Theorem pythagtriplem15 15534

Description: Lemma for pythagtrip 15539. 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 15531	. . . 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 15533	. . . 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 6668	. . 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 1063	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. NN )
7		simp1 1061	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. NN )
8	6, 7	nnaddcld 11067	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + A ) e. NN )
9	8	nncnd 11036	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + A ) e. CC )
10	9	3ad2ant1 1082	. . . 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 11399	. . . . . . . 8 |- ( C e. NN -> C e. ZZ )
12	11	3ad2ant3 1084	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. ZZ )
13		nnz 11399	. . . . . . . 8 |- ( A e. NN -> A e. ZZ )
14	13	3ad2ant1 1082	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. ZZ )
15	12, 14	zsubcld 11487	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - A ) e. ZZ )
16	15	zcnd 11483	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - A ) e. CC )
17	16	3ad2ant1 1082	. . . 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		2cnne0 11242	. . . . 5 |- ( 2 e. CC /\ 2 =/= 0 )
19		divsubdir 10721	. . . . 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 ) ) )
20	18, 19	mp3an3 1413	. . . 4 |- ( ( ( C + A ) e. CC /\ ( C - A ) e. CC ) -> ( ( ( C + A ) - ( C - A ) ) / 2 ) = ( ( ( C + A ) / 2 ) - ( ( C - A ) / 2 ) ) )
21	10, 17, 20	syl2anc 693	. . 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 2659	. 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 11028	. . . . . . 7 |- ( C e. NN -> C e. CC )
24	23	3ad2ant3 1084	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> C e. CC )
25	24	3ad2ant1 1082	. . . . 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 11028	. . . . . . 7 |- ( A e. NN -> A e. CC )
27	26	3ad2ant1 1082	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. CC )
28	27	3ad2ant1 1082	. . . . 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 10431	. . . 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 11275	. . . 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 2659	. . 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 6665	. 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		2cn 11091	. . . 4 |- 2 e. CC
34		2ne0 11113	. . . 4 |- 2 =/= 0
35		divcan3 10711	. . . 4 |- ( ( A e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( ( 2 x. A ) / 2 ) = A )
36	33, 34, 35	mp3an23 1416	. . 3 |- ( A e. CC -> ( ( 2 x. A ) / 2 ) = A )
37	28, 36	syl 17	. 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 )
38	22, 32, 37	3eqtrrd 2661	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   ZZcz 11377   ^cexp 12860   sqrcsqrt 13973   || cdvds 14983   gcd cgcd 15216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976

This theorem is referenced by:  pythagtriplem18  15537