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Theorem pythagtriplem17 14203
Description: Lemma for pythagtrip 14206. 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 14198 . . . 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 14200 . . . 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 6293 . . 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 10533 . . . . . . 7  |-  ( C  e.  NN  ->  C  e.  CC )
763ad2ant3 1014 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  CC )
873ad2ant1 1012 . . . . 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 10533 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  CC )
1093ad2ant1 1012 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  CC )
11103ad2ant1 1012 . . . . 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 9604 . . . 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 9919 . . . 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 2cnne0 10739 . . . . 5  |-  ( 2  e.  CC  /\  2  =/=  0 )
15 divdir 10219 . . . . 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 ) ) )
1614, 15mp3an3 1308 . . . 4  |-  ( ( ( C  +  A
)  e.  CC  /\  ( C  -  A
)  e.  CC )  ->  ( ( ( C  +  A )  +  ( C  -  A ) )  / 
2 )  =  ( ( ( C  +  A )  /  2
)  +  ( ( C  -  A )  /  2 ) ) )
1712, 13, 16syl2anc 661 . . 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
) ) )
185, 17eqtr4d 2504 . 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 ) )
198, 11, 8ppncand 9959 . . . 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 ) )
2082timesd 10770 . . . 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 ) )
2119, 20eqtr4d 2504 . . 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 ) )
2221oveq1d 6290 . 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 ) )
23 2cn 10595 . . . 4  |-  2  e.  CC
24 2ne0 10617 . . . 4  |-  2  =/=  0
25 divcan3 10220 . . . 4  |-  ( ( C  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  C
)  /  2 )  =  C )
2623, 24, 25mp3an23 1311 . . 3  |-  ( C  e.  CC  ->  (
( 2  x.  C
)  /  2 )  =  C )
278, 26syl 16 . 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 )
2818, 22, 273eqtrrd 2506 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9794    / cdiv 10195   NNcn 10525   2c2 10574   ^cexp 12122   sqrcsqr 13016    || cdivides 13836    gcd cgcd 13992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019
This theorem is referenced by:  pythagtriplem18  14204
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