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Theorem pell1qr1 26915
Description: 1 is a Pell solution and in the first quadrant as one. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
pell1qr1  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  (Pell1QR `  D
) )

Proof of Theorem pell1qr1
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1re 9082 . . 3  |-  1  e.  RR
21a1i 11 . 2  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  RR )
3 1nn0 10229 . . . 4  |-  1  e.  NN0
43a1i 11 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  NN0 )
5 0nn0 10228 . . . 4  |-  0  e.  NN0
65a1i 11 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
0  e.  NN0 )
7 eldifi 3461 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
87nncnd 10008 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
98sqrcld 12231 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( sqr `  D
)  e.  CC )
109mul01d 9257 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( ( sqr `  D
)  x.  0 )  =  0 )
1110oveq2d 6089 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( 1  +  ( ( sqr `  D
)  x.  0 ) )  =  ( 1  +  0 ) )
12 ax-1cn 9040 . . . . 5  |-  1  e.  CC
1312addid1i 9245 . . . 4  |-  ( 1  +  0 )  =  1
1411, 13syl6req 2484 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
1  =  ( 1  +  ( ( sqr `  D )  x.  0 ) ) )
15 sq1 11468 . . . . . 6  |-  ( 1 ^ 2 )  =  1
1615a1i 11 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( 1 ^ 2 )  =  1 )
17 sq0 11465 . . . . . . 7  |-  ( 0 ^ 2 )  =  0
1817oveq2i 6084 . . . . . 6  |-  ( D  x.  ( 0 ^ 2 ) )  =  ( D  x.  0 )
198mul01d 9257 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( D  x.  0 )  =  0 )
2018, 19syl5eq 2479 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( D  x.  (
0 ^ 2 ) )  =  0 )
2116, 20oveq12d 6091 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( ( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  ( 1  -  0 ) )
2212subid1i 9364 . . . 4  |-  ( 1  -  0 )  =  1
2321, 22syl6eq 2483 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  1 )
24 oveq1 6080 . . . . . 6  |-  ( a  =  1  ->  (
a  +  ( ( sqr `  D )  x.  b ) )  =  ( 1  +  ( ( sqr `  D
)  x.  b ) ) )
2524eqeq2d 2446 . . . . 5  |-  ( a  =  1  ->  (
1  =  ( a  +  ( ( sqr `  D )  x.  b
) )  <->  1  =  ( 1  +  ( ( sqr `  D
)  x.  b ) ) ) )
26 oveq1 6080 . . . . . . 7  |-  ( a  =  1  ->  (
a ^ 2 )  =  ( 1 ^ 2 ) )
2726oveq1d 6088 . . . . . 6  |-  ( a  =  1  ->  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( 1 ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) ) )
2827eqeq1d 2443 . . . . 5  |-  ( a  =  1  ->  (
( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( 1 ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
2925, 28anbi12d 692 . . . 4  |-  ( a  =  1  ->  (
( 1  =  ( a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 )  <->  ( 1  =  ( 1  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( 1 ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) )
30 oveq2 6081 . . . . . . 7  |-  ( b  =  0  ->  (
( sqr `  D
)  x.  b )  =  ( ( sqr `  D )  x.  0 ) )
3130oveq2d 6089 . . . . . 6  |-  ( b  =  0  ->  (
1  +  ( ( sqr `  D )  x.  b ) )  =  ( 1  +  ( ( sqr `  D
)  x.  0 ) ) )
3231eqeq2d 2446 . . . . 5  |-  ( b  =  0  ->  (
1  =  ( 1  +  ( ( sqr `  D )  x.  b
) )  <->  1  =  ( 1  +  ( ( sqr `  D
)  x.  0 ) ) ) )
33 oveq1 6080 . . . . . . . 8  |-  ( b  =  0  ->  (
b ^ 2 )  =  ( 0 ^ 2 ) )
3433oveq2d 6089 . . . . . . 7  |-  ( b  =  0  ->  ( D  x.  ( b ^ 2 ) )  =  ( D  x.  ( 0 ^ 2 ) ) )
3534oveq2d 6089 . . . . . 6  |-  ( b  =  0  ->  (
( 1 ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( 1 ^ 2 )  -  ( D  x.  (
0 ^ 2 ) ) ) )
3635eqeq1d 2443 . . . . 5  |-  ( b  =  0  ->  (
( ( 1 ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  1 ) )
3732, 36anbi12d 692 . . . 4  |-  ( b  =  0  ->  (
( 1  =  ( 1  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( 1 ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 )  <->  ( 1  =  ( 1  +  ( ( sqr `  D
)  x.  0 ) )  /\  ( ( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  1 ) ) )
3829, 37rspc2ev 3052 . . 3  |-  ( ( 1  e.  NN0  /\  0  e.  NN0  /\  (
1  =  ( 1  +  ( ( sqr `  D )  x.  0 ) )  /\  (
( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  1 ) )  ->  E. a  e.  NN0  E. b  e.  NN0  (
1  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
394, 6, 14, 23, 38syl112anc 1188 . 2  |-  ( D  e.  ( NN  \NN )  ->  E. a  e.  NN0  E. b  e.  NN0  (
1  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
40 elpell1qr 26891 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( 1  e.  (Pell1QR `  D )  <->  ( 1  e.  RR  /\  E. a  e.  NN0  E. b  e.  NN0  ( 1  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) ) )
412, 39, 40mpbir2and 889 1  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  (Pell1QR `  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698    \ cdif 3309   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283   NNcn 9992   2c2 10041   NN0cn0 10213   ^cexp 11374   sqrcsqr 12030  ◻NNcsquarenn 26880  Pell1QRcpell1qr 26881
This theorem is referenced by:  elpell1qr2  26916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-pell1qr 26886
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