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Theorem nonsq 11921
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 9098 . . . 4 |- ( B e. NN0 -> B e. ZZ )
21ad2antlr 481 . . 3 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> B e. ZZ )
3 simprl 521 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( B ^ 2 ) < A )
4 simpll 519 . . . . . . 7 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> A e. NN0 )
54nn0red 9055 . . . . . 6 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> A e. RR )
64nn0ge0d 9057 . . . . . 6 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> 0 <_ A )
7 resqrtth 10835 . . . . . 6 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqr ` A ) ^ 2 ) = A )
85, 6, 7syl2anc 409 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( ( sqr ` A ) ^ 2 ) = A )
93, 8breqtrrd 3964 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( B ^ 2 ) < ( ( sqr ` A ) ^ 2 ) )
10 simplr 520 . . . . . 6 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> B e. NN0 )
1110nn0red 9055 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> B e. RR )
12 nn0re 9010 . . . . . . 7 |- ( A e. NN0 -> A e. RR )
1312ad2antrr 480 . . . . . 6 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> A e. RR )
1413, 6resqrtcld 10967 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( sqr ` A ) e. RR )
1510nn0ge0d 9057 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> 0 <_ B )
1613, 6sqrtge0d 10970 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> 0 <_ ( sqr ` A ) )
1711, 14, 15, 16lt2sqd 10486 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( B < ( sqr ` A ) <-> ( B ^ 2 ) < ( ( sqr ` A ) ^ 2 ) ) )
189, 17mpbird 166 . . 3 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> B < ( sqr ` A ) )
19 simprr 522 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> A < ( ( B + 1 ) ^ 2 ) )
208, 19eqbrtrd 3958 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( ( sqr ` A ) ^ 2 ) < ( ( B + 1 ) ^ 2 ) )
21 peano2re 7922 . . . . . 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 9041 . . . . . . 7 |- ( B e. NN0 -> ( B + 1 ) e. NN0 )
2423ad2antlr 481 . . . . . 6 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( B + 1 ) e. NN0 )
2524nn0ge0d 9057 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> 0 <_ ( B + 1 ) )
2614, 22, 16, 25lt2sqd 10486 . . . 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 166 . . 3 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( sqr ` A ) < ( B + 1 ) )
28 btwnnz 9169 . . 3 |- ( ( B e. ZZ /\ B < ( sqr ` A ) /\ ( sqr ` A ) < ( B + 1 ) ) -> -. ( sqr ` A ) e. ZZ )
292, 18, 27, 28syl3anc 1217 . 2 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> -. ( sqr ` A ) e. ZZ )
30 nn0sqrtelqelz 11920 . . . 4 |- ( ( A e. NN0 /\ ( sqr ` A ) e. QQ ) -> ( sqr ` A ) e. ZZ )
3130ex 114 . . 3 |- ( A e. NN0 -> ( ( sqr ` A ) e. QQ -> ( sqr ` A ) e. ZZ ) )
3231ad2antrr 480 . 2 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( ( sqr ` A ) e. QQ -> ( sqr ` A ) e. ZZ ) )
3329, 32mtod 653 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 103   = wceq 1332   e. wcel 1481   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   RRcr 7643   0cc0 7644   1c1 7645   + caddc 7647   < clt 7824   <_ cle 7825   2c2 8795   NN0cn0 9001   ZZcz 9078   QQcq 9438   ^cexp 10323   sqrcsqrt 10800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-sup 6879  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-fl 10074  df-mod 10127  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-dvds 11530  df-gcd 11672  df-numer 11897  df-denom 11898
This theorem is referenced by: (None)
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