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Theorem pell1qr1 30969
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 1red 9628 . 2  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  RR )
2 1nn0 10832 . . . 4  |-  1  e.  NN0
32a1i 11 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  NN0 )
4 0nn0 10831 . . . 4  |-  0  e.  NN0
54a1i 11 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
0  e.  NN0 )
6 eldifi 3622 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
76nncnd 10572 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
87sqrtcld 13279 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( sqr `  D
)  e.  CC )
98mul01d 9796 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( ( sqr `  D
)  x.  0 )  =  0 )
109oveq2d 6312 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( 1  +  ( ( sqr `  D
)  x.  0 ) )  =  ( 1  +  0 ) )
11 1p0e1 10669 . . . 4  |-  ( 1  +  0 )  =  1
1210, 11syl6req 2515 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
1  =  ( 1  +  ( ( sqr `  D )  x.  0 ) ) )
13 sq1 12264 . . . . . 6  |-  ( 1 ^ 2 )  =  1
1413a1i 11 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( 1 ^ 2 )  =  1 )
15 sq0 12261 . . . . . . 7  |-  ( 0 ^ 2 )  =  0
1615oveq2i 6307 . . . . . 6  |-  ( D  x.  ( 0 ^ 2 ) )  =  ( D  x.  0 )
177mul01d 9796 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( D  x.  0 )  =  0 )
1816, 17syl5eq 2510 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( D  x.  (
0 ^ 2 ) )  =  0 )
1914, 18oveq12d 6314 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( ( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  ( 1  -  0 ) )
20 1m0e1 10667 . . . 4  |-  ( 1  -  0 )  =  1
2119, 20syl6eq 2514 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  1 )
22 oveq1 6303 . . . . . 6  |-  ( a  =  1  ->  (
a  +  ( ( sqr `  D )  x.  b ) )  =  ( 1  +  ( ( sqr `  D
)  x.  b ) ) )
2322eqeq2d 2471 . . . . 5  |-  ( a  =  1  ->  (
1  =  ( a  +  ( ( sqr `  D )  x.  b
) )  <->  1  =  ( 1  +  ( ( sqr `  D
)  x.  b ) ) ) )
24 oveq1 6303 . . . . . . 7  |-  ( a  =  1  ->  (
a ^ 2 )  =  ( 1 ^ 2 ) )
2524oveq1d 6311 . . . . . 6  |-  ( a  =  1  ->  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( 1 ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) ) )
2625eqeq1d 2459 . . . . 5  |-  ( a  =  1  ->  (
( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( 1 ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
2723, 26anbi12d 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 ) ) )
28 oveq2 6304 . . . . . . 7  |-  ( b  =  0  ->  (
( sqr `  D
)  x.  b )  =  ( ( sqr `  D )  x.  0 ) )
2928oveq2d 6312 . . . . . 6  |-  ( b  =  0  ->  (
1  +  ( ( sqr `  D )  x.  b ) )  =  ( 1  +  ( ( sqr `  D
)  x.  0 ) ) )
3029eqeq2d 2471 . . . . 5  |-  ( b  =  0  ->  (
1  =  ( 1  +  ( ( sqr `  D )  x.  b
) )  <->  1  =  ( 1  +  ( ( sqr `  D
)  x.  0 ) ) ) )
31 oveq1 6303 . . . . . . . 8  |-  ( b  =  0  ->  (
b ^ 2 )  =  ( 0 ^ 2 ) )
3231oveq2d 6312 . . . . . . 7  |-  ( b  =  0  ->  ( D  x.  ( b ^ 2 ) )  =  ( D  x.  ( 0 ^ 2 ) ) )
3332oveq2d 6312 . . . . . 6  |-  ( b  =  0  ->  (
( 1 ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( 1 ^ 2 )  -  ( D  x.  (
0 ^ 2 ) ) ) )
3433eqeq1d 2459 . . . . 5  |-  ( b  =  0  ->  (
( ( 1 ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( 1 ^ 2 )  -  ( D  x.  ( 0 ^ 2 ) ) )  =  1 ) )
3530, 34anbi12d 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 ) ) )
3627, 35rspc2ev 3221 . . 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 ) )
373, 5, 12, 21, 36syl112anc 1232 . 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 ) )
38 elpell1qr 30945 . 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 ) ) ) )
391, 37, 38mpbir2and 922 1  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  (Pell1QR `  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808    \ cdif 3468   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    - cmin 9824   NNcn 10556   2c2 10606   NN0cn0 10816   ^cexp 12168   sqrcsqrt 13077  ◻NNcsquarenn 30934  Pell1QRcpell1qr 30935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-pell1qr 30940
This theorem is referenced by:  elpell1qr2  30970
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