Theorem pythagtriplem4 12706

Description: Lemma for pythagtrip 12721. Show that C - B and C + B are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
pythagtriplem4	|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) = 1 )

Proof of Theorem pythagtriplem4

Step	Hyp	Ref	Expression
1		simp3r 1029	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> -. 2 || A )
2		nnz 9426	. . . . . . . . . . . . 13 |- ( C e. NN -> C e. ZZ )
3		nnz 9426	. . . . . . . . . . . . 13 |- ( B e. NN -> B e. ZZ )
4		zsubcl 9448	. . . . . . . . . . . . 13 |- ( ( C e. ZZ /\ B e. ZZ ) -> ( C - B ) e. ZZ )
5	2, 3, 4	syl2anr 290	. . . . . . . . . . . 12 |- ( ( B e. NN /\ C e. NN ) -> ( C - B ) e. ZZ )
6	5	3adant1 1018	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. ZZ )
7	6	3ad2ant1 1021	. . . . . . . . . 10 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( C - B ) e. ZZ )
8		simp13 1032	. . . . . . . . . . . 12 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> C e. NN )
9		simp12 1031	. . . . . . . . . . . 12 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> B e. NN )
10	8, 9	nnaddcld 9119	. . . . . . . . . . 11 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( C + B ) e. NN )
11	10	nnzd 9529	. . . . . . . . . 10 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( C + B ) e. ZZ )
12		gcddvds 12399	. . . . . . . . . 10 |- ( ( ( C - B ) e. ZZ /\ ( C + B ) e. ZZ ) -> ( ( ( C - B ) gcd ( C + B ) ) || ( C - B ) /\ ( ( C - B ) gcd ( C + B ) ) || ( C + B ) ) )
13	7, 11, 12	syl2anc 411	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( C - B ) gcd ( C + B ) ) || ( C - B ) /\ ( ( C - B ) gcd ( C + B ) ) || ( C + B ) ) )
14	13	simprd 114	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) || ( C + B ) )
15		breq1 4062	. . . . . . . . 9 |- ( ( ( C - B ) gcd ( C + B ) ) = 2 -> ( ( ( C - B ) gcd ( C + B ) ) || ( C + B ) <-> 2 || ( C + B ) ) )
16	15	biimpd 144	. . . . . . . 8 |- ( ( ( C - B ) gcd ( C + B ) ) = 2 -> ( ( ( C - B ) gcd ( C + B ) ) || ( C + B ) -> 2 || ( C + B ) ) )
17	14, 16	mpan9 281	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> 2 || ( C + B ) )
18		2z 9435	. . . . . . . 8 |- 2 e. ZZ
19		simpl13 1077	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> C e. NN )
20	19	nnzd 9529	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> C e. ZZ )
21		simpl12 1076	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> B e. NN )
22	21	nnzd 9529	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> B e. ZZ )
23	20, 22	zaddcld 9534	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( C + B ) e. ZZ )
24	20, 22	zsubcld 9535	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( C - B ) e. ZZ )
25		dvdsmultr1 12257	. . . . . . . 8 |- ( ( 2 e. ZZ /\ ( C + B ) e. ZZ /\ ( C - B ) e. ZZ ) -> ( 2 || ( C + B ) -> 2 || ( ( C + B ) x. ( C - B ) ) ) )
26	18, 23, 24, 25	mp3an2i 1355	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( 2 || ( C + B ) -> 2 || ( ( C + B ) x. ( C - B ) ) ) )
27	17, 26	mpd 13	. . . . . 6 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> 2 || ( ( C + B ) x. ( C - B ) ) )
28	19	nncnd 9085	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> C e. CC )
29	21	nncnd 9085	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> B e. CC )
30		subsq 10828	. . . . . . 7 |- ( ( C e. CC /\ B e. CC ) -> ( ( C ^ 2 ) - ( B ^ 2 ) ) = ( ( C + B ) x. ( C - B ) ) )
31	28, 29, 30	syl2anc 411	. . . . . 6 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( ( C ^ 2 ) - ( B ^ 2 ) ) = ( ( C + B ) x. ( C - B ) ) )
32	27, 31	breqtrrd 4087	. . . . 5 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> 2 || ( ( C ^ 2 ) - ( B ^ 2 ) ) )
33		simpl2 1004	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) )
