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Theorem pell1qr1 37435
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 10055 . 2 |- ( D e. ( NN \ ◻NN ) -> 1 e. RR )
2 1nn0 11308 . . . 4 |- 1 e. NN0
32a1i 11 . . 3 |- ( D e. ( NN \ ◻NN ) -> 1 e. NN0 )
4 0nn0 11307 . . . 4 |- 0 e. NN0
54a1i 11 . . 3 |- ( D e. ( NN \ ◻NN ) -> 0 e. NN0 )
6 eldifi 3732 . . . . . . . 8 |- ( D e. ( NN \ ◻NN ) -> D e. NN )
76nncnd 11036 . . . . . . 7 |- ( D e. ( NN \ ◻NN ) -> D e. CC )
87sqrtcld 14176 . . . . . 6 |- ( D e. ( NN \ ◻NN ) -> ( sqr ` D ) e. CC )
98mul01d 10235 . . . . 5 |- ( D e. ( NN \ ◻NN ) -> ( ( sqr ` D ) x. 0 ) = 0 )
109oveq2d 6666 . . . 4 |- ( D e. ( NN \ ◻NN ) -> ( 1 + ( ( sqr ` D ) x. 0 ) ) = ( 1 + 0 ) )
11 1p0e1 11133 . . . 4 |- ( 1 + 0 ) = 1
1210, 11syl6req 2673 . . 3 |- ( D e. ( NN \ ◻NN ) -> 1 = ( 1 + ( ( sqr ` D ) x. 0 ) ) )
13 sq1 12958 . . . . . 6 |- ( 1 ^ 2 ) = 1
1413a1i 11 . . . . 5 |- ( D e. ( NN \ ◻NN ) -> ( 1 ^ 2 ) = 1 )
15 sq0 12955 . . . . . . 7 |- ( 0 ^ 2 ) = 0
1615oveq2i 6661 . . . . . 6 |- ( D x. ( 0 ^ 2 ) ) = ( D x. 0 )
177mul01d 10235 . . . . . 6 |- ( D e. ( NN \ ◻NN ) -> ( D x. 0 ) = 0 )
1816, 17syl5eq 2668 . . . . 5 |- ( D e. ( NN \ ◻NN ) -> ( D x. ( 0 ^ 2 ) ) = 0 )
1914, 18oveq12d 6668 . . . 4 |- ( D e. ( NN \ ◻NN ) -> ( ( 1 ^ 2 ) - ( D x. ( 0 ^ 2 ) ) ) = ( 1 - 0 ) )
20 1m0e1 11131 . . . 4 |- ( 1 - 0 ) = 1
2119, 20syl6eq 2672 . . 3 |- ( D e. ( NN \ ◻NN ) -> ( ( 1 ^ 2 ) - ( D x. ( 0 ^ 2 ) ) ) = 1 )
22 oveq1 6657 . . . . . 6 |- ( a = 1 -> ( a + ( ( sqr ` D ) x. b ) ) = ( 1 + ( ( sqr ` D ) x. b ) ) )
2322eqeq2d 2632 . . . . 5 |- ( a = 1 -> ( 1 = ( a + ( ( sqr ` D ) x. b ) ) <-> 1 = ( 1 + ( ( sqr ` D ) x. b ) ) ) )
24 oveq1 6657 . . . . . . 7 |- ( a = 1 -> ( a ^ 2 ) = ( 1 ^ 2 ) )
2524oveq1d 6665 . . . . . 6 |- ( a = 1 -> ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = ( ( 1 ^ 2 ) - ( D x. ( b ^ 2 ) ) ) )
2625eqeq1d 2624 . . . . 5 |- ( a = 1 -> ( ( ( a ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 <-> ( ( 1 ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 ) )
2723, 26anbi12d 747 . . . 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 6658 . . . . . . 7 |- ( b = 0 -> ( ( sqr ` D ) x. b ) = ( ( sqr ` D ) x. 0 ) )
2928oveq2d 6666 . . . . . 6 |- ( b = 0 -> ( 1 + ( ( sqr ` D ) x. b ) ) = ( 1 + ( ( sqr ` D ) x. 0 ) ) )
3029eqeq2d 2632 . . . . 5 |- ( b = 0 -> ( 1 = ( 1 + ( ( sqr ` D ) x. b ) ) <-> 1 = ( 1 + ( ( sqr ` D ) x. 0 ) ) ) )
31 oveq1 6657 . . . . . . . 8 |- ( b = 0 -> ( b ^ 2 ) = ( 0 ^ 2 ) )
3231oveq2d 6666 . . . . . . 7 |- ( b = 0 -> ( D x. ( b ^ 2 ) ) = ( D x. ( 0 ^ 2 ) ) )
3332oveq2d 6666 . . . . . 6 |- ( b = 0 -> ( ( 1 ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = ( ( 1 ^ 2 ) - ( D x. ( 0 ^ 2 ) ) ) )
3433eqeq1d 2624 . . . . 5 |- ( b = 0 -> ( ( ( 1 ^ 2 ) - ( D x. ( b ^ 2 ) ) ) = 1 <-> ( ( 1 ^ 2 ) - ( D x. ( 0 ^ 2 ) ) ) = 1 ) )
3530, 34anbi12d 747 . . . 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 3324 . . 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 1330 . 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 37411 . 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 957 1 |- ( D e. ( NN \ ◻NN ) -> 1 e. (Pell1QR ` D ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ^cexp 12860   sqrcsqrt 13973  ◻NNcsquarenn 37400  Pell1QRcpell1qr 37401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-pell1qr 37406
This theorem is referenced by:  elpell1qr2  37436
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