Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rmspecsqrnq Structured version   Unicode version

Theorem rmspecsqrnq 29156
Description: The discriminant used to define the X and Y sequences has an irrational square root. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Assertion
Ref Expression
rmspecsqrnq  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  ( CC  \  QQ ) )

Proof of Theorem rmspecsqrnq
StepHypRef Expression
1 eluzelz 10866 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
21zcnd 10744 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  CC )
32sqcld 12002 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  CC )
4 ax-1cn 9336 . . . 4  |-  1  e.  CC
5 subcl 9605 . . . 4  |-  ( ( ( A ^ 2 )  e.  CC  /\  1  e.  CC )  ->  ( ( A ^
2 )  -  1 )  e.  CC )
63, 4, 5sylancl 657 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  CC )
76sqrcld 12919 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  CC )
8 eluz2b2 10923 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( A  e.  NN  /\  1  < 
A ) )
98biimpi 194 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  e.  NN  /\  1  < 
A ) )
109simpld 456 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  NN )
1110nnsqcld 12024 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  NN )
12 nnm1nn0 10617 . . . 4  |-  ( ( A ^ 2 )  e.  NN  ->  (
( A ^ 2 )  -  1 )  e.  NN0 )
1311, 12syl 16 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  NN0 )
14 nnm1nn0 10617 . . . 4  |-  ( A  e.  NN  ->  ( A  -  1 )  e.  NN0 )
1510, 14syl 16 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  -  1 )  e. 
NN0 )
16 eluzelre 10867 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  RR )
1716recnd 9408 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  CC )
18 binom2sub 11979 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 ) ^ 2 )  =  ( ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
1917, 4, 18sylancl 657 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 ) ^ 2 )  =  ( ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
20 2re 10387 . . . . . . . 8  |-  2  e.  RR
21 1re 9381 . . . . . . . . 9  |-  1  e.  RR
22 remulcl 9363 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  x.  1 )  e.  RR )
2316, 21, 22sylancl 657 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  x.  1 )  e.  RR )
24 remulcl 9363 . . . . . . . 8  |-  ( ( 2  e.  RR  /\  ( A  x.  1
)  e.  RR )  ->  ( 2  x.  ( A  x.  1 ) )  e.  RR )
2520, 23, 24sylancr 658 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  e.  RR )
2625recnd 9408 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  e.  CC )
2721resqcli 11947 . . . . . . . 8  |-  ( 1 ^ 2 )  e.  RR
2827recni 9394 . . . . . . 7  |-  ( 1 ^ 2 )  e.  CC
2928a1i 11 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  e.  CC )
303, 26, 29subsubd 9743 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( 2  x.  ( A  x.  1 ) )  -  (
1 ^ 2 ) ) )  =  ( ( ( A ^
2 )  -  (
2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
3119, 30eqtr4d 2476 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 ) ^ 2 )  =  ( ( A ^
2 )  -  (
( 2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) ) ) )
3221a1i 11 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  RR )
33 resubcl 9669 . . . . . 6  |-  ( ( ( 2  x.  ( A  x.  1 ) )  e.  RR  /\  ( 1 ^ 2 )  e.  RR )  ->  ( ( 2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  e.  RR )
3425, 27, 33sylancl 657 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  e.  RR )
3511nnred 10333 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  RR )
3642timesi 10438 . . . . . . . 8  |-  ( 2  x.  1 )  =  ( 1  +  1 )
379simprd 460 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  A )
3820a1i 11 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  2  e.  RR )
39 2pos 10409 . . . . . . . . . . 11  |-  0  <  2
4039a1i 11 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <  2 )
41 ltmul2 10176 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  A  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( 1  < 
A  <->  ( 2  x.  1 )  <  (
2  x.  A ) ) )
4232, 16, 38, 40, 41syl112anc 1217 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  A  <->  ( 2  x.  1 )  < 
( 2  x.  A
) ) )
4337, 42mpbid 210 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  1 )  < 
( 2  x.  A
) )
4436, 43syl5eqbrr 4323 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  1 )  < 
( 2  x.  A
) )
45 remulcl 9363 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  A  e.  RR )  ->  ( 2  x.  A
)  e.  RR )
4620, 16, 45sylancr 658 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  A )  e.  RR )
4732, 32, 46ltaddsubd 9935 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
1  +  1 )  <  ( 2  x.  A )  <->  1  <  ( ( 2  x.  A
)  -  1 ) ) )
4844, 47mpbid 210 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( ( 2  x.  A
)  -  1 ) )
4917mulid1d 9399 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  x.  1 )  =  A )
5049oveq2d 6106 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  =  ( 2  x.  A
) )
51 sq1 11956 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
5251a1i 11 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  =  1 )
5350, 52oveq12d 6108 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  =  ( ( 2  x.  A )  -  1 ) )
5448, 53breqtrrd 4315 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( ( 2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) ) )
5532, 34, 35, 54ltsub2dd 9948 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( 2  x.  ( A  x.  1 ) )  -  (
1 ^ 2 ) ) )  <  (
( A ^ 2 )  -  1 ) )
5631, 55eqbrtrd 4309 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 ) ^ 2 )  < 
( ( A ^
2 )  -  1 ) )
5735ltm1d 10261 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  <  ( A ^ 2 ) )
58 npcan 9615 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  +  1 )  =  A )
5917, 4, 58sylancl 657 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 )  +  1 )  =  A )
6059oveq1d 6105 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A  -  1 )  +  1 ) ^ 2 )  =  ( A ^ 2 ) )
6157, 60breqtrrd 4315 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  <  (
( ( A  - 
1 )  +  1 ) ^ 2 ) )
62 nonsq 13833 . . 3  |-  ( ( ( ( ( A ^ 2 )  - 
1 )  e.  NN0  /\  ( A  -  1 )  e.  NN0 )  /\  ( ( ( A  -  1 ) ^
2 )  <  (
( A ^ 2 )  -  1 )  /\  ( ( A ^ 2 )  - 
1 )  <  (
( ( A  - 
1 )  +  1 ) ^ 2 ) ) )  ->  -.  ( sqr `  ( ( A ^ 2 )  -  1 ) )  e.  QQ )
6313, 15, 56, 61, 62syl22anc 1214 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  -.  ( sqr `  ( ( A ^ 2 )  - 
1 ) )  e.  QQ )
647, 63eldifd 3336 1  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  ( CC  \  QQ ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761    \ cdif 3322   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    < clt 9414    - cmin 9591   NNcn 10318   2c2 10367   NN0cn0 10575   ZZ>=cuz 10857   QQcq 10949   ^cexp 11861   sqrcsqr 12718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687  df-numer 13809  df-denom 13810
This theorem is referenced by:  rmspecnonsq  29157  rmxypairf1o  29161  rmxycomplete  29167  rmxyneg  29170  rmxyadd  29171  rmxy1  29172  rmxy0  29173  jm2.22  29253
  Copyright terms: Public domain W3C validator