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Theorem nonsq 10964
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 8665 . . . 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 8618 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  A  e.  RR )
64nn0ge0d 8620 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  0  <_  A )
7 resqrtth 10290 . . . . . 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 3837 . . . 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 8618 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  B  e.  RR )
12 nn0re 8573 . . . . . . 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 10422 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( sqr `  A )  e.  RR )
1510nn0ge0d 8620 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  0  <_  B )
1613, 6sqrtge0d 10425 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  0  <_  ( sqr `  A ) )
1711, 14, 15, 16lt2sqd 9951 . . . 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 3831 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  ( ( sqr `  A ) ^
2 )  <  (
( B  +  1 ) ^ 2 ) )
21 peano2re 7520 . . . . . 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 8604 . . . . . . 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 8620 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  0  <_  ( B  +  1 ) )
2614, 22, 16, 25lt2sqd 9951 . . . 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 8735 . . 3  |-  ( ( B  e.  ZZ  /\  B  <  ( sqr `  A
)  /\  ( sqr `  A )  <  ( B  +  1 ) )  ->  -.  ( sqr `  A )  e.  ZZ )
292, 18, 27, 28syl3anc 1170 . 2  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( B ^
2 )  <  A  /\  A  <  ( ( B  +  1 ) ^ 2 ) ) )  ->  -.  ( sqr `  A )  e.  ZZ )
30 nn0sqrtelqelz 10963 . . . 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 622 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 1285    e. wcel 1434   class class class wbr 3811   ` cfv 4968  (class class class)co 5590   RRcr 7251   0cc0 7252   1c1 7253    + caddc 7255    < clt 7424    <_ cle 7425   2c2 8365   NN0cn0 8564   ZZcz 8645   QQcq 8998   ^cexp 9790   sqrcsqrt 10255
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7338  ax-resscn 7339  ax-1cn 7340  ax-1re 7341  ax-icn 7342  ax-addcl 7343  ax-addrcl 7344  ax-mulcl 7345  ax-mulrcl 7346  ax-addcom 7347  ax-mulcom 7348  ax-addass 7349  ax-mulass 7350  ax-distr 7351  ax-i2m1 7352  ax-0lt1 7353  ax-1rid 7354  ax-0id 7355  ax-rnegex 7356  ax-precex 7357  ax-cnre 7358  ax-pre-ltirr 7359  ax-pre-ltwlin 7360  ax-pre-lttrn 7361  ax-pre-apti 7362  ax-pre-ltadd 7363  ax-pre-mulgt0 7364  ax-pre-mulext 7365  ax-arch 7366  ax-caucvg 7367
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4406  df-rel 4407  df-cnv 4408  df-co 4409  df-dm 4410  df-rn 4411  df-res 4412  df-ima 4413  df-iota 4933  df-fun 4970  df-fn 4971  df-f 4972  df-f1 4973  df-fo 4974  df-f1o 4975  df-fv 4976  df-riota 5546  df-ov 5593  df-oprab 5594  df-mpt2 5595  df-1st 5845  df-2nd 5846  df-recs 6001  df-frec 6087  df-sup 6585  df-pnf 7426  df-mnf 7427  df-xr 7428  df-ltxr 7429  df-le 7430  df-sub 7557  df-neg 7558  df-reap 7951  df-ap 7958  df-div 8037  df-inn 8316  df-2 8374  df-3 8375  df-4 8376  df-n0 8565  df-z 8646  df-uz 8914  df-q 8999  df-rp 9029  df-fz 9319  df-fzo 9443  df-fl 9565  df-mod 9618  df-iseq 9740  df-iexp 9791  df-cj 10102  df-re 10103  df-im 10104  df-rsqrt 10257  df-abs 10258  df-dvds 10576  df-gcd 10718  df-numer 10940  df-denom 10941
This theorem is referenced by: (None)
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