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Theorem nonsq 12859
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 9560 . . . 4 |- ( B e. NN0 -> B e. ZZ )
21ad2antlr 489 . . 3 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> B e. ZZ )
3 simprl 531 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( B ^ 2 ) < A )
4 simpll 527 . . . . . . 7 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> A e. NN0 )
54nn0red 9517 . . . . . 6 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> A e. RR )
64nn0ge0d 9519 . . . . . 6 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> 0 <_ A )
7 resqrtth 11671 . . . . . 6 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqr ` A ) ^ 2 ) = A )
85, 6, 7syl2anc 411 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( ( sqr ` A ) ^ 2 ) = A )
93, 8breqtrrd 4121 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( B ^ 2 ) < ( ( sqr ` A ) ^ 2 ) )
10 simplr 529 . . . . . 6 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> B e. NN0 )
1110nn0red 9517 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> B e. RR )
12 nn0re 9470 . . . . . . 7 |- ( A e. NN0 -> A e. RR )
1312ad2antrr 488 . . . . . 6 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> A e. RR )
1413, 6resqrtcld 11803 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( sqr ` A ) e. RR )
1510nn0ge0d 9519 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> 0 <_ B )
1613, 6sqrtge0d 11806 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> 0 <_ ( sqr ` A ) )
1711, 14, 15, 16lt2sqd 11029 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( B < ( sqr ` A ) <-> ( B ^ 2 ) < ( ( sqr ` A ) ^ 2 ) ) )
189, 17mpbird 167 . . 3 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> B < ( sqr ` A ) )
19 simprr 533 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> A < ( ( B + 1 ) ^ 2 ) )
208, 19eqbrtrd 4115 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( ( sqr ` A ) ^ 2 ) < ( ( B + 1 ) ^ 2 ) )
21 peano2re 8374 . . . . . 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 9501 . . . . . . 7 |- ( B e. NN0 -> ( B + 1 ) e. NN0 )
2423ad2antlr 489 . . . . . 6 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( B + 1 ) e. NN0 )
2524nn0ge0d 9519 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> 0 <_ ( B + 1 ) )
2614, 22, 16, 25lt2sqd 11029 . . . 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 167 . . 3 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( sqr ` A ) < ( B + 1 ) )
28 btwnnz 9635 . . 3 |- ( ( B e. ZZ /\ B < ( sqr ` A ) /\ ( sqr ` A ) < ( B + 1 ) ) -> -. ( sqr ` A ) e. ZZ )
292, 18, 27, 28syl3anc 1274 . 2 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> -. ( sqr ` A ) e. ZZ )
30 nn0sqrtelqelz 12858 . . . 4 |- ( ( A e. NN0 /\ ( sqr ` A ) e. QQ ) -> ( sqr ` A ) e. ZZ )
3130ex 115 . . 3 |- ( A e. NN0 -> ( ( sqr ` A ) e. QQ -> ( sqr ` A ) e. ZZ ) )
3231ad2antrr 488 . 2 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> ( ( sqr ` A ) e. QQ -> ( sqr ` A ) e. ZZ ) )
3329, 32mtod 669 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 104   = wceq 1398   e. wcel 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   RRcr 8091   0cc0 8092   1c1 8093   + caddc 8095   < clt 8273   <_ cle 8274   2c2 9253   NN0cn0 9461   ZZcz 9540   QQcq 9914   ^cexp 10863   sqrcsqrt 11636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212
This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-sup 7243  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fz 10306  df-fzo 10440  df-fl 10593  df-mod 10648  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-dvds 12429  df-gcd 12605  df-numer 12835  df-denom 12836
This theorem is referenced by: (None)
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