Theorem pythagtriplem4 12959

Description: Lemma for pythagtrip 12974. 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 1053	. . 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 9592	. . . . . . . . . . . . 13 |- ( C e. NN -> C e. ZZ )
3		nnz 9592	. . . . . . . . . . . . 13 |- ( B e. NN -> B e. ZZ )
4		zsubcl 9614	. . . . . . . . . . . . 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 1042	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. ZZ )
7	6	3ad2ant1 1045	. . . . . . . . . 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 1056	. . . . . . . . . . . 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 1055	. . . . . . . . . . . 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 9281	. . . . . . . . . . 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 9695	. . . . . . . . . 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 12652	. . . . . . . . . 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 4111	. . . . . . . . 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 9601	. . . . . . . 8 |- 2 e. ZZ
19		simpl13 1101	. . . . . . . . . 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 9695	. . . . . . . . 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 1100	. . . . . . . . . 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 9695	. . . . . . . . 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 9700	. . . . . . . 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 9701	. . . . . . . 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 12510	. . . . . . . 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 1379	. . . . . . 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 9247	. . . . . . 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 9247	. . . . . . 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 11004	. . . . . . 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 4136	. . . . 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 1028	. . . . . . 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 6064	. . . . . 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 1099	. . . . . . . . 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 11052	. . . . . . . 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 9247	. . . . . . 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 11052	. . . . . . . 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 9247	. . . . . . 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 8581	. . . . . 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 2267	. . . . 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 4134	. . . 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 9592	. . . . . . . 8 |- ( A e. NN -> A e. ZZ )
44	43	3ad2ant1 1045	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. ZZ )
45	44	3ad2ant1 1045	. . . . . 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 12817	. . . . . 6 |- 2 e. Prime
48		2nn 9395	. . . . . 6 |- 2 e. NN
49		prmdvdsexp 12838	. . . . . 6 |- ( ( 2 e. Prime /\ A e. ZZ /\ 2 e. NN ) -> ( 2 || ( A ^ 2 ) <-> 2 || A ) )
50	47, 48, 49	mp3an13 1365	. . . . 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 671	. 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 9605	. . . . . . . 8 |- -u 1 e. ZZ
55		gcdaddm 12673	. . . . . . . 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 1379	. . . . . . 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 9247	. . . . . . . 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 9247	. . . . . . . 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 8510	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC /\ B e. CC ) -> ( ( C + B ) - ( C - B ) ) = ( B + B ) )
60	59	3anidm23 1334	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) - ( C - B ) ) = ( B + B ) )
61		subcl 8468	. . . . . . . . . . . . 13 |- ( ( C e. CC /\ B e. CC ) -> ( C - B ) e. CC )
62	61	mulm1d 8679	. . . . . . . . . . . 12 |- ( ( C e. CC /\ B e. CC ) -> ( -u 1 x. ( C - B ) ) = -u ( C - B ) )
63	62	oveq2d 6065	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) = ( ( C + B ) + -u ( C - B ) ) )
64		addcl 8248	. . . . . . . . . . . 12 |- ( ( C e. CC /\ B e. CC ) -> ( C + B ) e. CC )
65	64, 61	negsubd 8586	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + -u ( C - B ) ) = ( ( C + B ) - ( C - B ) ) )
66	63, 65	eqtrd 2265	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) = ( ( C + B ) - ( C - B ) ) )
67		2times 9361	. . . . . . . . . . 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 2275	. . . . . . . . 9 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) = ( 2 x. B ) )
70	69	oveq2d 6065	. . . . . . . 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 2265	. . . . . 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 9695	. . . . . . . . 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 9627	. . . . . . . . 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 12652	. . . . . . . 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 4130	. . . . 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 9599	. . . . . . . 8 |- 1 e. ZZ
81		gcdaddm 12673	. . . . . . . 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 1379	. . . . . . 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 8511	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC /\ C e. CC ) -> ( ( C + B ) + ( C - B ) ) = ( C + C ) )
84	83	3anidm13 1333	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( C - B ) ) = ( C + C ) )
85	61	mullidd 8288	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC ) -> ( 1 x. ( C - B ) ) = ( C - B ) )
86	85	oveq2d 6065	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( 1 x. ( C - B ) ) ) = ( ( C + B ) + ( C - B ) ) )
87		2times 9361	. . . . . . . . . . 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 2275	. . . . . . . . 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 6065	. . . . . . 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 2265	. . . . . 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 9695	. . . . . . . . 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 9627	. . . . . . . . 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 12652	. . . . . . . 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 4130	. . . . 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 9253	. . . . . . . . . . . . . 14 |- ( ( C e. NN /\ B e. NN ) -> ( C + B ) e. NN )
101	100	nnne0d 9278	. . . . . . . . . . . . 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 1042	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) =/= 0 )
104	103	3ad2ant1 1045	. . . . . . . . . 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 2433	. . . . . . . . 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 939	. . . . . . . 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 12651	. . . . . . . 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 1273	. . . . . . 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 9695	. . . . . 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 12701	. . . . . 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 1274	. . . . 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 9509	. . . . . 6 |- 2 e. NN0
114		mulgcd 12705	. . . . . 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 1379	. . . . 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 12958	. . . . . . 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 6065	. . . . . 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 9387	. . . . . 6 |- ( 2 x. 1 ) = 2
119	117, 118	eqtrdi 2281	. . . . 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 2265	. . . 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 4134	. . 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 12812	. . . 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 733	. 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 716   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   class class class wbr 4108  (class class class)co 6049   CCcc 8121   0cc0 8123   1c1 8124   + caddc 8126   x. cmul 8128   - cmin 8440   -ucneg 8441   NNcn 9233   2c2 9284   NN0cn0 9492   ZZcz 9573   ^cexp 10896   || cdvds 12466   gcd cgcd 12642   Primecprime 12797

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242  ax-caucvg 8243

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-1o 6646  df-2o 6647  df-er 6766  df-en 6975  df-sup 7274  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-n0 9493  df-z 9574  df-uz 9850  df-q 9948  df-rp 9983  df-fz 10339  df-fzo 10473  df-fl 10626  df-mod 10681  df-seqfrec 10806  df-exp 10897  df-cj 11520  df-re 11521  df-im 11522  df-rsqrt 11676  df-abs 11677  df-dvds 12467  df-gcd 12643  df-prm 12798

This theorem is referenced by:  pythagtriplem6  12961  pythagtriplem7  12962