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Theorem rmspecsqrnq 26969
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 10496 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
21zcnd 10376 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  CC )
32sqcld 11521 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  CC )
4 ax-1cn 9048 . . . 4  |-  1  e.  CC
5 subcl 9305 . . . 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 12239 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  CC )
8 eluz2b2 10548 . . . . . . 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 11543 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  NN )
12 nnm1nn0 10261 . . . 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 10261 . . . 4  |-  ( A  e.  NN  ->  ( A  -  1 )  e.  NN0 )
1510, 14syl 16 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  -  1 )  e. 
NN0 )
16 eluzelre 10497 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  RR )
1716recnd 9114 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  CC )
18 binom2sub 11498 . . . . . 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 10069 . . . . . . . 8  |-  2  e.  RR
21 1re 9090 . . . . . . . . 9  |-  1  e.  RR
22 remulcl 9075 . . . . . . . . 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 9075 . . . . . . . 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 9114 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  e.  CC )
2721resqcli 11467 . . . . . . . 8  |-  ( 1 ^ 2 )  e.  RR
2827recni 9102 . . . . . . 7  |-  ( 1 ^ 2 )  e.  CC
2928a1i 11 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  e.  CC )
303, 26, 29subsubd 9439 . . . . 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 2471 . . . 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 9365 . . . . . 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 10015 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  RR )
3642timesi 10101 . . . . . . . 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 10082 . . . . . . . . . . 11  |-  0  <  2
4039a1i 11 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <  2 )
41 ltmul2 9861 . . . . . . . . . 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 4246 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  1 )  < 
( 2  x.  A
) )
45 remulcl 9075 . . . . . . . . 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 9626 . . . . . . 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 9105 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  x.  1 )  =  A )
5049oveq2d 6097 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 2  x.  ( A  x.  1 ) )  =  ( 2  x.  A
) )
51 sq1 11476 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
5251a1i 11 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  =  1 )
5350, 52oveq12d 6099 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) )  =  ( ( 2  x.  A )  -  1 ) )
5448, 53breqtrrd 4238 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( ( 2  x.  ( A  x.  1 ) )  -  ( 1 ^ 2 ) ) )
5532, 34, 35, 54ltsub2dd 9639 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( 2  x.  ( A  x.  1 ) )  -  (
1 ^ 2 ) ) )  <  (
( A ^ 2 )  -  1 ) )
5631, 55eqbrtrd 4232 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  -  1 ) ^ 2 )  < 
( ( A ^
2 )  -  1 ) )
5735ltm1d 9943 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  <  ( A ^ 2 ) )
58 npcan 9314 . . . . . 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 6096 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A  -  1 )  +  1 ) ^ 2 )  =  ( A ^ 2 ) )
6157, 60breqtrrd 4238 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  <  (
( ( A  - 
1 )  +  1 ) ^ 2 ) )
62 nonsq 13151 . . 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 3331 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 1652    e. wcel 1725    \ cdif 3317   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    - cmin 9291   NNcn 10000   2c2 10049   NN0cn0 10221   ZZ>=cuz 10488   QQcq 10574   ^cexp 11382   sqrcsqr 12038
This theorem is referenced by:  rmspecnonsq  26970  rmxypairf1o  26974  rmxycomplete  26980  rmxyneg  26983  rmxyadd  26984  rmxy1  26985  rmxy0  26986  jm2.22  27066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-numer 13127  df-denom 13128
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