MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sqr2irrlem Structured version   Unicode version

Theorem sqr2irrlem 13983
Description: Lemma for irrationality of square root of 2. The core of the proof - if  A  /  B  =  sqr ( 2 ), then 
A and  B are even, so  A  /  2 and  B  /  2 are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). This is Metamath 100 proof #1. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)
Hypotheses
Ref Expression
sqr2irrlem.1  |-  ( ph  ->  A  e.  ZZ )
sqr2irrlem.2  |-  ( ph  ->  B  e.  NN )
sqrt2irrlem.3  |-  ( ph  ->  ( sqr `  2
)  =  ( A  /  B ) )
Assertion
Ref Expression
sqr2irrlem  |-  ( ph  ->  ( ( A  / 
2 )  e.  ZZ  /\  ( B  /  2
)  e.  NN ) )

Proof of Theorem sqr2irrlem
StepHypRef Expression
1 2cn 10523 . . . . . . . . . . . 12  |-  2  e.  CC
2 sqrtth 13199 . . . . . . . . . . . 12  |-  ( 2  e.  CC  ->  (
( sqr `  2
) ^ 2 )  =  2 )
31, 2ax-mp 5 . . . . . . . . . . 11  |-  ( ( sqr `  2 ) ^ 2 )  =  2
4 sqrt2irrlem.3 . . . . . . . . . . . 12  |-  ( ph  ->  ( sqr `  2
)  =  ( A  /  B ) )
54oveq1d 6211 . . . . . . . . . . 11  |-  ( ph  ->  ( ( sqr `  2
) ^ 2 )  =  ( ( A  /  B ) ^
2 ) )
63, 5syl5eqr 2437 . . . . . . . . . 10  |-  ( ph  ->  2  =  ( ( A  /  B ) ^ 2 ) )
7 sqr2irrlem.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ZZ )
87zcnd 10885 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
9 sqr2irrlem.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  NN )
109nncnd 10468 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
119nnne0d 10497 . . . . . . . . . . 11  |-  ( ph  ->  B  =/=  0 )
128, 10, 11sqdivd 12225 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  /  B ) ^ 2 )  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
136, 12eqtrd 2423 . . . . . . . . 9  |-  ( ph  ->  2  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
1413oveq1d 6211 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( ( ( A ^ 2 )  /  ( B ^
2 ) )  x.  ( B ^ 2 ) ) )
158sqcld 12210 . . . . . . . . 9  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
169nnsqcld 12232 . . . . . . . . . 10  |-  ( ph  ->  ( B ^ 2 )  e.  NN )
1716nncnd 10468 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
1816nnne0d 10497 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  =/=  0 )
1915, 17, 18divcan1d 10238 . . . . . . . 8  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
( B ^ 2 ) )  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2014, 19eqtrd 2423 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2120oveq1d 6211 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( ( A ^ 2 )  /  2 ) )
221a1i 11 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
23 2ne0 10545 . . . . . . . 8  |-  2  =/=  0
2423a1i 11 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
2517, 22, 24divcan3d 10242 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( B ^ 2 ) )
2621, 25eqtr3d 2425 . . . . 5  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  =  ( B ^ 2 ) )
2726, 16eqeltrd 2470 . . . 4  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  NN )
2827nnzd 10883 . . 3  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  ZZ )
29 zesq 12191 . . . 4  |-  ( A  e.  ZZ  ->  (
( A  /  2
)  e.  ZZ  <->  ( ( A ^ 2 )  / 
2 )  e.  ZZ ) )
307, 29syl 16 . . 3  |-  ( ph  ->  ( ( A  / 
2 )  e.  ZZ  <->  ( ( A ^ 2 )  /  2 )  e.  ZZ ) )
3128, 30mpbird 232 . 2  |-  ( ph  ->  ( A  /  2
)  e.  ZZ )
321sqvali 12150 . . . . . . . 8  |-  ( 2 ^ 2 )  =  ( 2  x.  2 )
3332oveq2i 6207 . . . . . . 7  |-  ( ( A ^ 2 )  /  ( 2 ^ 2 ) )  =  ( ( A ^
2 )  /  (
2  x.  2 ) )
348, 22, 24sqdivd 12225 . . . . . . 7  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( A ^ 2 )  /  ( 2 ^ 2 ) ) )
3515, 22, 22, 24, 24divdiv1d 10268 . . . . . . 7  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( A ^ 2 )  /  ( 2  x.  2 ) ) )
3633, 34, 353eqtr4a 2449 . . . . . 6  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( ( A ^ 2 )  /  2 )  /  2 ) )
3726oveq1d 6211 . . . . . 6  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( B ^ 2 )  /  2 ) )
3836, 37eqtrd 2423 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( B ^ 2 )  /  2 ) )
39 zsqcl 12141 . . . . . 6  |-  ( ( A  /  2 )  e.  ZZ  ->  (
( A  /  2
) ^ 2 )  e.  ZZ )
4031, 39syl 16 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  e.  ZZ )
4138, 40eqeltrrd 2471 . . . 4  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  ZZ )
4216nnrpd 11175 . . . . . 6  |-  ( ph  ->  ( B ^ 2 )  e.  RR+ )
4342rphalfcld 11189 . . . . 5  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  RR+ )
4443rpgt0d 11180 . . . 4  |-  ( ph  ->  0  <  ( ( B ^ 2 )  /  2 ) )
45 elnnz 10791 . . . 4  |-  ( ( ( B ^ 2 )  /  2 )  e.  NN  <->  ( (
( B ^ 2 )  /  2 )  e.  ZZ  /\  0  <  ( ( B ^
2 )  /  2
) ) )
4641, 44, 45sylanbrc 662 . . 3  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  NN )
47 nnesq 12192 . . . 4  |-  ( B  e.  NN  ->  (
( B  /  2
)  e.  NN  <->  ( ( B ^ 2 )  / 
2 )  e.  NN ) )
489, 47syl 16 . . 3  |-  ( ph  ->  ( ( B  / 
2 )  e.  NN  <->  ( ( B ^ 2 )  /  2 )  e.  NN ) )
4946, 48mpbird 232 . 2  |-  ( ph  ->  ( B  /  2
)  e.  NN )
5031, 49jca 530 1  |-  ( ph  ->  ( ( A  / 
2 )  e.  ZZ  /\  ( B  /  2
)  e.  NN ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   CCcc 9401   0cc0 9403    x. cmul 9408    < clt 9539    / cdiv 10123   NNcn 10452   2c2 10502   ZZcz 10781   ^cexp 12069   sqrcsqrt 13068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071
This theorem is referenced by:  sqrt2irr  13984
  Copyright terms: Public domain W3C validator