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Theorem sqrt2irrap 12121
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 12103. (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 9568 . . 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 530 . . . . . . . . 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 531 . . . . . . . . 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 9570 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  NN )  ->  ( a  /  b
)  e.  QQ )
8 qre 9571 . . . . . . . . 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 409 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  e.  RR )
11 sqrt2re 12104 . . . . . . . 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 7908 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  0  e.  RR )
144zcnd 9322 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  e.  CC )
156nncnd 8879 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b  e.  CC )
166nnap0d 8911 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b #  0
)
1714, 15, 16divrecapd 8697 . . . . . . . . 9  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  =  ( a  x.  (
1  /  b ) ) )
184zred 9321 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  e.  RR )
196nnrecred 8912 . . . . . . . . . 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 7922 . . . . . . . . . . 11  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  1  e.  RR )
226nnrpd 9638 . . . . . . . . . . 11  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b  e.  RR+ )
23 0le1 8387 . . . . . . . . . . . 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 9681 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  0  <_  ( 1  /  b ) )
26 mulle0r 8847 . . . . . . . . . 10  |-  ( ( ( a  e.  RR  /\  ( 1  /  b
)  e.  RR )  /\  ( a  <_ 
0  /\  0  <_  ( 1  /  b ) ) )  ->  (
a  x.  ( 1  /  b ) )  <_  0 )
2718, 19, 20, 25, 26syl22anc 1234 . . . . . . . . 9  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  x.  ( 1  /  b
) )  <_  0
)
2817, 27eqbrtrd 4009 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  <_ 
0 )
29 2re 8935 . . . . . . . . . 10  |-  2  e.  RR
30 2pos 8956 . . . . . . . . . 10  |-  0  <  2
3129, 30sqrtgt0ii 11082 . . . . . . . . 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 8031 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  < 
( sqr `  2
) )
3410, 12, 33gtapd 8543 . . . . . 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 9209 . . . . . . . 8  |-  ( a  e.  NN  <->  ( a  e.  ZZ  /\  0  < 
a ) )
3835, 36, 37sylanbrc 415 . . . . . . 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 12120 . . . . . . 7  |-  ( ( a  e.  NN  /\  b  e.  NN )  ->  ( sqr `  2
) #  ( a  / 
b ) )
4138, 39, 40syl2anc 409 . . . . . 6  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  0  <  a
)  ->  ( sqr `  2 ) #  ( a  /  b ) )
42 0z 9210 . . . . . . . . 9  |-  0  e.  ZZ
43 zlelttric 9244 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  0  e.  ZZ )  ->  ( a  <_  0  \/  0  <  a ) )
4442, 43mpan2 423 . . . . . . . 8  |-  ( a  e.  ZZ  ->  (
a  <_  0  \/  0  <  a ) )
4544ad2antrl 487 . . . . . . 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 793 . . . . 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 4015 . . . 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 2595 . 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 703    = wceq 1348    e. wcel 2141   E.wrex 2449   class class class wbr 3987   ` cfv 5196  (class class class)co 5850   RRcr 7760   0cc0 7761   1c1 7762    x. cmul 7766    < clt 7941    <_ cle 7942   # cap 8487    / cdiv 8576   NNcn 8865   2c2 8916   ZZcz 9199   QQcq 9565   sqrcsqrt 10947
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-xor 1371  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-1o 6392  df-2o 6393  df-er 6509  df-en 6715  df-sup 6957  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-fz 9953  df-fzo 10086  df-fl 10213  df-mod 10266  df-seqfrec 10389  df-exp 10463  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-dvds 11737  df-gcd 11885  df-prm 12049
This theorem is referenced by:  2irrexpqap  13649
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