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Theorem sqr2irrlem 12542
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 9832 . . . . . . . . . . . 12  |-  2  e.  CC
2 sqrth 11864 . . . . . . . . . . . 12  |-  ( 2  e.  CC  ->  (
( sqr `  2
) ^ 2 )  =  2 )
31, 2ax-mp 8 . . . . . . . . . . 11  |-  ( ( sqr `  2 ) ^ 2 )  =  2
4 sqr2irrlem.3 . . . . . . . . . . . 12  |-  ( ph  ->  ( sqr `  2
)  =  ( A  /  B ) )
54oveq1d 5889 . . . . . . . . . . 11  |-  ( ph  ->  ( ( sqr `  2
) ^ 2 )  =  ( ( A  /  B ) ^
2 ) )
63, 5syl5eqr 2342 . . . . . . . . . 10  |-  ( ph  ->  2  =  ( ( A  /  B ) ^ 2 ) )
7 sqr2irrlem.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ZZ )
87zcnd 10134 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
9 sqr2irrlem.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  NN )
109nncnd 9778 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
119nnne0d 9806 . . . . . . . . . . 11  |-  ( ph  ->  B  =/=  0 )
128, 10, 11sqdivd 11274 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  /  B ) ^ 2 )  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
136, 12eqtrd 2328 . . . . . . . . 9  |-  ( ph  ->  2  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
1413oveq1d 5889 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( ( ( A ^ 2 )  /  ( B ^
2 ) )  x.  ( B ^ 2 ) ) )
158sqcld 11259 . . . . . . . . 9  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
169nnsqcld 11281 . . . . . . . . . 10  |-  ( ph  ->  ( B ^ 2 )  e.  NN )
1716nncnd 9778 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
1816nnne0d 9806 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  =/=  0 )
1915, 17, 18divcan1d 9553 . . . . . . . 8  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
( B ^ 2 ) )  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2014, 19eqtrd 2328 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2120oveq1d 5889 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( ( A ^ 2 )  /  2 ) )
221a1i 10 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
23 2ne0 9845 . . . . . . . 8  |-  2  =/=  0
2423a1i 10 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
2517, 22, 24divcan3d 9557 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( B ^ 2 ) )
2621, 25eqtr3d 2330 . . . . 5  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  =  ( B ^ 2 ) )
2726, 16eqeltrd 2370 . . . 4  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  NN )
2827nnzd 10132 . . 3  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  ZZ )
29 zesq 11240 . . . 4  |-  ( A  e.  ZZ  ->  (
( A  /  2
)  e.  ZZ  <->  ( ( A ^ 2 )  / 
2 )  e.  ZZ ) )
307, 29syl 15 . . 3  |-  ( ph  ->  ( ( A  / 
2 )  e.  ZZ  <->  ( ( A ^ 2 )  /  2 )  e.  ZZ ) )
3128, 30mpbird 223 . 2  |-  ( ph  ->  ( A  /  2
)  e.  ZZ )
321sqvali 11199 . . . . . . . 8  |-  ( 2 ^ 2 )  =  ( 2  x.  2 )
3332oveq2i 5885 . . . . . . 7  |-  ( ( A ^ 2 )  /  ( 2 ^ 2 ) )  =  ( ( A ^
2 )  /  (
2  x.  2 ) )
348, 22, 24sqdivd 11274 . . . . . . 7  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( A ^ 2 )  /  ( 2 ^ 2 ) ) )
3515, 22, 22, 24, 24divdiv1d 9583 . . . . . . 7  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( A ^ 2 )  /  ( 2  x.  2 ) ) )
3633, 34, 353eqtr4a 2354 . . . . . 6  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( ( A ^ 2 )  /  2 )  /  2 ) )
3726oveq1d 5889 . . . . . 6  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( B ^ 2 )  /  2 ) )
3836, 37eqtrd 2328 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( B ^ 2 )  /  2 ) )
39 zsqcl 11190 . . . . . 6  |-  ( ( A  /  2 )  e.  ZZ  ->  (
( A  /  2
) ^ 2 )  e.  ZZ )
4031, 39syl 15 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  e.  ZZ )
4138, 40eqeltrrd 2371 . . . 4  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  ZZ )
4216nnrpd 10405 . . . . . 6  |-  ( ph  ->  ( B ^ 2 )  e.  RR+ )
4342rphalfcld 10418 . . . . 5  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  RR+ )
4443rpgt0d 10409 . . . 4  |-  ( ph  ->  0  <  ( ( B ^ 2 )  /  2 ) )
45 elnnz 10050 . . . 4  |-  ( ( ( B ^ 2 )  /  2 )  e.  NN  <->  ( (
( B ^ 2 )  /  2 )  e.  ZZ  /\  0  <  ( ( B ^
2 )  /  2
) ) )
4641, 44, 45sylanbrc 645 . . 3  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  NN )
47 nnesq 11241 . . . 4  |-  ( B  e.  NN  ->  (
( B  /  2
)  e.  NN  <->  ( ( B ^ 2 )  / 
2 )  e.  NN ) )
489, 47syl 15 . . 3  |-  ( ph  ->  ( ( B  / 
2 )  e.  NN  <->  ( ( B ^ 2 )  /  2 )  e.  NN ) )
4946, 48mpbird 223 . 2  |-  ( ph  ->  ( B  /  2
)  e.  NN )
5031, 49jca 518 1  |-  ( ph  ->  ( ( A  / 
2 )  e.  ZZ  /\  ( B  /  2
)  e.  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753    x. cmul 8758    < clt 8883    / cdiv 9439   NNcn 9762   2c2 9811   ZZcz 10040   ^cexp 11120   sqrcsqr 11734
This theorem is referenced by:  sqr2irr  12543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737
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