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Theorem sqrt2irrap 12198
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 12180. (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 9640 . . 3  |-  ( Q  e.  QQ  <->  E. a  e.  ZZ  E. b  e.  NN  Q  =  ( a  /  b ) )
21biimpi 120 . 2  |-  ( Q  e.  QQ  ->  E. a  e.  ZZ  E. b  e.  NN  Q  =  ( a  /  b ) )
3 simplrl 535 . . . . . . . . 9  |-  ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b
) )  ->  a  e.  ZZ )
43adantr 276 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  e.  ZZ )
5 simplrr 536 . . . . . . . . 9  |-  ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b
) )  ->  b  e.  NN )
65adantr 276 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b  e.  NN )
7 znq 9642 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  NN )  ->  ( a  /  b
)  e.  QQ )
8 qre 9643 . . . . . . . . 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 411 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  e.  RR )
11 sqrt2re 12181 . . . . . . . 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 7976 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  0  e.  RR )
144zcnd 9394 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  e.  CC )
156nncnd 8951 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b  e.  CC )
166nnap0d 8983 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b #  0
)
1714, 15, 16divrecapd 8768 . . . . . . . . 9  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  =  ( a  x.  (
1  /  b ) ) )
184zred 9393 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  e.  RR )
196nnrecred 8984 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( 1  /  b )  e.  RR )
20 simpr 110 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  <_  0 )
21 1red 7990 . . . . . . . . . . 11  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  1  e.  RR )
226nnrpd 9712 . . . . . . . . . . 11  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b  e.  RR+ )
23 0le1 8456 . . . . . . . . . . . 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 9755 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  0  <_  ( 1  /  b ) )
26 mulle0r 8919 . . . . . . . . . 10  |-  ( ( ( a  e.  RR  /\  ( 1  /  b
)  e.  RR )  /\  ( a  <_ 
0  /\  0  <_  ( 1  /  b ) ) )  ->  (
a  x.  ( 1  /  b ) )  <_  0 )
2718, 19, 20, 25, 26syl22anc 1250 . . . . . . . . 9  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  x.  ( 1  /  b
) )  <_  0
)
2817, 27eqbrtrd 4040 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  <_ 
0 )
29 2re 9007 . . . . . . . . . 10  |-  2  e.  RR
30 2pos 9028 . . . . . . . . . 10  |-  0  <  2
3129, 30sqrtgt0ii 11158 . . . . . . . . 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 8100 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  < 
( sqr `  2
) )
3410, 12, 33gtapd 8612 . . . . . 6  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( sqr `  2 ) #  ( a  /  b ) )
353adantr 276 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  0  <  a
)  ->  a  e.  ZZ )
36 simpr 110 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  0  <  a
)  ->  0  <  a )
37 elnnz 9281 . . . . . . . 8  |-  ( a  e.  NN  <->  ( a  e.  ZZ  /\  0  < 
a ) )
3835, 36, 37sylanbrc 417 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  0  <  a
)  ->  a  e.  NN )
395adantr 276 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  0  <  a
)  ->  b  e.  NN )
40 sqrt2irraplemnn 12197 . . . . . . 7  |-  ( ( a  e.  NN  /\  b  e.  NN )  ->  ( sqr `  2
) #  ( a  / 
b ) )
4138, 39, 40syl2anc 411 . . . . . 6  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  0  <  a
)  ->  ( sqr `  2 ) #  ( a  /  b ) )
42 0z 9282 . . . . . . . . 9  |-  0  e.  ZZ
43 zlelttric 9316 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  0  e.  ZZ )  ->  ( a  <_  0  \/  0  <  a ) )
4442, 43mpan2 425 . . . . . . . 8  |-  ( a  e.  ZZ  ->  (
a  <_  0  \/  0  <  a ) )
4544ad2antrl 490 . . . . . . 7  |-  ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  ->  ( a  <_  0  \/  0  < 
a ) )
4645adantr 276 . . . . . 6  |-  ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b
) )  ->  (
a  <_  0  \/  0  <  a ) )
4734, 41, 46mpjaodan 799 . . . . 5  |-  ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b
) )  ->  ( sqr `  2 ) #  ( a  /  b ) )
48 simpr 110 . . . . 5  |-  ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b
) )  ->  Q  =  ( a  / 
b ) )
4947, 48breqtrrd 4046 . . . 4  |-  ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b
) )  ->  ( sqr `  2 ) #  Q
)
5049ex 115 . . 3  |-  ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  ->  ( Q  =  ( a  / 
b )  ->  ( sqr `  2 ) #  Q
) )
5150rexlimdvva 2615 . 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 104    \/ wo 709    = wceq 1364    e. wcel 2160   E.wrex 2469   class class class wbr 4018   ` cfv 5231  (class class class)co 5891   RRcr 7828   0cc0 7829   1c1 7830    x. cmul 7834    < clt 8010    <_ cle 8011   # cap 8556    / cdiv 8647   NNcn 8937   2c2 8988   ZZcz 9271   QQcq 9637   sqrcsqrt 11023
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947  ax-arch 7948  ax-caucvg 7949
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-1o 6435  df-2o 6436  df-er 6553  df-en 6759  df-sup 7001  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-inn 8938  df-2 8996  df-3 8997  df-4 8998  df-n0 9195  df-z 9272  df-uz 9547  df-q 9638  df-rp 9672  df-fz 10027  df-fzo 10161  df-fl 10288  df-mod 10341  df-seqfrec 10464  df-exp 10538  df-cj 10869  df-re 10870  df-im 10871  df-rsqrt 11025  df-abs 11026  df-dvds 11813  df-gcd 11962  df-prm 12126
This theorem is referenced by:  2irrexpqap  14793
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