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Theorem pell1qr1 26368
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 8837 . . 3  |-  1  e.  RR
21a1i 10 . 2  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  RR )
3 1nn0 9981 . . . 4  |-  1  e.  NN0
43a1i 10 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  NN0 )
5 0nn0 9980 . . . 4  |-  0  e.  NN0
65a1i 10 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
0  e.  NN0 )
7 eldifi 3298 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
87nncnd 9762 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
98sqrcld 11919 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( sqr `  D
)  e.  CC )
109mul01d 9011 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( ( sqr `  D
)  x.  0 )  =  0 )
1110oveq2d 5874 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( 1  +  ( ( sqr `  D
)  x.  0 ) )  =  ( 1  +  0 ) )
12 ax-1cn 8795 . . . . 5  |-  1  e.  CC
1312addid1i 8999 . . . 4  |-  ( 1  +  0 )  =  1
1411, 13syl6req 2332 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
1  =  ( 1  +  ( ( sqr `  D )  x.  0 ) ) )
15 sq1 11198 . . . . . 6  |-  ( 1 ^ 2 )  =  1
1615a1i 10 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( 1 ^ 2 )  =  1 )
17 sq0 11195 . . . . . . 7  |-  ( 0 ^ 2 )  =  0
1817oveq2i 5869 . . . . . 6  |-  ( D  x.  ( 0 ^ 2 ) )  =  ( D  x.  0 )
198mul01d 9011 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( D  x.  0 )  =  0 )
2018, 19syl5eq 2327 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( D  x.  (
0 ^ 2 ) )  =  0 )
2116, 20oveq12d 5876 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( ( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  ( 1  -  0 ) )
2212subid1i 9118 . . . 4  |-  ( 1  -  0 )  =  1
2321, 22syl6eq 2331 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  1 )
24 oveq1 5865 . . . . . 6  |-  ( a  =  1  ->  (
a  +  ( ( sqr `  D )  x.  b ) )  =  ( 1  +  ( ( sqr `  D
)  x.  b ) ) )
2524eqeq2d 2294 . . . . 5  |-  ( a  =  1  ->  (
1  =  ( a  +  ( ( sqr `  D )  x.  b
) )  <->  1  =  ( 1  +  ( ( sqr `  D
)  x.  b ) ) ) )
26 oveq1 5865 . . . . . . 7  |-  ( a  =  1  ->  (
a ^ 2 )  =  ( 1 ^ 2 ) )
2726oveq1d 5873 . . . . . 6  |-  ( a  =  1  ->  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( 1 ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) ) )
2827eqeq1d 2291 . . . . 5  |-  ( a  =  1  ->  (
( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( 1 ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
2925, 28anbi12d 691 . . . 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 5866 . . . . . . 7  |-  ( b  =  0  ->  (
( sqr `  D
)  x.  b )  =  ( ( sqr `  D )  x.  0 ) )
3130oveq2d 5874 . . . . . 6  |-  ( b  =  0  ->  (
1  +  ( ( sqr `  D )  x.  b ) )  =  ( 1  +  ( ( sqr `  D
)  x.  0 ) ) )
3231eqeq2d 2294 . . . . 5  |-  ( b  =  0  ->  (
1  =  ( 1  +  ( ( sqr `  D )  x.  b
) )  <->  1  =  ( 1  +  ( ( sqr `  D
)  x.  0 ) ) ) )
33 oveq1 5865 . . . . . . . 8  |-  ( b  =  0  ->  (
b ^ 2 )  =  ( 0 ^ 2 ) )
3433oveq2d 5874 . . . . . . 7  |-  ( b  =  0  ->  ( D  x.  ( b ^ 2 ) )  =  ( D  x.  ( 0 ^ 2 ) ) )
3534oveq2d 5874 . . . . . 6  |-  ( b  =  0  ->  (
( 1 ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( 1 ^ 2 )  -  ( D  x.  (
0 ^ 2 ) ) ) )
3635eqeq1d 2291 . . . . 5  |-  ( b  =  0  ->  (
( ( 1 ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  1 ) )
3732, 36anbi12d 691 . . . 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 2892 . . 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 1186 . 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 26344 . 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 888 1  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  (Pell1QR `  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    \ cdif 3149   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   2c2 9795   NN0cn0 9965   ^cexp 11104   sqrcsqr 11718  ◻NNcsquarenn 26333  Pell1QRcpell1qr 26334
This theorem is referenced by:  elpell1qr2  26369
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-abs 11721  df-pell1qr 26339
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