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Theorem pell1qr1 29210
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 9384 . . 3  |-  1  e.  RR
21a1i 11 . 2  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  RR )
3 1nn0 10594 . . . 4  |-  1  e.  NN0
43a1i 11 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  NN0 )
5 0nn0 10593 . . . 4  |-  0  e.  NN0
65a1i 11 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
0  e.  NN0 )
7 eldifi 3477 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
87nncnd 10337 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
98sqrcld 12922 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( sqr `  D
)  e.  CC )
109mul01d 9567 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( ( sqr `  D
)  x.  0 )  =  0 )
1110oveq2d 6106 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( 1  +  ( ( sqr `  D
)  x.  0 ) )  =  ( 1  +  0 ) )
12 1p0e1 10433 . . . 4  |-  ( 1  +  0 )  =  1
1311, 12syl6req 2491 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
1  =  ( 1  +  ( ( sqr `  D )  x.  0 ) ) )
14 sq1 11959 . . . . . 6  |-  ( 1 ^ 2 )  =  1
1514a1i 11 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( 1 ^ 2 )  =  1 )
16 sq0 11956 . . . . . . 7  |-  ( 0 ^ 2 )  =  0
1716oveq2i 6101 . . . . . 6  |-  ( D  x.  ( 0 ^ 2 ) )  =  ( D  x.  0 )
188mul01d 9567 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( D  x.  0 )  =  0 )
1917, 18syl5eq 2486 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( D  x.  (
0 ^ 2 ) )  =  0 )
2015, 19oveq12d 6108 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( ( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  ( 1  -  0 ) )
21 1m0e1 10431 . . . 4  |-  ( 1  -  0 )  =  1
2220, 21syl6eq 2490 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  1 )
23 oveq1 6097 . . . . . 6  |-  ( a  =  1  ->  (
a  +  ( ( sqr `  D )  x.  b ) )  =  ( 1  +  ( ( sqr `  D
)  x.  b ) ) )
2423eqeq2d 2453 . . . . 5  |-  ( a  =  1  ->  (
1  =  ( a  +  ( ( sqr `  D )  x.  b
) )  <->  1  =  ( 1  +  ( ( sqr `  D
)  x.  b ) ) ) )
25 oveq1 6097 . . . . . . 7  |-  ( a  =  1  ->  (
a ^ 2 )  =  ( 1 ^ 2 ) )
2625oveq1d 6105 . . . . . 6  |-  ( a  =  1  ->  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( 1 ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) ) )
2726eqeq1d 2450 . . . . 5  |-  ( a  =  1  ->  (
( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( 1 ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
2824, 27anbi12d 710 . . . 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 ) ) )
29 oveq2 6098 . . . . . . 7  |-  ( b  =  0  ->  (
( sqr `  D
)  x.  b )  =  ( ( sqr `  D )  x.  0 ) )
3029oveq2d 6106 . . . . . 6  |-  ( b  =  0  ->  (
1  +  ( ( sqr `  D )  x.  b ) )  =  ( 1  +  ( ( sqr `  D
)  x.  0 ) ) )
3130eqeq2d 2453 . . . . 5  |-  ( b  =  0  ->  (
1  =  ( 1  +  ( ( sqr `  D )  x.  b
) )  <->  1  =  ( 1  +  ( ( sqr `  D
)  x.  0 ) ) ) )
32 oveq1 6097 . . . . . . . 8  |-  ( b  =  0  ->  (
b ^ 2 )  =  ( 0 ^ 2 ) )
3332oveq2d 6106 . . . . . . 7  |-  ( b  =  0  ->  ( D  x.  ( b ^ 2 ) )  =  ( D  x.  ( 0 ^ 2 ) ) )
3433oveq2d 6106 . . . . . 6  |-  ( b  =  0  ->  (
( 1 ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( 1 ^ 2 )  -  ( D  x.  (
0 ^ 2 ) ) ) )
3534eqeq1d 2450 . . . . 5  |-  ( b  =  0  ->  (
( ( 1 ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  1 ) )
3631, 35anbi12d 710 . . . 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 ) ) )
3728, 36rspc2ev 3080 . . 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 ) )
384, 6, 13, 22, 37syl112anc 1222 . 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 ) )
39 elpell1qr 29186 . 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 ) ) ) )
402, 38, 39mpbir2and 913 1  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  (Pell1QR `  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2715    \ cdif 3324   ` cfv 5417  (class class class)co 6090   RRcr 9280   0cc0 9281   1c1 9282    + caddc 9284    x. cmul 9286    - cmin 9594   NNcn 10321   2c2 10370   NN0cn0 10578   ^cexp 11864   sqrcsqr 12721  ◻NNcsquarenn 29175  Pell1QRcpell1qr 29176
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-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-pell1qr 29181
This theorem is referenced by:  elpell1qr2  29211
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