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Theorem sqrt2irrap 11871
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 11853. (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 9428 . . 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 524 . . . . . . . . 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 525 . . . . . . . . 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 9430 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  NN )  ->  ( a  /  b
)  e.  QQ )
8 qre 9431 . . . . . . . . 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 11854 . . . . . . . 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 7781 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  0  e.  RR )
144zcnd 9188 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  e.  CC )
156nncnd 8748 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b  e.  CC )
166nnap0d 8780 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b #  0
)
1714, 15, 16divrecapd 8567 . . . . . . . . 9  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  =  ( a  x.  (
1  /  b ) ) )
184zred 9187 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  a  e.  RR )
196nnrecred 8781 . . . . . . . . . 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 7795 . . . . . . . . . . 11  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  1  e.  RR )
226nnrpd 9496 . . . . . . . . . . 11  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  b  e.  RR+ )
23 0le1 8257 . . . . . . . . . . . 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 9538 . . . . . . . . . 10  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  0  <_  ( 1  /  b ) )
26 mulle0r 8716 . . . . . . . . . 10  |-  ( ( ( a  e.  RR  /\  ( 1  /  b
)  e.  RR )  /\  ( a  <_ 
0  /\  0  <_  ( 1  /  b ) ) )  ->  (
a  x.  ( 1  /  b ) )  <_  0 )
2718, 19, 20, 25, 26syl22anc 1217 . . . . . . . . 9  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  x.  ( 1  /  b
) )  <_  0
)
2817, 27eqbrtrd 3950 . . . . . . . 8  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  <_ 
0 )
29 2re 8804 . . . . . . . . . 10  |-  2  e.  RR
30 2pos 8825 . . . . . . . . . 10  |-  0  <  2
3129, 30sqrtgt0ii 10917 . . . . . . . . 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 7901 . . . . . . 7  |-  ( ( ( ( Q  e.  QQ  /\  ( a  e.  ZZ  /\  b  e.  NN ) )  /\  Q  =  ( a  /  b ) )  /\  a  <_  0
)  ->  ( a  /  b )  < 
( sqr `  2
) )
3410, 12, 33gtapd 8413 . . . . . 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 9078 . . . . . . . 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 11870 . . . . . . 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 9079 . . . . . . . . 9  |-  0  e.  ZZ
43 zlelttric 9113 . . . . . . . . 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 787 . . . . 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 3956 . . . 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 2557 . 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 697    = wceq 1331    e. wcel 1480   E.wrex 2417   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   RRcr 7633   0cc0 7634   1c1 7635    x. cmul 7639    < clt 7814    <_ cle 7815   # cap 8357    / cdiv 8446   NNcn 8734   2c2 8785   ZZcz 9068   QQcq 9425   sqrcsqrt 10782
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7725  ax-resscn 7726  ax-1cn 7727  ax-1re 7728  ax-icn 7729  ax-addcl 7730  ax-addrcl 7731  ax-mulcl 7732  ax-mulrcl 7733  ax-addcom 7734  ax-mulcom 7735  ax-addass 7736  ax-mulass 7737  ax-distr 7738  ax-i2m1 7739  ax-0lt1 7740  ax-1rid 7741  ax-0id 7742  ax-rnegex 7743  ax-precex 7744  ax-cnre 7745  ax-pre-ltirr 7746  ax-pre-ltwlin 7747  ax-pre-lttrn 7748  ax-pre-apti 7749  ax-pre-ltadd 7750  ax-pre-mulgt0 7751  ax-pre-mulext 7752  ax-arch 7753  ax-caucvg 7754
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-xor 1354  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-1o 6313  df-2o 6314  df-er 6429  df-en 6635  df-sup 6871  df-pnf 7816  df-mnf 7817  df-xr 7818  df-ltxr 7819  df-le 7820  df-sub 7949  df-neg 7950  df-reap 8351  df-ap 8358  df-div 8447  df-inn 8735  df-2 8793  df-3 8794  df-4 8795  df-n0 8992  df-z 9069  df-uz 9341  df-q 9426  df-rp 9456  df-fz 9805  df-fzo 9934  df-fl 10057  df-mod 10110  df-seqfrec 10233  df-exp 10307  df-cj 10628  df-re 10629  df-im 10630  df-rsqrt 10784  df-abs 10785  df-dvds 11507  df-gcd 11649  df-prm 11802
This theorem is referenced by: (None)
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