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Theorem sqrt2irrap 11769
Description: The square root of 2 is irrational. That is, for any rational number,  ( sqr `  2
) is apart from it. In the absence of excluded middle, we can distinguish between this and "the square root of 2 is not rational" which is sqrt2irr 11752. (Contributed by Jim Kingdon, 2-Oct-2021.)
Assertion
Ref Expression
sqrt2irrap  |-  ( Q  e.  QQ  ->  ( sqr `  2 ) #  Q
)

Proof of Theorem sqrt2irrap
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elq 9370 . . 3  |-  ( Q  e.  QQ  <->  E. a  e.  ZZ  E. b  e.  NN  Q  =  ( a  /  b ) )
21biimpi 119 . 2  |-  ( Q  e.  QQ  ->  E. a  e.  ZZ  E. b  e.  NN  Q  =  ( a  /  b ) )
3 simplrl 509 . . . . . . . . 9  |-  ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b
) )  ->  a  e.  ZZ )
43adantr 274 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  e.  ZZ )
5 simplrr 510 . . . . . . . . 9  |-  ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b
) )  ->  b  e.  NN )
65adantr 274 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b  e.  NN )
7 znq 9372 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  NN )  ->  ( a  /  b
)  e.  QQ )
8 qre 9373 . . . . . . . . 9  |-  ( ( a  /  b )  e.  QQ  ->  (
a  /  b )  e.  RR )
97, 8syl 14 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  NN )  ->  ( a  /  b
)  e.  RR )
104, 6, 9syl2anc 408 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  e.  RR )
11 sqrt2re 11753 . . . . . . . 8  |-  ( sqr `  2 )  e.  RR
1211a1i 9 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( sqr `  2 )  e.  RR )
13 0red 7735 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  0  e.  RR )
144zcnd 9132 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  e.  CC )
156nncnd 8698 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b  e.  CC )
166nnap0d 8730 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b #  0
)
1714, 15, 16divrecapd 8520 . . . . . . . . 9  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  =  ( a  x.  (
1  /  b ) ) )
184zred 9131 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  e.  RR )
196nnrecred 8731 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( 1  /  b )  e.  RR )
20 simpr 109 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  <_  0 )
21 1red 7749 . . . . . . . . . . 11  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  1  e.  RR )
226nnrpd 9437 . . . . . . . . . . 11  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b  e.  RR+ )
23 0le1 8211 . . . . . . . . . . . 12  |-  0  <_  1
2423a1i 9 . . . . . . . . . . 11  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  0  <_  1 )
2521, 22, 24divge0d 9479 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  0  <_  ( 1  /  b ) )
26 mulle0r 8666 . . . . . . . . . 10  |-  ( ( ( a  e.  RR  /\  ( 1  /  b
)  e.  RR )  /\  ( a  <_ 
0  /\  0  <_  ( 1  /  b ) ) )  ->  (
a  x.  ( 1  /  b ) )  <_  0 )
2718, 19, 20, 25, 26syl22anc 1202 . . . . . . . . 9  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  x.  ( 1  /  b
) )  <_  0
)
2817, 27eqbrtrd 3920 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  <_ 
0 )
29 2re 8754 . . . . . . . . . 10  |-  2  e.  RR
30 2pos 8775 . . . . . . . . . 10  |-  0  <  2
3129, 30sqrtgt0ii 10858 . . . . . . . . 9  |-  0  <  ( sqr `  2
)
3231a1i 9 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  0  <  ( sqr `  2 ) )
3310, 13, 12, 28, 32lelttrd 7855 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  < 
( sqr `  2
) )
3410, 12, 33gtapd 8366 . . . . . 6  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( sqr `  2 ) #  ( a  /  b ) )
353adantr 274 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  0  <  a
)  ->  a  e.  ZZ )
36 simpr 109 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  0  <  a
)  ->  0  <  a )
37 elnnz 9022 . . . . . . . 8  |-  ( a  e.  NN  <->  ( a  e.  ZZ  /\  0  < 
a ) )
3835, 36, 37sylanbrc 413 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  0  <  a
)  ->  a  e.  NN )
395adantr 274 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  0  <  a
)  ->  b  e.  NN )
40 sqrt2irraplemnn 11768 . . . . . . 7  |-  ( ( a  e.  NN  /\  b  e.  NN )  ->  ( sqr `  2
) #  ( a  / 
b ) )
4138, 39, 40syl2anc 408 . . . . . 6  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  0  <  a
)  ->  ( sqr `  2 ) #  ( a  /  b ) )
42 0z 9023 . . . . . . . . 9  |-  0  e.  ZZ
43 zlelttric 9057 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  0  e.  ZZ )  ->  ( a  <_  0  \/  0  <  a ) )
4442, 43mpan2 421 . . . . . . . 8  |-  ( a  e.  ZZ  ->  (
a  <_  0  \/  0  <  a ) )
4544ad2antrl 481 . . . . . . 7  |-  ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  ->  ( a  <_  0  \/  0  < 
a ) )
4645adantr 274 . . . . . 6  |-  ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b
) )  ->  (
a  <_  0  \/  0  <  a ) )
4734, 41, 46mpjaodan 772 . . . . 5  |-  ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b
) )  ->  ( sqr `  2 ) #  ( a  /  b ) )
48 simpr 109 . . . . 5  |-  ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b
) )  ->  Q  =  ( a  / 
b ) )
4947, 48breqtrrd 3926 . . . 4  |-  ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b
) )  ->  ( sqr `  2 ) #  Q
)
5049ex 114 . . 3  |-  ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  ->  ( Q  =  ( a  / 
b )  ->  ( sqr `  2 ) #  Q
) )
5150rexlimdvva 2534 . 2  |-  ( Q  e.  QQ  ->  ( E. a  e.  ZZ  E. b  e.  NN  Q  =  ( a  / 
b )  ->  ( sqr `  2 ) #  Q
) )
522, 51mpd 13 1  |-  ( Q  e.  QQ  ->  ( sqr `  2 ) #  Q
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 682    = wceq 1316    e. wcel 1465   E.wrex 2394   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   RRcr 7587   0cc0 7588   1c1 7589    x. cmul 7593    < clt 7768    <_ cle 7769   # cap 8310    / cdiv 8399   NNcn 8684   2c2 8735   ZZcz 9012   QQcq 9367   sqrcsqrt 10723
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-stab 801  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-xor 1339  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-1o 6281  df-2o 6282  df-er 6397  df-en 6603  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-2 8743  df-3 8744  df-4 8745  df-n0 8936  df-z 9013  df-uz 9283  df-q 9368  df-rp 9398  df-fz 9746  df-fzo 9875  df-fl 9998  df-mod 10051  df-seqfrec 10174  df-exp 10248  df-cj 10569  df-re 10570  df-im 10571  df-rsqrt 10725  df-abs 10726  df-dvds 11406  df-gcd 11548  df-prm 11701
This theorem is referenced by: (None)
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