Theorem pythagtriplem15 12169

Description: Lemma for pythagtrip 12174. Show the relationship between M, N, and A. (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
pythagtriplem15	|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> A = ( ( M ^ 2 ) - ( N ^ 2 ) ) )

Proof of Theorem pythagtriplem15

Step	Hyp	Ref	Expression
1		pythagtriplem15.1	. . . . 5 |- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )
2	1	pythagtriplem12 12166	. . . 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 )
4	3	pythagtriplem14 12168	. . . 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 ) )
5	2, 4	oveq12d 5845	. . 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		simp3 984	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. NN )
7		simp1 982	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. NN )
8	6, 7	nnaddcld 8887	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + A ) e. NN )
9	8	nncnd 8853	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + A ) e. CC )
10	9	3ad2ant1 1003	. . . 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 )
11		nnz 9192	. . . . . . . 8 |- ( C e. NN -> C e. ZZ )
12	11	3ad2ant3 1005	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. ZZ )
13		nnz 9192	. . . . . . . 8 |- ( A e. NN -> A e. ZZ )
14	13	3ad2ant1 1003	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. ZZ )
15	12, 14	zsubcld 9297	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - A ) e. ZZ )
16	15	zcnd 9293	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - A ) e. CC )
17	16	3ad2ant1 1003	. . . 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 )
18		2cnd 8912	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> 2 e. CC )
19		2ap0 8932	. . . . 5 |- 2 # 0
20	19	a1i 9	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> 2 # 0 )
21	10, 17, 18, 20	divsubdirapd 8708	. . 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 ) ) )
22	5, 21	eqtr4d 2193	. 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 ) )
23		nncn 8847	. . . . . . 7 |- ( C e. NN -> C e. CC )
24	23	3ad2ant3 1005	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. CC )
25	24	3ad2ant1 1003	. . . . 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 )
26		nncn 8847	. . . . . . 7 |- ( A e. NN -> A e. CC )
27	26	3ad2ant1 1003	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. CC )
28	27	3ad2ant1 1003	. . . . 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 )
29	25, 28, 28	pnncand 8230	. . . 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 ) ) = ( A + A ) )
30	28	2timesd 9081	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( 2 x. A ) = ( A + A ) )
31	29, 30	eqtr4d 2193	. . 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. A ) )
32	31	oveq1d 5842	. 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. A ) / 2 ) )
33	28, 18, 20	divcanap3d 8673	. 2 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( 2 x. A ) / 2 ) = A )
34	22, 32, 33	3eqtrrd 2195	1 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> A = ( ( M ^ 2 ) - ( N ^ 2 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1335   e. wcel 2128   class class class wbr 3967   ` cfv 5173  (class class class)co 5827   CCcc 7733   0cc0 7735   1c1 7736   + caddc 7738   x. cmul 7740   - cmin 8051   # cap 8461   / cdiv 8550   NNcn 8839   2c2 8890   ZZcz 9173   ^cexp 10428   sqrcsqrt 10908   || cdvds 11695   gcd cgcd 11842

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-mulrcl 7834  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-mulass 7838  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-1rid 7842  ax-0id 7843  ax-rnegex 7844  ax-precex 7845  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-apti 7850  ax-pre-ltadd 7851  ax-pre-mulgt0 7852  ax-pre-mulext 7853  ax-arch 7854  ax-caucvg 7855

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-po 4259  df-iso 4260  df-iord 4329  df-on 4331  df-ilim 4332  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-frec 6341  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-reap 8455  df-ap 8462  df-div 8551  df-inn 8840  df-2 8898  df-3 8899  df-4 8900  df-n0 9097  df-z 9174  df-uz 9446  df-rp 9568  df-seqfrec 10355  df-exp 10429  df-rsqrt 10910

This theorem is referenced by:  pythagtriplem18  12172