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Theorem rmspecsqrnq 26991
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 10238 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
21zcnd 10118 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  CC )
32sqcld 11243 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  CC )
4 ax-1cn 8795 . . . 4  |-  1  e.  CC
5 subcl 9051 . . . 4  |-  ( ( ( A ^ 2 )  e.  CC  /\  1  e.  CC )  ->  ( ( A ^
2 )  -  1 )  e.  CC )
63, 4, 5sylancl 643 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  CC )
76sqrcld 11919 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  CC )
8 eluz2b2 10290 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( A  e.  NN  /\  1  < 
A ) )
98biimpi 186 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  e.  NN  /\  1  < 
A ) )
109simpld 445 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  NN )
1110nnsqcld 11265 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  NN )
12 nnm1nn0 10005 . . . 4  |-  ( ( A ^ 2 )  e.  NN  ->  (
( A ^ 2 )  -  1 )  e.  NN0 )
1311, 12syl 15 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  NN0 )
14 nnm1nn0 10005 . . . 4  |-  ( A  e.  NN  ->  ( A  -  1 )  e.  NN0 )
1510, 14syl 15 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  -  1 )  e. 
NN0 )
16 eluzelre 10239 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  RR )
1716recnd 8861 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  CC )
18 binom2sub 11220 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 ) ^ 2 )  =  ( ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
1917, 4, 18sylancl 643 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 ) ^ 2 )  =  ( ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
20 2re 9815 . . . . . . . 8  |-  2  e.  RR
21 1re 8837 . . . . . . . . 9  |-  1  e.  RR
22 remulcl 8822 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  x.  1 )  e.  RR )
2316, 21, 22sylancl 643 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  x.  1 )  e.  RR )
24 remulcl 8822 . . . . . . . 8  |-  ( ( 2  e.  RR  /\  ( A  x.  1
)  e.  RR )  ->  ( 2  x.  ( A  x.  1 ) )  e.  RR )
2520, 23, 24sylancr 644 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  e.  RR )
2625recnd 8861 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  e.  CC )
2721resqcli 11189 . . . . . . . 8  |-  ( 1 ^ 2 )  e.  RR
2827recni 8849 . . . . . . 7  |-  ( 1 ^ 2 )  e.  CC
2928a1i 10 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  e.  CC )
303, 26, 29subsubd 9185 . . . . 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 2318 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 ) ^ 2 )  =  ( ( A ^
2 )  -  (
( 2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) ) ) )
3221a1i 10 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  RR )
33 resubcl 9111 . . . . . 6  |-  ( ( ( 2  x.  ( A  x.  1 ) )  e.  RR  /\  ( 1 ^ 2 )  e.  RR )  ->  ( ( 2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  e.  RR )
3425, 27, 33sylancl 643 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  e.  RR )
3511nnred 9761 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  RR )
3642timesi 9845 . . . . . . . 8  |-  ( 2  x.  1 )  =  ( 1  +  1 )
379simprd 449 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  A )
3820a1i 10 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  2  e.  RR )
39 2pos 9828 . . . . . . . . . . 11  |-  0  <  2
4039a1i 10 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <  2 )
41 ltmul2 9607 . . . . . . . . . 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 1186 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  A  <->  ( 2  x.  1 )  < 
( 2  x.  A
) ) )
4337, 42mpbid 201 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  1 )  < 
( 2  x.  A
) )
4436, 43syl5eqbrr 4057 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  1 )  < 
( 2  x.  A
) )
45 remulcl 8822 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  A  e.  RR )  ->  ( 2  x.  A
)  e.  RR )
4620, 16, 45sylancr 644 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  A )  e.  RR )
4732, 32, 46ltaddsubd 9372 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
1  +  1 )  <  ( 2  x.  A )  <->  1  <  ( ( 2  x.  A
)  -  1 ) ) )
4844, 47mpbid 201 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( ( 2  x.  A
)  -  1 ) )
4917mulid1d 8852 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  x.  1 )  =  A )
5049oveq2d 5874 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  =  ( 2  x.  A
) )
51 sq1 11198 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
5251a1i 10 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  =  1 )
5350, 52oveq12d 5876 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  =  ( ( 2  x.  A )  -  1 ) )
5448, 53breqtrrd 4049 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( ( 2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) ) )
5532, 34, 35, 54ltsub2dd 9385 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( 2  x.  ( A  x.  1 ) )  -  (
1 ^ 2 ) ) )  <  (
( A ^ 2 )  -  1 ) )
5631, 55eqbrtrd 4043 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 ) ^ 2 )  < 
( ( A ^
2 )  -  1 ) )
5735ltm1d 9689 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  <  ( A ^ 2 ) )
58 npcan 9060 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  +  1 )  =  A )
5917, 4, 58sylancl 643 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 )  +  1 )  =  A )
6059oveq1d 5873 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A  -  1 )  +  1 ) ^ 2 )  =  ( A ^ 2 ) )
6157, 60breqtrrd 4049 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  <  (
( ( A  - 
1 )  +  1 ) ^ 2 ) )
62 nonsq 12830 . . 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 1183 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  -.  ( sqr `  ( ( A ^ 2 )  - 
1 ) )  e.  QQ )
64 eldif 3162 . 2  |-  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  e.  ( CC  \  QQ )  <->  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  CC  /\ 
-.  ( sqr `  (
( A ^ 2 )  -  1 ) )  e.  QQ ) )
657, 63, 64sylanbrc 645 1  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  ( CC  \  QQ ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    - cmin 9037   NNcn 9746   2c2 9795   NN0cn0 9965   ZZ>=cuz 10230   QQcq 10316   ^cexp 11104   sqrcsqr 11718
This theorem is referenced by:  rmspecnonsq  26992  rmxypairf1o  26996  rmxycomplete  27002  rmxyneg  27005  rmxyadd  27006  rmxy1  27007  rmxy0  27008  jm2.22  27088
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-1st 6122  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-q 10317  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-numer 12806  df-denom 12807
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