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Theorem sqrt2irrlem 10934
Description: Lemma for sqrt2irr 10935. This is 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
sqrt2irrlem.1  |-  ( ph  ->  A  e.  ZZ )
sqrt2irrlem.2  |-  ( ph  ->  B  e.  NN )
sqrt2irrlem.3  |-  ( ph  ->  ( sqr `  2
)  =  ( A  /  B ) )
Assertion
Ref Expression
sqrt2irrlem  |-  ( ph  ->  ( ( A  / 
2 )  e.  ZZ  /\  ( B  /  2
)  e.  NN ) )

Proof of Theorem sqrt2irrlem
StepHypRef Expression
1 2re 8404 . . . . . . . . . . . 12  |-  2  e.  RR
2 0le2 8424 . . . . . . . . . . . 12  |-  0  <_  2
3 resqrtth 10305 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR  /\  0  <_  2 )  -> 
( ( sqr `  2
) ^ 2 )  =  2 )
41, 2, 3mp2an 417 . . . . . . . . . . 11  |-  ( ( sqr `  2 ) ^ 2 )  =  2
5 sqrt2irrlem.3 . . . . . . . . . . . 12  |-  ( ph  ->  ( sqr `  2
)  =  ( A  /  B ) )
65oveq1d 5609 . . . . . . . . . . 11  |-  ( ph  ->  ( ( sqr `  2
) ^ 2 )  =  ( ( A  /  B ) ^
2 ) )
74, 6syl5eqr 2131 . . . . . . . . . 10  |-  ( ph  ->  2  =  ( ( A  /  B ) ^ 2 ) )
8 sqrt2irrlem.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ZZ )
98zcnd 8779 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
10 sqrt2irrlem.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  NN )
1110nncnd 8348 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
1210nnap0d 8379 . . . . . . . . . . 11  |-  ( ph  ->  B #  0 )
139, 11, 12sqdivapd 9948 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  /  B ) ^ 2 )  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
147, 13eqtrd 2117 . . . . . . . . 9  |-  ( ph  ->  2  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
1514oveq1d 5609 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( ( ( A ^ 2 )  /  ( B ^
2 ) )  x.  ( B ^ 2 ) ) )
169sqcld 9933 . . . . . . . . 9  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
1710nnsqcld 9956 . . . . . . . . . 10  |-  ( ph  ->  ( B ^ 2 )  e.  NN )
1817nncnd 8348 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
1917nnap0d 8379 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 ) #  0 )
2016, 18, 19divcanap1d 8173 . . . . . . . 8  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
( B ^ 2 ) )  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2115, 20eqtrd 2117 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2221oveq1d 5609 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( ( A ^ 2 )  /  2 ) )
23 2cnd 8407 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
24 2ap0 8427 . . . . . . . 8  |-  2 #  0
2524a1i 9 . . . . . . 7  |-  ( ph  ->  2 #  0 )
2618, 23, 25divcanap3d 8177 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( B ^ 2 ) )
2722, 26eqtr3d 2119 . . . . 5  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  =  ( B ^ 2 ) )
2827, 17eqeltrd 2161 . . . 4  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  NN )
2928nnzd 8777 . . 3  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  ZZ )
30 zesq 9921 . . . 4  |-  ( A  e.  ZZ  ->  (
( A  /  2
)  e.  ZZ  <->  ( ( A ^ 2 )  / 
2 )  e.  ZZ ) )
318, 30syl 14 . . 3  |-  ( ph  ->  ( ( A  / 
2 )  e.  ZZ  <->  ( ( A ^ 2 )  /  2 )  e.  ZZ ) )
3229, 31mpbird 165 . 2  |-  ( ph  ->  ( A  /  2
)  e.  ZZ )
33 2cn 8405 . . . . . . . . 9  |-  2  e.  CC
3433sqvali 9885 . . . . . . . 8  |-  ( 2 ^ 2 )  =  ( 2  x.  2 )
3534oveq2i 5605 . . . . . . 7  |-  ( ( A ^ 2 )  /  ( 2 ^ 2 ) )  =  ( ( A ^
2 )  /  (
2  x.  2 ) )
369, 23, 25sqdivapd 9948 . . . . . . 7  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( A ^ 2 )  /  ( 2 ^ 2 ) ) )
3716, 23, 23, 25, 25divdivap1d 8203 . . . . . . 7  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( A ^ 2 )  /  ( 2  x.  2 ) ) )
3835, 36, 373eqtr4a 2143 . . . . . 6  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( ( A ^ 2 )  /  2 )  /  2 ) )
3927oveq1d 5609 . . . . . 6  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( B ^ 2 )  /  2 ) )
4038, 39eqtrd 2117 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( B ^ 2 )  /  2 ) )
41 zsqcl 9876 . . . . . 6  |-  ( ( A  /  2 )  e.  ZZ  ->  (
( A  /  2
) ^ 2 )  e.  ZZ )
4232, 41syl 14 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  e.  ZZ )
4340, 42eqeltrrd 2162 . . . 4  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  ZZ )
4417nnrpd 9081 . . . . . 6  |-  ( ph  ->  ( B ^ 2 )  e.  RR+ )
4544rphalfcld 9095 . . . . 5  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  RR+ )
4645rpgt0d 9085 . . . 4  |-  ( ph  ->  0  <  ( ( B ^ 2 )  /  2 ) )
47 elnnz 8670 . . . 4  |-  ( ( ( B ^ 2 )  /  2 )  e.  NN  <->  ( (
( B ^ 2 )  /  2 )  e.  ZZ  /\  0  <  ( ( B ^
2 )  /  2
) ) )
4843, 46, 47sylanbrc 408 . . 3  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  NN )
49 nnesq 9922 . . . 4  |-  ( B  e.  NN  ->  (
( B  /  2
)  e.  NN  <->  ( ( B ^ 2 )  / 
2 )  e.  NN ) )
5010, 49syl 14 . . 3  |-  ( ph  ->  ( ( B  / 
2 )  e.  NN  <->  ( ( B ^ 2 )  /  2 )  e.  NN ) )
5148, 50mpbird 165 . 2  |-  ( ph  ->  ( B  /  2
)  e.  NN )
5232, 51jca 300 1  |-  ( ph  ->  ( ( A  / 
2 )  e.  ZZ  /\  ( B  /  2
)  e.  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436   class class class wbr 3814   ` cfv 4972  (class class class)co 5594   RRcr 7270   0cc0 7271    x. cmul 7276    < clt 7443    <_ cle 7444   # cap 7976    / cdiv 8055   NNcn 8334   2c2 8384   ZZcz 8660   ^cexp 9805   sqrcsqrt 10270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-mulrcl 7365  ax-addcom 7366  ax-mulcom 7367  ax-addass 7368  ax-mulass 7369  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-1rid 7373  ax-0id 7374  ax-rnegex 7375  ax-precex 7376  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382  ax-pre-mulgt0 7383  ax-pre-mulext 7384  ax-arch 7385  ax-caucvg 7386
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-if 3377  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-ilim 4163  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-frec 6091  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-reap 7970  df-ap 7977  df-div 8056  df-inn 8335  df-2 8393  df-3 8394  df-4 8395  df-n0 8584  df-z 8661  df-uz 8929  df-rp 9044  df-iseq 9755  df-iexp 9806  df-rsqrt 10272
This theorem is referenced by:  sqrt2irr  10935
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