34	33	oveq1d 5982	. . . . . 6 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( B ^ 2 ) ) = ( ( C ^ 2 ) - ( B ^ 2 ) ) )
35		simpl11 1075	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> A e. NN )
36	35	nnsqcld 10876	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( A ^ 2 ) e. NN )
37	36	nncnd 9085	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( A ^ 2 ) e. CC )
38	21	nnsqcld 10876	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( B ^ 2 ) e. NN )
39	38	nncnd 9085	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( B ^ 2 ) e. CC )
40	37, 39	pncand 8419	. . . . . 6 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( B ^ 2 ) ) = ( A ^ 2 ) )
41	34, 40	eqtr3d 2242	. . . . 5 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( ( C ^ 2 ) - ( B ^ 2 ) ) = ( A ^ 2 ) )
42	32, 41	breqtrd 4085	. . . 4 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> 2 || ( A ^ 2 ) )
43		nnz 9426	. . . . . . . 8 |- ( A e. NN -> A e. ZZ )
44	43	3ad2ant1 1021	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. ZZ )
45	44	3ad2ant1 1021	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> A e. ZZ )
46	45	adantr 276	. . . . 5 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> A e. ZZ )
47		2prm 12564	. . . . . 6 |- 2 e. Prime
48		2nn 9233	. . . . . 6 |- 2 e. NN
49		prmdvdsexp 12585	. . . . . 6 |- ( ( 2 e. Prime /\ A e. ZZ /\ 2 e. NN ) -> ( 2 || ( A ^ 2 ) <-> 2 || A ) )
50	47, 48, 49	mp3an13 1341	. . . . 5 |- ( A e. ZZ -> ( 2 || ( A ^ 2 ) <-> 2 || A ) )
51	46, 50	syl 14	. . . 4 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> ( 2 || ( A ^ 2 ) <-> 2 || A ) )
52	42, 51	mpbid 147	. . 3 |- ( ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) /\ ( ( C - B ) gcd ( C + B ) ) = 2 ) -> 2 || A )
53	1, 52	mtand 667	. 2 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> -. ( ( C - B ) gcd ( C + B ) ) = 2 )
54		neg1z 9439	. . . . . . . 8 |- -u 1 e. ZZ
55		gcdaddm 12420	. . . . . . . 8 |- ( ( -u 1 e. ZZ /\ ( C - B ) e. ZZ /\ ( C + B ) e. ZZ ) -> ( ( C - B ) gcd ( C + B ) ) = ( ( C - B ) gcd ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) ) )
56	54, 7, 11, 55	mp3an2i 1355	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) = ( ( C - B ) gcd ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) ) )
57	8	nncnd 9085	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> C e. CC )
58	9	nncnd 9085	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> B e. CC )
59		pnncan 8348	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC /\ B e. CC ) -> ( ( C + B ) - ( C - B ) ) = ( B + B ) )
60	59	3anidm23 1310	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) - ( C - B ) ) = ( B + B ) )
61		subcl 8306	. . . . . . . . . . . . 13 |- ( ( C e. CC /\ B e. CC ) -> ( C - B ) e. CC )
62	61	mulm1d 8517	. . . . . . . . . . . 12 |- ( ( C e. CC /\ B e. CC ) -> ( -u 1 x. ( C - B ) ) = -u ( C - B ) )
63	62	oveq2d 5983	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) = ( ( C + B ) + -u ( C - B ) ) )
64		addcl 8085	. . . . . . . . . . . 12 |- ( ( C e. CC /\ B e. CC ) -> ( C + B ) e. CC )
65	64, 61	negsubd 8424	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + -u ( C - B ) ) = ( ( C + B ) - ( C - B ) ) )
66	63, 65	eqtrd 2240	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) = ( ( C + B ) - ( C - B ) ) )
67		2times 9199	. . . . . . . . . . 11 |- ( B e. CC -> ( 2 x. B ) = ( B + B ) )
68	67	adantl 277	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( 2 x. B ) = ( B + B ) )
69	60, 66, 68	3eqtr4d 2250	. . . . . . . . 9 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) = ( 2 x. B ) )
70	69	oveq2d 5983	. . . . . . . 8 |- ( ( C e. CC /\ B e. CC ) -> ( ( C - B ) gcd ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) ) = ( ( C - B ) gcd ( 2 x. B ) ) )
