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Theorem pythagtriplem17 12884
Description: Lemma for pythagtrip 12887. Show the relationship between  M,  N, and  C. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
pythagtriplem15.1  |-  M  =  ( ( ( sqr `  ( C  +  B
) )  +  ( sqr `  ( C  -  B ) ) )  /  2 )
pythagtriplem15.2  |-  N  =  ( ( ( sqr `  ( C  +  B
) )  -  ( sqr `  ( C  -  B ) ) )  /  2 )
Assertion
Ref Expression
pythagtriplem17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  C  =  ( ( M ^ 2 )  +  ( N ^ 2 ) ) )

Proof of Theorem pythagtriplem17
StepHypRef Expression
1 pythagtriplem15.1 . . . . 5  |-  M  =  ( ( ( sqr `  ( C  +  B
) )  +  ( sqr `  ( C  -  B ) ) )  /  2 )
21pythagtriplem12 12879 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( M ^ 2 )  =  ( ( C  +  A )  /  2
) )
3 pythagtriplem15.2 . . . . 5  |-  N  =  ( ( ( sqr `  ( C  +  B
) )  -  ( sqr `  ( C  -  B ) ) )  /  2 )
43pythagtriplem14 12881 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( N ^ 2 )  =  ( ( C  -  A )  /  2
) )
52, 4oveq12d 5876 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( M ^ 2 )  +  ( N ^ 2 ) )  =  ( ( ( C  +  A )  /  2 )  +  ( ( C  -  A )  /  2
) ) )
6 nncn 9754 . . . . . . 7  |-  ( C  e.  NN  ->  C  e.  CC )
763ad2ant3 978 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  CC )
873ad2ant1 976 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  C  e.  CC )
9 nncn 9754 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  CC )
1093ad2ant1 976 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  CC )
11103ad2ant1 976 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  A  e.  CC )
128, 11addcld 8854 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  +  A )  e.  CC )
138, 11subcld 9157 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  -  A )  e.  CC )
14 2cn 9816 . . . . . 6  |-  2  e.  CC
15 2ne0 9829 . . . . . 6  |-  2  =/=  0
1614, 15pm3.2i 441 . . . . 5  |-  ( 2  e.  CC  /\  2  =/=  0 )
17 divdir 9447 . . . . 5  |-  ( ( ( C  +  A
)  e.  CC  /\  ( C  -  A
)  e.  CC  /\  ( 2  e.  CC  /\  2  =/=  0 ) )  ->  ( (
( C  +  A
)  +  ( C  -  A ) )  /  2 )  =  ( ( ( C  +  A )  / 
2 )  +  ( ( C  -  A
)  /  2 ) ) )
1816, 17mp3an3 1266 . . . 4  |-  ( ( ( C  +  A
)  e.  CC  /\  ( C  -  A
)  e.  CC )  ->  ( ( ( C  +  A )  +  ( C  -  A ) )  / 
2 )  =  ( ( ( C  +  A )  /  2
)  +  ( ( C  -  A )  /  2 ) ) )
1912, 13, 18syl2anc 642 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( C  +  A )  +  ( C  -  A ) )  /  2 )  =  ( ( ( C  +  A )  /  2 )  +  ( ( C  -  A )  /  2
) ) )
205, 19eqtr4d 2318 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( M ^ 2 )  +  ( N ^ 2 ) )  =  ( ( ( C  +  A )  +  ( C  -  A ) )  / 
2 ) )
218, 11, 8ppncand 9197 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  +  A
)  +  ( C  -  A ) )  =  ( C  +  C ) )
2282timesd 9954 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
2  x.  C )  =  ( C  +  C ) )
2321, 22eqtr4d 2318 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  +  A
)  +  ( C  -  A ) )  =  ( 2  x.  C ) )
2423oveq1d 5873 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( C  +  A )  +  ( C  -  A ) )  /  2 )  =  ( ( 2  x.  C )  / 
2 ) )
25 divcan3 9448 . . . 4  |-  ( ( C  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  C
)  /  2 )  =  C )
2614, 15, 25mp3an23 1269 . . 3  |-  ( C  e.  CC  ->  (
( 2  x.  C
)  /  2 )  =  C )
278, 26syl 15 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( 2  x.  C
)  /  2 )  =  C )
2820, 24, 273eqtrrd 2320 1  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  C  =  ( ( M ^ 2 )  +  ( N ^ 2 ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   ^cexp 11104   sqrcsqr 11718    || cdivides 12531    gcd cgcd 12685
This theorem is referenced by:  pythagtriplem18  12885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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