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Theorem sqr2irrlem 12400
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 9696 . . . . . . . . . . . 12  |-  2  e.  CC
2 sqrth 11725 . . . . . . . . . . . 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 5725 . . . . . . . . . . 11  |-  ( ph  ->  ( ( sqr `  2
) ^ 2 )  =  ( ( A  /  B ) ^
2 ) )
63, 5syl5eqr 2299 . . . . . . . . . 10  |-  ( ph  ->  2  =  ( ( A  /  B ) ^ 2 ) )
7 sqr2irrlem.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ZZ )
87zcnd 9997 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
9 sqr2irrlem.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  NN )
109nncnd 9642 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
119nnne0d 9670 . . . . . . . . . . 11  |-  ( ph  ->  B  =/=  0 )
128, 10, 11sqdivd 11136 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  /  B ) ^ 2 )  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
136, 12eqtrd 2285 . . . . . . . . 9  |-  ( ph  ->  2  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
1413oveq1d 5725 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( ( ( A ^ 2 )  /  ( B ^
2 ) )  x.  ( B ^ 2 ) ) )
158sqcld 11121 . . . . . . . . 9  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
169nnsqcld 11143 . . . . . . . . . 10  |-  ( ph  ->  ( B ^ 2 )  e.  NN )
1716nncnd 9642 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
1816nnne0d 9670 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  =/=  0 )
1915, 17, 18divcan1d 9417 . . . . . . . 8  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
( B ^ 2 ) )  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2014, 19eqtrd 2285 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2120oveq1d 5725 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( ( A ^ 2 )  /  2 ) )
221a1i 12 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
23 2ne0 9709 . . . . . . . 8  |-  2  =/=  0
2423a1i 12 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
2517, 22, 24divcan3d 9421 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( B ^ 2 ) )
2621, 25eqtr3d 2287 . . . . 5  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  =  ( B ^ 2 ) )
2726, 16eqeltrd 2327 . . . 4  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  NN )
2827nnzd 9995 . . 3  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  ZZ )
29 zesq 11102 . . . 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 11061 . . . . . . . 8  |-  ( 2 ^ 2 )  =  ( 2  x.  2 )
3332oveq2i 5721 . . . . . . 7  |-  ( ( A ^ 2 )  /  ( 2 ^ 2 ) )  =  ( ( A ^
2 )  /  (
2  x.  2 ) )
348, 22, 24sqdivd 11136 . . . . . . 7  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( A ^ 2 )  /  ( 2 ^ 2 ) ) )
3515, 22, 22, 24, 24divdiv1d 9447 . . . . . . 7  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( A ^ 2 )  /  ( 2  x.  2 ) ) )
3633, 34, 353eqtr4a 2311 . . . . . 6  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( ( A ^ 2 )  /  2 )  /  2 ) )
3726oveq1d 5725 . . . . . 6  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( B ^ 2 )  /  2 ) )
3836, 37eqtrd 2285 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( B ^ 2 )  /  2 ) )
39 zsqcl 11052 . . . . . 6  |-  ( ( A  /  2 )  e.  ZZ  ->  (
( A  /  2
) ^ 2 )  e.  ZZ )
4031, 39syl 17 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  e.  ZZ )
4138, 40eqeltrrd 2328 . . . 4  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  ZZ )
4216nnrpd 10268 . . . . . 6  |-  ( ph  ->  ( B ^ 2 )  e.  RR+ )
4342rphalfcld 10281 . . . . 5  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  RR+ )
4443rpgt0d 10272 . . . 4  |-  ( ph  ->  0  <  ( ( B ^ 2 )  /  2 ) )
45 elnnz 9913 . . . 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 11103 . . . 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 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   CCcc 8615   0cc0 8617    x. cmul 8622    < clt 8747    / cdiv 9303   NNcn 9626   2c2 9675   ZZcz 9903   ^cexp 10982   sqrcsqr 11595
This theorem is referenced by:  sqr2irr  12401
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
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 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-sup 7078  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-seq 10925  df-exp 10983  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598
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