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Theorem sqr2irrlem 13632
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 )
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 10493 . . . . . . . . . . . 12  |-  2  e.  CC
2 sqrth 12954 . . . . . . . . . . . 12  |-  ( 2  e.  CC  ->  (
( sqr `  2
) ^ 2 )  =  2 )
31, 2ax-mp 5 . . . . . . . . . . 11  |-  ( ( sqr `  2 ) ^ 2 )  =  2
4 sqr2irrlem.3 . . . . . . . . . . . 12  |-  ( ph  ->  ( sqr `  2
)  =  ( A  /  B ) )
54oveq1d 6205 . . . . . . . . . . 11  |-  ( ph  ->  ( ( sqr `  2
) ^ 2 )  =  ( ( A  /  B ) ^
2 ) )
63, 5syl5eqr 2506 . . . . . . . . . 10  |-  ( ph  ->  2  =  ( ( A  /  B ) ^ 2 ) )
7 sqr2irrlem.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ZZ )
87zcnd 10849 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
9 sqr2irrlem.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  NN )
109nncnd 10439 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
119nnne0d 10467 . . . . . . . . . . 11  |-  ( ph  ->  B  =/=  0 )
128, 10, 11sqdivd 12122 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  /  B ) ^ 2 )  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
136, 12eqtrd 2492 . . . . . . . . 9  |-  ( ph  ->  2  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
1413oveq1d 6205 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( ( ( A ^ 2 )  /  ( B ^
2 ) )  x.  ( B ^ 2 ) ) )
158sqcld 12107 . . . . . . . . 9  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
169nnsqcld 12129 . . . . . . . . . 10  |-  ( ph  ->  ( B ^ 2 )  e.  NN )
1716nncnd 10439 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
1816nnne0d 10467 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  =/=  0 )
1915, 17, 18divcan1d 10209 . . . . . . . 8  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
( B ^ 2 ) )  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2014, 19eqtrd 2492 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2120oveq1d 6205 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( ( A ^ 2 )  /  2 ) )
221a1i 11 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
23 2ne0 10515 . . . . . . . 8  |-  2  =/=  0
2423a1i 11 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
2517, 22, 24divcan3d 10213 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( B ^ 2 ) )
2621, 25eqtr3d 2494 . . . . 5  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  =  ( B ^ 2 ) )
2726, 16eqeltrd 2539 . . . 4  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  NN )
2827nnzd 10847 . . 3  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  ZZ )
29 zesq 12088 . . . 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 12046 . . . . . . . 8  |-  ( 2 ^ 2 )  =  ( 2  x.  2 )
3332oveq2i 6201 . . . . . . 7  |-  ( ( A ^ 2 )  /  ( 2 ^ 2 ) )  =  ( ( A ^
2 )  /  (
2  x.  2 ) )
348, 22, 24sqdivd 12122 . . . . . . 7  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( A ^ 2 )  /  ( 2 ^ 2 ) ) )
3515, 22, 22, 24, 24divdiv1d 10239 . . . . . . 7  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( A ^ 2 )  /  ( 2  x.  2 ) ) )
3633, 34, 353eqtr4a 2518 . . . . . 6  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( ( A ^ 2 )  /  2 )  /  2 ) )
3726oveq1d 6205 . . . . . 6  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( B ^ 2 )  /  2 ) )
3836, 37eqtrd 2492 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( B ^ 2 )  /  2 ) )
39 zsqcl 12037 . . . . . 6  |-  ( ( A  /  2 )  e.  ZZ  ->  (
( A  /  2
) ^ 2 )  e.  ZZ )
4031, 39syl 16 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  e.  ZZ )
4138, 40eqeltrrd 2540 . . . 4  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  ZZ )
4216nnrpd 11127 . . . . . 6  |-  ( ph  ->  ( B ^ 2 )  e.  RR+ )
4342rphalfcld 11140 . . . . 5  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  RR+ )
4443rpgt0d 11131 . . . 4  |-  ( ph  ->  0  <  ( ( B ^ 2 )  /  2 ) )
45 elnnz 10757 . . . 4  |-  ( ( ( B ^ 2 )  /  2 )  e.  NN  <->  ( (
( B ^ 2 )  /  2 )  e.  ZZ  /\  0  <  ( ( B ^
2 )  /  2
) ) )
4641, 44, 45sylanbrc 664 . . 3  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  NN )
47 nnesq 12089 . . . 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 532 1  |-  ( ph  ->  ( ( A  / 
2 )  e.  ZZ  /\  ( B  /  2
)  e.  NN ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   CCcc 9381   0cc0 9383    x. cmul 9388    < clt 9519    / cdiv 10094   NNcn 10423   2c2 10472   ZZcz 10747   ^cexp 11966   sqrcsqr 12824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-seq 11908  df-exp 11967  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827
This theorem is referenced by:  sqr2irr  13633
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