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Theorem sqrt2irrlem 11751
Description: Lemma for sqrt2irr 11752. 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 8754 . . . . . . . . . . . 12  |-  2  e.  RR
2 0le2 8774 . . . . . . . . . . . 12  |-  0  <_  2
3 resqrtth 10758 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR  /\  0  <_  2 )  -> 
( ( sqr `  2
) ^ 2 )  =  2 )
41, 2, 3mp2an 422 . . . . . . . . . . 11  |-  ( ( sqr `  2 ) ^ 2 )  =  2
5 sqrt2irrlem.3 . . . . . . . . . . . 12  |-  ( ph  ->  ( sqr `  2
)  =  ( A  /  B ) )
65oveq1d 5757 . . . . . . . . . . 11  |-  ( ph  ->  ( ( sqr `  2
) ^ 2 )  =  ( ( A  /  B ) ^
2 ) )
74, 6syl5eqr 2164 . . . . . . . . . 10  |-  ( ph  ->  2  =  ( ( A  /  B ) ^ 2 ) )
8 sqrt2irrlem.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ZZ )
98zcnd 9132 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
10 sqrt2irrlem.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  NN )
1110nncnd 8698 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
1210nnap0d 8730 . . . . . . . . . . 11  |-  ( ph  ->  B #  0 )
139, 11, 12sqdivapd 10392 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  /  B ) ^ 2 )  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
147, 13eqtrd 2150 . . . . . . . . 9  |-  ( ph  ->  2  =  ( ( A ^ 2 )  /  ( B ^
2 ) ) )
1514oveq1d 5757 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( ( ( A ^ 2 )  /  ( B ^
2 ) )  x.  ( B ^ 2 ) ) )
169sqcld 10377 . . . . . . . . 9  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
1710nnsqcld 10400 . . . . . . . . . 10  |-  ( ph  ->  ( B ^ 2 )  e.  NN )
1817nncnd 8698 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
1917nnap0d 8730 . . . . . . . . 9  |-  ( ph  ->  ( B ^ 2 ) #  0 )
2016, 18, 19divcanap1d 8518 . . . . . . . 8  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
( B ^ 2 ) )  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2115, 20eqtrd 2150 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( B ^ 2 ) )  =  ( A ^
2 ) )
2221oveq1d 5757 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( ( A ^ 2 )  /  2 ) )
23 2cnd 8757 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
24 2ap0 8777 . . . . . . . 8  |-  2 #  0
2524a1i 9 . . . . . . 7  |-  ( ph  ->  2 #  0 )
2618, 23, 25divcanap3d 8522 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( B ^ 2 ) )  /  2
)  =  ( B ^ 2 ) )
2722, 26eqtr3d 2152 . . . . 5  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  =  ( B ^ 2 ) )
2827, 17eqeltrd 2194 . . . 4  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  NN )
2928nnzd 9130 . . 3  |-  ( ph  ->  ( ( A ^
2 )  /  2
)  e.  ZZ )
30 zesq 10365 . . . 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 166 . 2  |-  ( ph  ->  ( A  /  2
)  e.  ZZ )
33 2cn 8755 . . . . . . . . 9  |-  2  e.  CC
3433sqvali 10327 . . . . . . . 8  |-  ( 2 ^ 2 )  =  ( 2  x.  2 )
3534oveq2i 5753 . . . . . . 7  |-  ( ( A ^ 2 )  /  ( 2 ^ 2 ) )  =  ( ( A ^
2 )  /  (
2  x.  2 ) )
369, 23, 25sqdivapd 10392 . . . . . . 7  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( A ^ 2 )  /  ( 2 ^ 2 ) ) )
3716, 23, 23, 25, 25divdivap1d 8549 . . . . . . 7  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( A ^ 2 )  /  ( 2  x.  2 ) ) )
3835, 36, 373eqtr4a 2176 . . . . . 6  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( ( A ^ 2 )  /  2 )  /  2 ) )
3927oveq1d 5757 . . . . . 6  |-  ( ph  ->  ( ( ( A ^ 2 )  / 
2 )  /  2
)  =  ( ( B ^ 2 )  /  2 ) )
4038, 39eqtrd 2150 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  =  ( ( B ^ 2 )  /  2 ) )
41 zsqcl 10318 . . . . . 6  |-  ( ( A  /  2 )  e.  ZZ  ->  (
( A  /  2
) ^ 2 )  e.  ZZ )
4232, 41syl 14 . . . . 5  |-  ( ph  ->  ( ( A  / 
2 ) ^ 2 )  e.  ZZ )
4340, 42eqeltrrd 2195 . . . 4  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  ZZ )
4417nnrpd 9437 . . . . . 6  |-  ( ph  ->  ( B ^ 2 )  e.  RR+ )
4544rphalfcld 9451 . . . . 5  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  RR+ )
4645rpgt0d 9441 . . . 4  |-  ( ph  ->  0  <  ( ( B ^ 2 )  /  2 ) )
47 elnnz 9022 . . . 4  |-  ( ( ( B ^ 2 )  /  2 )  e.  NN  <->  ( (
( B ^ 2 )  /  2 )  e.  ZZ  /\  0  <  ( ( B ^
2 )  /  2
) ) )
4843, 46, 47sylanbrc 413 . . 3  |-  ( ph  ->  ( ( B ^
2 )  /  2
)  e.  NN )
49 nnesq 10366 . . . 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 166 . 2  |-  ( ph  ->  ( B  /  2
)  e.  NN )
5232, 51jca 304 1  |-  ( ph  ->  ( ( A  / 
2 )  e.  ZZ  /\  ( B  /  2
)  e.  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   RRcr 7587   0cc0 7588    x. cmul 7593    < clt 7768    <_ cle 7769   # cap 8310    / cdiv 8399   NNcn 8684   2c2 8735   ZZcz 9012   ^cexp 10247   sqrcsqrt 10723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-2 8743  df-3 8744  df-4 8745  df-n0 8936  df-z 9013  df-uz 9283  df-rp 9398  df-seqfrec 10174  df-exp 10248  df-rsqrt 10725
This theorem is referenced by:  sqrt2irr  11752
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