71	57, 58, 70	syl2anc 411	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) ) = ( ( C - B ) gcd ( 2 x. B ) ) )
72	56, 71	eqtrd 2240	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) = ( ( C - B ) gcd ( 2 x. B ) ) )
73	9	nnzd 9529	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> B e. ZZ )
74		zmulcl 9461	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ B e. ZZ ) -> ( 2 x. B ) e. ZZ )
75	18, 73, 74	sylancr 414	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( 2 x. B ) e. ZZ )
76		gcddvds 12399	. . . . . . . 8 |- ( ( ( C - B ) e. ZZ /\ ( 2 x. B ) e. ZZ ) -> ( ( ( C - B ) gcd ( 2 x. B ) ) || ( C - B ) /\ ( ( C - B ) gcd ( 2 x. B ) ) || ( 2 x. B ) ) )
77	7, 75, 76	syl2anc 411	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( C - B ) gcd ( 2 x. B ) ) || ( C - B ) /\ ( ( C - B ) gcd ( 2 x. B ) ) || ( 2 x. B ) ) )
78	77	simprd 114	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( 2 x. B ) ) || ( 2 x. B ) )
79	72, 78	eqbrtrd 4081	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) || ( 2 x. B ) )
80		1z 9433	. . . . . . . 8 |- 1 e. ZZ
81		gcdaddm 12420	. . . . . . . 8 |- ( ( 1 e. ZZ /\ ( C - B ) e. ZZ /\ ( C + B ) e. ZZ ) -> ( ( C - B ) gcd ( C + B ) ) = ( ( C - B ) gcd ( ( C + B ) + ( 1 x. ( C - B ) ) ) ) )
82	80, 7, 11, 81	mp3an2i 1355	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) = ( ( C - B ) gcd ( ( C + B ) + ( 1 x. ( C - B ) ) ) ) )
83		ppncan 8349	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC /\ C e. CC ) -> ( ( C + B ) + ( C - B ) ) = ( C + C ) )
84	83	3anidm13 1309	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( C - B ) ) = ( C + C ) )
85	61	mulid2d 8126	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC ) -> ( 1 x. ( C - B ) ) = ( C - B ) )
86	85	oveq2d 5983	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( 1 x. ( C - B ) ) ) = ( ( C + B ) + ( C - B ) ) )
87		2times 9199	. . . . . . . . . . 11 |- ( C e. CC -> ( 2 x. C ) = ( C + C ) )
88	87	adantr 276	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( 2 x. C ) = ( C + C ) )
89	84, 86, 88	3eqtr4d 2250	. . . . . . . . 9 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( 1 x. ( C - B ) ) ) = ( 2 x. C ) )
90	57, 58, 89	syl2anc 411	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C + B ) + ( 1 x. ( C - B ) ) ) = ( 2 x. C ) )
91	90	oveq2d 5983	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( ( C + B ) + ( 1 x. ( C - B ) ) ) ) = ( ( C - B ) gcd ( 2 x. C ) ) )
92	82, 91	eqtrd 2240	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) = ( ( C - B ) gcd ( 2 x. C ) ) )
93	8	nnzd 9529	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> C e. ZZ )
94		zmulcl 9461	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ C e. ZZ ) -> ( 2 x. C ) e. ZZ )
95	18, 93, 94	sylancr 414	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( 2 x. C ) e. ZZ )
96		gcddvds 12399	. . . . . . . 8 |- ( ( ( C - B ) e. ZZ /\ ( 2 x. C ) e. ZZ ) -> ( ( ( C - B ) gcd ( 2 x. C ) ) || ( C - B ) /\ ( ( C - B ) gcd ( 2 x. C ) ) || ( 2 x. C ) ) )
97	7, 95, 96	syl2anc 411	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( C - B ) gcd ( 2 x. C ) ) || ( C - B ) /\ ( ( C - B ) gcd ( 2 x. C ) ) || ( 2 x. C ) ) )
98	97	simprd 114	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( 2 x. C ) ) || ( 2 x. C ) )
99	92, 98	eqbrtrd 4081	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) || ( 2 x. C ) )
100		nnaddcl 9091	. . . . . . . . . . . . . 14 |- ( ( C e. NN /\ B e. NN ) -> ( C + B ) e. NN )
101	100	nnne0d 9116	. . . . . . . . . . . . 13 |- ( ( C e. NN /\ B e. NN ) -> ( C + B ) =/= 0 )
102	101	ancoms 268	. . . . . . . . . . . 12 |- ( ( B e. NN /\ C e. NN ) -> ( C + B ) =/= 0 )
