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Theorem sqrt2irrlem 11046
Description: Lemma for sqrt2irr 11047. 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 8430 . . . . . . . . . . . 12  |-  2  e.  RR
2 0le2 8450 . . . . . . . . . . . 12  |-  0  <_  2
3 resqrtth 10363 . . . . . . . . . . . 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 5630 . . . . . . . . . . 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 8805 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
10 sqrt2irrlem.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  NN )
1110nncnd 8374 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
1210nnap0d 8405 . . . . . . . . . . 11  |-  ( ph  ->  B #  0 )
139, 11, 12sqdivapd 9999 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  /  B ) ^ 2 )  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
147, 13eqtrd 2117 . . . . . . . . 9  |-  ( ph  ->  2  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
1514oveq1d 5630 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( ( ( A ^ 2 )  /  ( B ^
2 ) )  x.  ( B ^ 2 ) ) )
169sqcld 9984 . . . . . . . . 9  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
1710nnsqcld 10007 . . . . . . . . . 10  |-  ( ph  ->  ( B ^ 2 )  e.  NN )
1817nncnd 8374 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
1917nnap0d 8405 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 ) #  0 )
2016, 18, 19divcanap1d 8199 . . . . . . . 8  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
( B ^ 2 ) )  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2115, 20eqtrd 2117 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2221oveq1d 5630 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( ( A ^ 2 )  /  2 ) )
23 2cnd 8433 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
24 2ap0 8453 . . . . . . . 8  |-  2 #  0
2524a1i 9 . . . . . . 7  |-  ( ph  ->  2 #  0 )
2618, 23, 25divcanap3d 8203 . . . . . 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 8803 . . 3  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  ZZ )
30 zesq 9972 . . . 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 8431 . . . . . . . . 9  |-  2  e.  CC
3433sqvali 9936 . . . . . . . 8  |-  ( 2 ^ 2 )  =  ( 2  x.  2 )
3534oveq2i 5626 . . . . . . 7  |-  ( ( A ^ 2 )  /  ( 2 ^ 2 ) )  =  ( ( A ^
2 )  /  (
2  x.  2 ) )
369, 23, 25sqdivapd 9999 . . . . . . 7  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( A ^ 2 )  /  ( 2 ^ 2 ) ) )
3716, 23, 23, 25, 25divdivap1d 8229 . . . . . . 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 5630 . . . . . 6  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( B ^ 2 )  /  2 ) )
4038, 39eqtrd 2117 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( B ^ 2 )  /  2 ) )
41 zsqcl 9927 . . . . . 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 9107 . . . . . 6  |-  ( ph  ->  ( B ^ 2 )  e.  RR+ )
4544rphalfcld 9121 . . . . 5  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  RR+ )
4645rpgt0d 9111 . . . 4  |-  ( ph  ->  0  <  ( ( B ^ 2 )  /  2 ) )
47 elnnz 8696 . . . 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 9973 . . . 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 3822   ` cfv 4983  (class class class)co 5615   RRcr 7296   0cc0 7297    x. cmul 7302    < clt 7469    <_ cle 7470   # cap 8002    / cdiv 8081   NNcn 8360   2c2 8410   ZZcz 8686   ^cexp 9856   sqrcsqrt 10328
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 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378  ax-cnex 7383  ax-resscn 7384  ax-1cn 7385  ax-1re 7386  ax-icn 7387  ax-addcl 7388  ax-addrcl 7389  ax-mulcl 7390  ax-mulrcl 7391  ax-addcom 7392  ax-mulcom 7393  ax-addass 7394  ax-mulass 7395  ax-distr 7396  ax-i2m1 7397  ax-0lt1 7398  ax-1rid 7399  ax-0id 7400  ax-rnegex 7401  ax-precex 7402  ax-cnre 7403  ax-pre-ltirr 7404  ax-pre-ltwlin 7405  ax-pre-lttrn 7406  ax-pre-apti 7407  ax-pre-ltadd 7408  ax-pre-mulgt0 7409  ax-pre-mulext 7410  ax-arch 7411  ax-caucvg 7412
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 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-id 4096  df-po 4099  df-iso 4100  df-iord 4169  df-on 4171  df-ilim 4172  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-riota 5571  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-frec 6112  df-pnf 7471  df-mnf 7472  df-xr 7473  df-ltxr 7474  df-le 7475  df-sub 7602  df-neg 7603  df-reap 7996  df-ap 8003  df-div 8082  df-inn 8361  df-2 8419  df-3 8420  df-4 8421  df-n0 8610  df-z 8687  df-uz 8955  df-rp 9070  df-iseq 9783  df-iexp 9857  df-rsqrt 10330
This theorem is referenced by:  sqrt2irr  11047
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