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Theorem sqrt2irrap 12193
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 12175. (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 9635 . . 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 9637 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  NN )  ->  ( a  /  b
)  e.  QQ )
8 qre 9638 . . . . . . . . 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 12176 . . . . . . . 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 7971 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  0  e.  RR )
144zcnd 9389 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  e.  CC )
156nncnd 8946 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b  e.  CC )
166nnap0d 8978 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b #  0
)
1714, 15, 16divrecapd 8763 . . . . . . . . 9  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  =  ( a  x.  (
1  /  b ) ) )
184zred 9388 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  e.  RR )
196nnrecred 8979 . . . . . . . . . 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 7985 . . . . . . . . . . 11  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  1  e.  RR )
226nnrpd 9707 . . . . . . . . . . 11  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b  e.  RR+ )
23 0le1 8451 . . . . . . . . . . . 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 9750 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  0  <_  ( 1  /  b ) )
26 mulle0r 8914 . . . . . . . . . 10  |-  ( ( ( a  e.  RR  /\  ( 1  /  b
)  e.  RR )  /\  ( a  <_ 
0  /\  0  <_  ( 1  /  b ) ) )  ->  (
a  x.  ( 1  /  b ) )  <_  0 )
2718, 19, 20, 25, 26syl22anc 1249 . . . . . . . . 9  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  x.  ( 1  /  b
) )  <_  0
)
2817, 27eqbrtrd 4037 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  <_ 
0 )
29 2re 9002 . . . . . . . . . 10  |-  2  e.  RR
30 2pos 9023 . . . . . . . . . 10  |-  0  <  2
3129, 30sqrtgt0ii 11153 . . . . . . . . 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 8095 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  < 
( sqr `  2
) )
3410, 12, 33gtapd 8607 . . . . . 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 9276 . . . . . . . 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 12192 . . . . . . 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 9277 . . . . . . . . 9  |-  0  e.  ZZ
43 zlelttric 9311 . . . . . . . . 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 4043 . . . 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 2612 . 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 1363    e. wcel 2158   E.wrex 2466   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   RRcr 7823   0cc0 7824   1c1 7825    x. cmul 7829    < clt 8005    <_ cle 8006   # cap 8551    / cdiv 8642   NNcn 8932   2c2 8983   ZZcz 9266   QQcq 9632   sqrcsqrt 11018
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-xor 1386  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-1o 6430  df-2o 6431  df-er 6548  df-en 6754  df-sup 6996  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-fz 10022  df-fzo 10156  df-fl 10283  df-mod 10336  df-seqfrec 10459  df-exp 10533  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-dvds 11808  df-gcd 11957  df-prm 12121
This theorem is referenced by:  2irrexpqap  14636
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