld 8863 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  ( ( sqr `  D
)  x.  B )  e.  RR )
112, 10readdcld 8862 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  +  ( ( sqr `  D
)  x.  B ) )  e.  RR )
1211adantr 451 . 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 959 . . 3  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e. 
NN0  /\  B  e.  NN0 )  /\  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 )  ->  A  e.  NN0 )
14 simpl3 960 . . 3  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e. 
NN0  /\  B  e.  NN0 )  /\  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) )  =  1 )  ->  B  e.  NN0 )
15 eqidd 2284 . . 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 447 . . 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 5865 . . . . . 6  |-  ( a  =  A  ->  (
a  +  ( ( sqr `  D )  x.  b ) )  =  ( A  +  ( ( sqr `  D
)  x.  b ) ) )
1817eqeq2d 2294 . . . . 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 5865 . . . . . . 7  |-  ( a  =  A  ->  (
a ^ 2 )  =  ( A ^
2 ) )
2019oveq1d 5873 . . . . . 6  |-  ( a  =  A  ->  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( A ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) ) )
2120eqeq1d 2291 . . . . 5  |-  ( a  =  A  ->  (
( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( A ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
2218, 21anbi12d 691 . . . 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 5866 . . . . . . 7  |-  ( b  =  B  ->  (
( sqr `  D
)  x.  b )  =  ( ( sqr `  D )  x.  B
) )
2423oveq2d 5874 . . . . . 6  |-  ( b  =  B  ->  ( A  +  ( ( sqr `  D )  x.  b ) )  =  ( A  +  ( ( sqr `  D
)  x.  B ) ) )
2524eqeq2d 2294 . . . . 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 5865 . . . . . . . 8  |-  ( b  =  B  ->  (
b ^ 2 )  =  ( B ^
2 ) )
2726oveq2d 5874 . . . . . . 7  |-  ( b  =  B  ->  ( D  x.  ( b ^ 2 ) )  =  ( D  x.  ( B ^ 2 ) ) )
2827oveq2d 5874 . . . . . 6  |-  ( b  =  B  ->  (
( A ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) ) )
2928eqeq1d 2291 . . . . 5  |-  ( b  =  B  ->  (
( ( A ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( A ^ 2 )  -  ( D  x.  ( B ^
2 ) ) )  =  1 ) )
3025, 29anbi12d 691 . . . 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 2892 . . 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 1186 . 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 26932 . . . 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 976 . . 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 451 . 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 888 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    \ cdif 3149   ` cfv 5255  (class class class)co 5858   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   2c2 9795   NN0cn0 9965   ^cexp 11104   sqrcsqr 11718  ◻NNcsquarenn 26921  Pell1QRcpell1qr 26922
This theorem is referenced by:  pellqrex  26964  rmspecfund  26994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-pell1qr 26927
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