103	102	3adant1 1018	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) =/= 0 )
104	103	3ad2ant1 1021	. . . . . . . . . 10 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( C + B ) =/= 0 )
105	104	neneqd 2399	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> -. ( C + B ) = 0 )
106	105	intnand 933	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> -. ( ( C - B ) = 0 /\ ( C + B ) = 0 ) )
107		gcdn0cl 12398	. . . . . . . 8 |- ( ( ( ( C - B ) e. ZZ /\ ( C + B ) e. ZZ ) /\ -. ( ( C - B ) = 0 /\ ( C + B ) = 0 ) ) -> ( ( C - B ) gcd ( C + B ) ) e. NN )
108	7, 11, 106, 107	syl21anc 1249	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) e. NN )
109	108	nnzd 9529	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) e. ZZ )
110		dvdsgcd 12448	. . . . . 6 |- ( ( ( ( C - B ) gcd ( C + B ) ) e. ZZ /\ ( 2 x. B ) e. ZZ /\ ( 2 x. C ) e. ZZ ) -> ( ( ( ( C - B ) gcd ( C + B ) ) || ( 2 x. B ) /\ ( ( C - B ) gcd ( C + B ) ) || ( 2 x. C ) ) -> ( ( C - B ) gcd ( C + B ) ) || ( ( 2 x. B ) gcd ( 2 x. C ) ) ) )
111	109, 75, 95, 110	syl3anc 1250	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( ( C - B ) gcd ( C + B ) ) || ( 2 x. B ) /\ ( ( C - B ) gcd ( C + B ) ) || ( 2 x. C ) ) -> ( ( C - B ) gcd ( C + B ) ) || ( ( 2 x. B ) gcd ( 2 x. C ) ) ) )
112	79, 99, 111	mp2and 433	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) || ( ( 2 x. B ) gcd ( 2 x. C ) ) )
113		2nn0 9347	. . . . . 6 |- 2 e. NN0
114		mulgcd 12452	. . . . . 6 |- ( ( 2 e. NN0 /\ B e. ZZ /\ C e. ZZ ) -> ( ( 2 x. B ) gcd ( 2 x. C ) ) = ( 2 x. ( B gcd C ) ) )
115	113, 73, 93, 114	mp3an2i 1355	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( 2 x. B ) gcd ( 2 x. C ) ) = ( 2 x. ( B gcd C ) ) )
116		pythagtriplem3 12705	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( B gcd C ) = 1 )
117	116	oveq2d 5983	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( 2 x. ( B gcd C ) ) = ( 2 x. 1 ) )
118		2t1e2 9225	. . . . . 6 |- ( 2 x. 1 ) = 2
119	117, 118	eqtrdi 2256	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( 2 x. ( B gcd C ) ) = 2 )
120	115, 119	eqtrd 2240	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( 2 x. B ) gcd ( 2 x. C ) ) = 2 )
121	112, 120	breqtrd 4085	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) || 2 )
122		dvdsprime 12559	. . . 4 |- ( ( 2 e. Prime /\ ( ( C - B ) gcd ( C + B ) ) e. NN ) -> ( ( ( C - B ) gcd ( C + B ) ) || 2 <-> ( ( ( C - B ) gcd ( C + B ) ) = 2 \/ ( ( C - B ) gcd ( C + B ) ) = 1 ) ) )
123	47, 108, 122	sylancr 414	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( C - B ) gcd ( C + B ) ) || 2 <-> ( ( ( C - B ) gcd ( C + B ) ) = 2 \/ ( ( C - B ) gcd ( C + B ) ) = 1 ) ) )
124	121, 123	mpbid 147	. 2 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( ( C - B ) gcd ( C + B ) ) = 2 \/ ( ( C - B ) gcd ( C + B ) ) = 1 ) )
125		orel1 727	. 2 |- ( -. ( ( C - B ) gcd ( C + B ) ) = 2 -> ( ( ( ( C - B ) gcd ( C + B ) ) = 2 \/ ( ( C - B ) gcd ( C + B ) ) = 1 ) -> ( ( C - B ) gcd ( C + B ) ) = 1 ) )
126	53, 124, 125	sylc 62	1 |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) = 1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2178   =/= wne 2378   class class class wbr 4059  (class class class)co 5967   CCcc 7958   0cc0 7960   1c1 7961   + caddc 7963   x. cmul 7965   - cmin 8278   -ucneg 8279   NNcn 9071   2c2 9122   NN0cn0 9330   ZZcz 9407   ^cexp 10720   || cdvds 12213   gcd cgcd 12389   Primecprime 12544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-2o 6526  df-er 6643  df-en 6851  df-sup 7112  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-gcd 12390  df-prm 12545

This theorem is referenced by:  pythagtriplem6  12708  pythagtriplem7  12709