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Theorem nonsq 11459
Description: Any integer strictly between two adjacent squares has a non-rational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.)
Assertion
Ref Expression
nonsq  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  -.  ( sqr `  A )  e.  QQ )

Proof of Theorem nonsq
StepHypRef Expression
1 nn0z 8768 . . . 4  |-  ( B  e.  NN0  ->  B  e.  ZZ )
21ad2antlr 473 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  B  e.  ZZ )
3 simprl 498 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( B ^ 2 )  < 
A )
4 simpll 496 . . . . . . 7  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  A  e.  NN0 )
54nn0red 8725 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  A  e.  RR )
64nn0ge0d 8727 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  0  <_  A )
7 resqrtth 10460 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
85, 6, 7syl2anc 403 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( ( sqr `  A ) ^
2 )  =  A )
93, 8breqtrrd 3871 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( B ^ 2 )  < 
( ( sqr `  A
) ^ 2 ) )
10 simplr 497 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  B  e.  NN0 )
1110nn0red 8725 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  B  e.  RR )
12 nn0re 8680 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  RR )
1312ad2antrr 472 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  A  e.  RR )
1413, 6resqrtcld 10592 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( sqr `  A )  e.  RR )
1510nn0ge0d 8727 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  0  <_  B )
1613, 6sqrtge0d 10595 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  0  <_  ( sqr `  A ) )
1711, 14, 15, 16lt2sqd 10113 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( B  <  ( sqr `  A
)  <->  ( B ^
2 )  <  (
( sqr `  A
) ^ 2 ) ) )
189, 17mpbird 165 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  B  <  ( sqr `  A ) )
19 simprr 499 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  A  <  ( ( B  +  1 ) ^ 2 ) )
208, 19eqbrtrd 3865 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( ( sqr `  A ) ^
2 )  <  (
( B  +  1 ) ^ 2 ) )
21 peano2re 7616 . . . . . 6  |-  ( B  e.  RR  ->  ( B  +  1 )  e.  RR )
2211, 21syl 14 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( B  +  1 )  e.  RR )
23 peano2nn0 8711 . . . . . . 7  |-  ( B  e.  NN0  ->  ( B  +  1 )  e. 
NN0 )
2423ad2antlr 473 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( B  +  1 )  e. 
NN0 )
2524nn0ge0d 8727 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  0  <_  ( B  +  1 ) )
2614, 22, 16, 25lt2sqd 10113 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( ( sqr `  A )  < 
( B  +  1 )  <->  ( ( sqr `  A ) ^ 2 )  <  ( ( B  +  1 ) ^ 2 ) ) )
2720, 26mpbird 165 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( sqr `  A )  <  ( B  +  1 ) )
28 btwnnz 8838 . . 3  |-  ( ( B  e.  ZZ  /\  B  <  ( sqr `  A
)  /\  ( sqr `  A )  <  ( B  +  1 ) )  ->  -.  ( sqr `  A )  e.  ZZ )
292, 18, 27, 28syl3anc 1174 . 2  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  -.  ( sqr `  A )  e.  ZZ )
30 nn0sqrtelqelz 11458 . . . 4  |-  ( ( A  e.  NN0  /\  ( sqr `  A )  e.  QQ )  -> 
( sqr `  A
)  e.  ZZ )
3130ex 113 . . 3  |-  ( A  e.  NN0  ->  ( ( sqr `  A )  e.  QQ  ->  ( sqr `  A )  e.  ZZ ) )
3231ad2antrr 472 . 2  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( ( sqr `  A )  e.  QQ  ->  ( sqr `  A )  e.  ZZ ) )
3329, 32mtod 624 1  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  -.  ( sqr `  A )  e.  QQ )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   RRcr 7347   0cc0 7348   1c1 7349    + caddc 7351    < clt 7520    <_ cle 7521   2c2 8471   NN0cn0 8671   ZZcz 8748   QQcq 9102   ^cexp 9950   sqrcsqrt 10425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462  ax-caucvg 7463
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-sup 6677  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-3 8480  df-4 8481  df-n0 8672  df-z 8749  df-uz 9018  df-q 9103  df-rp 9133  df-fz 9423  df-fzo 9550  df-fl 9673  df-mod 9726  df-iseq 9849  df-seq3 9850  df-exp 9951  df-cj 10272  df-re 10273  df-im 10274  df-rsqrt 10427  df-abs 10428  df-dvds 11071  df-gcd 11213  df-numer 11435  df-denom 11436
This theorem is referenced by: (None)
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