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Theorem rmspecsqrnq 29247
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 10870 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
21zcnd 10748 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  CC )
32sqcld 12006 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  CC )
4 ax-1cn 9340 . . . 4  |-  1  e.  CC
5 subcl 9609 . . . 4  |-  ( ( ( A ^ 2 )  e.  CC  /\  1  e.  CC )  ->  ( ( A ^
2 )  -  1 )  e.  CC )
63, 4, 5sylancl 662 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  CC )
76sqrcld 12923 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  CC )
8 eluz2b2 10927 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( A  e.  NN  /\  1  < 
A ) )
98biimpi 194 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  e.  NN  /\  1  < 
A ) )
109simpld 459 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  NN )
1110nnsqcld 12028 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  NN )
12 nnm1nn0 10621 . . . 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 10621 . . . 4  |-  ( A  e.  NN  ->  ( A  -  1 )  e.  NN0 )
1510, 14syl 16 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  -  1 )  e. 
NN0 )
16 eluzelre 10871 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  RR )
1716recnd 9412 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  CC )
18 binom2sub 11983 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 ) ^ 2 )  =  ( ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
1917, 4, 18sylancl 662 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 ) ^ 2 )  =  ( ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
20 2re 10391 . . . . . . . 8  |-  2  e.  RR
21 1re 9385 . . . . . . . . 9  |-  1  e.  RR
22 remulcl 9367 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  x.  1 )  e.  RR )
2316, 21, 22sylancl 662 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  x.  1 )  e.  RR )
24 remulcl 9367 . . . . . . . 8  |-  ( ( 2  e.  RR  /\  ( A  x.  1
)  e.  RR )  ->  ( 2  x.  ( A  x.  1 ) )  e.  RR )
2520, 23, 24sylancr 663 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  e.  RR )
2625recnd 9412 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  e.  CC )
2721resqcli 11951 . . . . . . . 8  |-  ( 1 ^ 2 )  e.  RR
2827recni 9398 . . . . . . 7  |-  ( 1 ^ 2 )  e.  CC
2928a1i 11 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  e.  CC )
303, 26, 29subsubd 9747 . . . . 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 2478 . . . 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 9673 . . . . . 6  |-  ( ( ( 2  x.  ( A  x.  1 ) )  e.  RR  /\  ( 1 ^ 2 )  e.  RR )  ->  ( ( 2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  e.  RR )
3425, 27, 33sylancl 662 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  e.  RR )
3511nnred 10337 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  RR )
3642timesi 10442 . . . . . . . 8  |-  ( 2  x.  1 )  =  ( 1  +  1 )
379simprd 463 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  A )
3820a1i 11 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  2  e.  RR )
39 2pos 10413 . . . . . . . . . . 11  |-  0  <  2
4039a1i 11 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <  2 )
41 ltmul2 10180 . . . . . . . . . 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 1222 . . . . . . . . 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 4326 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  1 )  < 
( 2  x.  A
) )
45 remulcl 9367 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  A  e.  RR )  ->  ( 2  x.  A
)  e.  RR )
4620, 16, 45sylancr 663 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  A )  e.  RR )
4732, 32, 46ltaddsubd 9939 . . . . . . 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 9403 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  x.  1 )  =  A )
5049oveq2d 6107 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  =  ( 2  x.  A
) )
51 sq1 11960 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
5251a1i 11 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  =  1 )
5350, 52oveq12d 6109 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  =  ( ( 2  x.  A )  -  1 ) )
5448, 53breqtrrd 4318 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( ( 2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) ) )
5532, 34, 35, 54ltsub2dd 9952 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( 2  x.  ( A  x.  1 ) )  -  (
1 ^ 2 ) ) )  <  (
( A ^ 2 )  -  1 ) )
5631, 55eqbrtrd 4312 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 ) ^ 2 )  < 
( ( A ^
2 )  -  1 ) )
5735ltm1d 10265 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  <  ( A ^ 2 ) )
58 npcan 9619 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  +  1 )  =  A )
5917, 4, 58sylancl 662 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 )  +  1 )  =  A )
6059oveq1d 6106 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A  -  1 )  +  1 ) ^ 2 )  =  ( A ^ 2 ) )
6157, 60breqtrrd 4318 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  <  (
( ( A  - 
1 )  +  1 ) ^ 2 ) )
62 nonsq 13837 . . 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 1219 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  -.  ( sqr `  ( ( A ^ 2 )  - 
1 ) )  e.  QQ )
647, 63eldifd 3339 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 1369    e. wcel 1756    \ cdif 3325   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    < clt 9418    - cmin 9595   NNcn 10322   2c2 10371   NN0cn0 10579   ZZ>=cuz 10861   QQcq 10953   ^cexp 11865   sqrcsqr 12722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536  df-gcd 13691  df-numer 13813  df-denom 13814
This theorem is referenced by:  rmspecnonsq  29248  rmxypairf1o  29252  rmxycomplete  29258  rmxyneg  29261  rmxyadd  29262  rmxy1  29263  rmxy0  29264  jm2.22  29344
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