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Theorem rmspecsqrnq 26859
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 10452 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
21zcnd 10332 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  CC )
32sqcld 11476 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  CC )
4 ax-1cn 9004 . . . 4  |-  1  e.  CC
5 subcl 9261 . . . 4  |-  ( ( ( A ^ 2 )  e.  CC  /\  1  e.  CC )  ->  ( ( A ^
2 )  -  1 )  e.  CC )
63, 4, 5sylancl 644 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  CC )
76sqrcld 12194 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  CC )
8 eluz2b2 10504 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( A  e.  NN  /\  1  < 
A ) )
98biimpi 187 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  e.  NN  /\  1  < 
A ) )
109simpld 446 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  NN )
1110nnsqcld 11498 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  NN )
12 nnm1nn0 10217 . . . 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 10217 . . . 4  |-  ( A  e.  NN  ->  ( A  -  1 )  e.  NN0 )
1510, 14syl 16 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  -  1 )  e. 
NN0 )
16 eluzelre 10453 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  RR )
1716recnd 9070 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  CC )
18 binom2sub 11453 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 ) ^ 2 )  =  ( ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
1917, 4, 18sylancl 644 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 ) ^ 2 )  =  ( ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
20 2re 10025 . . . . . . . 8  |-  2  e.  RR
21 1re 9046 . . . . . . . . 9  |-  1  e.  RR
22 remulcl 9031 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  x.  1 )  e.  RR )
2316, 21, 22sylancl 644 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  x.  1 )  e.  RR )
24 remulcl 9031 . . . . . . . 8  |-  ( ( 2  e.  RR  /\  ( A  x.  1
)  e.  RR )  ->  ( 2  x.  ( A  x.  1 ) )  e.  RR )
2520, 23, 24sylancr 645 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  e.  RR )
2625recnd 9070 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  e.  CC )
2721resqcli 11422 . . . . . . . 8  |-  ( 1 ^ 2 )  e.  RR
2827recni 9058 . . . . . . 7  |-  ( 1 ^ 2 )  e.  CC
2928a1i 11 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  e.  CC )
303, 26, 29subsubd 9395 . . . . 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 2439 . . . 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 9321 . . . . . 6  |-  ( ( ( 2  x.  ( A  x.  1 ) )  e.  RR  /\  ( 1 ^ 2 )  e.  RR )  ->  ( ( 2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  e.  RR )
3425, 27, 33sylancl 644 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  e.  RR )
3511nnred 9971 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  RR )
3642timesi 10057 . . . . . . . 8  |-  ( 2  x.  1 )  =  ( 1  +  1 )
379simprd 450 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  A )
3820a1i 11 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  2  e.  RR )
39 2pos 10038 . . . . . . . . . . 11  |-  0  <  2
4039a1i 11 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <  2 )
41 ltmul2 9817 . . . . . . . . . 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 1188 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  A  <->  ( 2  x.  1 )  < 
( 2  x.  A
) ) )
4337, 42mpbid 202 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  1 )  < 
( 2  x.  A
) )
4436, 43syl5eqbrr 4206 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  1 )  < 
( 2  x.  A
) )
45 remulcl 9031 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  A  e.  RR )  ->  ( 2  x.  A
)  e.  RR )
4620, 16, 45sylancr 645 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  A )  e.  RR )
4732, 32, 46ltaddsubd 9582 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
1  +  1 )  <  ( 2  x.  A )  <->  1  <  ( ( 2  x.  A
)  -  1 ) ) )
4844, 47mpbid 202 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( ( 2  x.  A
)  -  1 ) )
4917mulid1d 9061 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  x.  1 )  =  A )
5049oveq2d 6056 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  =  ( 2  x.  A
) )
51 sq1 11431 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
5251a1i 11 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  =  1 )
5350, 52oveq12d 6058 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  =  ( ( 2  x.  A )  -  1 ) )
5448, 53breqtrrd 4198 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( ( 2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) ) )
5532, 34, 35, 54ltsub2dd 9595 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( 2  x.  ( A  x.  1 ) )  -  (
1 ^ 2 ) ) )  <  (
( A ^ 2 )  -  1 ) )
5631, 55eqbrtrd 4192 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 ) ^ 2 )  < 
( ( A ^
2 )  -  1 ) )
5735ltm1d 9899 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  <  ( A ^ 2 ) )
58 npcan 9270 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  +  1 )  =  A )
5917, 4, 58sylancl 644 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 )  +  1 )  =  A )
6059oveq1d 6055 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A  -  1 )  +  1 ) ^ 2 )  =  ( A ^ 2 ) )
6157, 60breqtrrd 4198 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  <  (
( ( A  - 
1 )  +  1 ) ^ 2 ) )
62 nonsq 13106 . . 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 1185 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  -.  ( sqr `  ( ( A ^ 2 )  - 
1 ) )  e.  QQ )
647, 63eldifd 3291 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    \ cdif 3277   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    - cmin 9247   NNcn 9956   2c2 10005   NN0cn0 10177   ZZ>=cuz 10444   QQcq 10530   ^cexp 11337   sqrcsqr 11993
This theorem is referenced by:  rmspecnonsq  26860  rmxypairf1o  26864  rmxycomplete  26870  rmxyneg  26873  rmxyadd  26874  rmxy1  26875  rmxy0  26876  jm2.22  26956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-numer 13082  df-denom 13083
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