Theorem pell1234qrval 37414

Description: Value of the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)

Assertion

Ref	Expression
pell1234qrval	|- ( D e. ( NN \ ◻NN ) -> (Pell1234QR ` D ) = { y e. RR | E. z e. ZZ E. w e. ZZ ( y = ( z + ( ( sqr ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) } )

Distinct variable group:   y, z, w, D

Proof of Theorem pell1234qrval

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . . 8 |- ( d = D -> ( sqr ` d ) = ( sqr ` D ) )
2	1	oveq1d 6665	. . . . . . 7 |- ( d = D -> ( ( sqr ` d ) x. w ) = ( ( sqr ` D ) x. w ) )
3	2	oveq2d 6666	. . . . . 6 |- ( d = D -> ( z + ( ( sqr ` d ) x. w ) ) = ( z + ( ( sqr ` D ) x. w ) ) )
4	3	eqeq2d 2632	. . . . 5 |- ( d = D -> ( y = ( z + ( ( sqr ` d ) x. w ) ) <-> y = ( z + ( ( sqr ` D ) x. w ) ) ) )
5		oveq1 6657	. . . . . . 7 |- ( d = D -> ( d x. ( w ^ 2 ) ) = ( D x. ( w ^ 2 ) ) )
6	5	oveq2d 6666	. . . . . 6 |- ( d = D -> ( ( z ^ 2 ) - ( d x. ( w ^ 2 ) ) ) = ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) )
7	6	eqeq1d 2624	. . . . 5 |- ( d = D -> ( ( ( z ^ 2 ) - ( d x. ( w ^ 2 ) ) ) = 1 <-> ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) )
8	4, 7	anbi12d 747	. . . 4 |- ( d = D -> ( ( y = ( z + ( ( sqr ` d ) x. w ) ) /\ ( ( z ^ 2 ) - ( d x. ( w ^ 2 ) ) ) = 1 ) <-> ( y = ( z + ( ( sqr ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) ) )
9	8	2rexbidv 3057	. . 3 |- ( d = D -> ( E. z e. ZZ E. w e. ZZ ( y = ( z + ( ( sqr ` d ) x. w ) ) /\ ( ( z ^ 2 ) - ( d x. ( w ^ 2 ) ) ) = 1 ) <-> E. z e. ZZ E. w e. ZZ ( y = ( z + ( ( sqr ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) ) )
10	9	rabbidv 3189	. 2 |- ( d = D -> { y e. RR | E. z e. ZZ E. w e. ZZ ( y = ( z + ( ( sqr ` d ) x. w ) ) /\ ( ( z ^ 2 ) - ( d x. ( w ^ 2 ) ) ) = 1 ) } = { y e. RR | E. z e. ZZ E. w e. ZZ ( y = ( z + ( ( sqr ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) } )
11		df-pell1234qr 37408	. 2 |- Pell1234QR = ( d e. ( NN \ ◻NN ) |-> { y e. RR | E. z e. ZZ E. w e. ZZ ( y = ( z + ( ( sqr ` d ) x. w ) ) /\ ( ( z ^ 2 ) - ( d x. ( w ^ 2 ) ) ) = 1 ) } )
12		reex 10027	. . 3 |- RR e. _V
13	12	rabex 4813	. 2 |- { y e. RR | E. z e. ZZ E. w e. ZZ ( y = ( z + ( ( sqr ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) } e. _V
14	10, 11, 13	fvmpt 6282	1 |- ( D e. ( NN \ ◻NN ) -> (Pell1234QR ` D ) = { y e. RR | E. z e. ZZ E. w e. ZZ ( y = ( z + ( ( sqr ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   \ cdif 3571   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   2c2 11070   ZZcz 11377   ^cexp 12860   sqrcsqrt 13973  ◻_NNcsquarenn 37400  Pell1234QRcpell1234qr 37402

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-pell1234qr 37408

This theorem is referenced by:  elpell1234qr  37415