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

Theorem sqr2irrlem 12520
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). (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 )
sqr2irrlem.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 9811 . . . . . . . . . . . 12  |-  2  e.  CC
2 sqrth 11842 . . . . . . . . . . . 12  |-  ( 2  e.  CC  ->  (
( sqr `  2
) ^ 2 )  =  2 )
31, 2ax-mp 10 . . . . . . . . . . 11  |-  ( ( sqr `  2 ) ^ 2 )  =  2
4 sqr2irrlem.3 . . . . . . . . . . . 12  |-  ( ph  ->  ( sqr `  2
)  =  ( A  /  B ) )
54oveq1d 5834 . . . . . . . . . . 11  |-  ( ph  ->  ( ( sqr `  2
) ^ 2 )  =  ( ( A  /  B ) ^
2 ) )
63, 5syl5eqr 2330 . . . . . . . . . 10  |-  ( ph  ->  2  =  ( ( A  /  B ) ^ 2 ) )
7 sqr2irrlem.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ZZ )
87zcnd 10113 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
9 sqr2irrlem.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  NN )
109nncnd 9757 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
119nnne0d 9785 . . . . . . . . . . 11  |-  ( ph  ->  B  =/=  0 )
128, 10, 11sqdivd 11252 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  /  B ) ^ 2 )  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
136, 12eqtrd 2316 . . . . . . . . 9  |-  ( ph  ->  2  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
1413oveq1d 5834 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( ( ( A ^ 2 )  /  ( B ^
2 ) )  x.  ( B ^ 2 ) ) )
158sqcld 11237 . . . . . . . . 9  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
169nnsqcld 11259 . . . . . . . . . 10  |-  ( ph  ->  ( B ^ 2 )  e.  NN )
1716nncnd 9757 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
1816nnne0d 9785 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  =/=  0 )
1915, 17, 18divcan1d 9532 . . . . . . . 8  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
( B ^ 2 ) )  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2014, 19eqtrd 2316 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2120oveq1d 5834 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( ( A ^ 2 )  /  2 ) )
221a1i 12 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
23 2ne0 9824 . . . . . . . 8  |-  2  =/=  0
2423a1i 12 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
2517, 22, 24divcan3d 9536 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( B ^ 2 ) )
2621, 25eqtr3d 2318 . . . . 5  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  =  ( B ^ 2 ) )
2726, 16eqeltrd 2358 . . . 4  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  NN )
2827nnzd 10111 . . 3  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  ZZ )
29 zesq 11218 . . . 4  |-  ( A  e.  ZZ  ->  (
( A  /  2
)  e.  ZZ  <->  ( ( A ^ 2 )  / 
2 )  e.  ZZ ) )
307, 29syl 17 . . 3  |-  ( ph  ->  ( ( A  / 
2 )  e.  ZZ  <->  ( ( A ^ 2 )  /  2 )  e.  ZZ ) )
3128, 30mpbird 225 . 2  |-  ( ph  ->  ( A  /  2
)  e.  ZZ )
321sqvali 11177 . . . . . . . 8  |-  ( 2 ^ 2 )  =  ( 2  x.  2 )
3332oveq2i 5830 . . . . . . 7  |-  ( ( A ^ 2 )  /  ( 2 ^ 2 ) )  =  ( ( A ^
2 )  /  (
2  x.  2 ) )
348, 22, 24sqdivd 11252 . . . . . . 7  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( A ^ 2 )  /  ( 2 ^ 2 ) ) )
3515, 22, 22, 24, 24divdiv1d 9562 . . . . . . 7  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( A ^ 2 )  /  ( 2  x.  2 ) ) )
3633, 34, 353eqtr4a 2342 . . . . . 6  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( ( A ^ 2 )  /  2 )  /  2 ) )
3726oveq1d 5834 . . . . . 6  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( B ^ 2 )  /  2 ) )
3836, 37eqtrd 2316 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( B ^ 2 )  /  2 ) )
39 zsqcl 11168 . . . . . 6  |-  ( ( A  /  2 )  e.  ZZ  ->  (
( A  /  2
) ^ 2 )  e.  ZZ )
4031, 39syl 17 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  e.  ZZ )
4138, 40eqeltrrd 2359 . . . 4  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  ZZ )
4216nnrpd 10384 . . . . . 6  |-  ( ph  ->  ( B ^ 2 )  e.  RR+ )
4342rphalfcld 10397 . . . . 5  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  RR+ )
4443rpgt0d 10388 . . . 4  |-  ( ph  ->  0  <  ( ( B ^ 2 )  /  2 ) )
45 elnnz 10029 . . . 4  |-  ( ( ( B ^ 2 )  /  2 )  e.  NN  <->  ( (
( B ^ 2 )  /  2 )  e.  ZZ  /\  0  <  ( ( B ^
2 )  /  2
) ) )
4641, 44, 45sylanbrc 648 . . 3  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  NN )
47 nnesq 11219 . . . 4  |-  ( B  e.  NN  ->  (
( B  /  2
)  e.  NN  <->  ( ( B ^ 2 )  / 
2 )  e.  NN ) )
489, 47syl 17 . . 3  |-  ( ph  ->  ( ( B  / 
2 )  e.  NN  <->  ( ( B ^ 2 )  /  2 )  e.  NN ) )
4946, 48mpbird 225 . 2  |-  ( ph  ->  ( B  /  2
)  e.  NN )
5031, 49jca 520 1  |-  ( ph  ->  ( ( A  / 
2 )  e.  ZZ  /\  ( B  /  2
)  e.  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1628    e. wcel 1688    =/= wne 2447   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   CCcc 8730   0cc0 8732    x. cmul 8737    < clt 8862    / cdiv 9418   NNcn 9741   2c2 9790   ZZcz 10019   ^cexp 11098   sqrcsqr 11712
This theorem is referenced by:  sqr2irr  12521
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-sup 7189  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-seq 11041  df-exp 11099  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715
  Copyright terms: Public domain W3C validator