Theorem pythagtriplem4 12196

Description: Lemma for pythagtrip 12211. 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 1016	. . 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 9206	. . . . . . . . . . . . 13 |- ( C e. NN -> C e. ZZ )
3		nnz 9206	. . . . . . . . . . . . 13 |- ( B e. NN -> B e. ZZ )
4		zsubcl 9228	. . . . . . . . . . . . 13 |- ( ( C e. ZZ /\ B e. ZZ ) -> ( C - B ) e. ZZ )
5	2, 3, 4	syl2anr 288	. . . . . . . . . . . 12 |- ( ( B e. NN /\ C e. NN ) -> ( C - B ) e. ZZ )
6	5	3adant1 1005	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. ZZ )
7	6	3ad2ant1 1008	. . . . . . . . . 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 1019	. . . . . . . . . . . 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 1018	. . . . . . . . . . . 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 8901	. . . . . . . . . . 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 9308	. . . . . . . . . 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 11892	. . . . . . . . . 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 409	. . . . . . . . 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 113	. . . . . . . 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 3984	. . . . . . . . 9 |- ( ( ( C - B ) gcd ( C + B ) ) = 2 -> ( ( ( C - B ) gcd ( C + B ) ) || ( C + B ) <-> 2 || ( C + B ) ) )
16	15	biimpd 143	. . . . . . . 8 |- ( ( ( C - B ) gcd ( C + B ) ) = 2 -> ( ( ( C - B ) gcd ( C + B ) ) || ( C + B ) -> 2 || ( C + B ) ) )
17	14, 16	mpan9 279	. . . . . . 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 9215	. . . . . . . 8 |- 2 e. ZZ
19		simpl13 1064	. . . . . . . . . 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 9308	. . . . . . . . 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 1063	. . . . . . . . . 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 9308	. . . . . . . . 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 9313	. . . . . . . 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 9314	. . . . . . . 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 11767	. . . . . . . 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 1332	. . . . . . 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 8867	. . . . . . 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 8867	. . . . . . 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 10557	. . . . . . 7 |- ( ( C e. CC /\ B e. CC ) -> ( ( C ^ 2 ) - ( B ^ 2 ) ) = ( ( C + B ) x. ( C - B ) ) )
31	28, 29, 30	syl2anc 409	. . . . . 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 4009	. . . . 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 991	. . . . . . 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 5856	. . . . . 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 1062	. . . . . . . . 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 10605	. . . . . . . 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 8867	. . . . . . 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 10605	. . . . . . . 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 8867	. . . . . . 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 8206	. . . . . 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 2200	. . . . 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 4007	. . . 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 9206	. . . . . . . 8 |- ( A e. NN -> A e. ZZ )
44	43	3ad2ant1 1008	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. ZZ )
45	44	3ad2ant1 1008	. . . . . 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 274	. . . . 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 12055	. . . . . 6 |- 2 e. Prime
48		2nn 9014	. . . . . 6 |- 2 e. NN
49		prmdvdsexp 12076	. . . . . 6 |- ( ( 2 e. Prime /\ A e. ZZ /\ 2 e. NN ) -> ( 2 || ( A ^ 2 ) <-> 2 || A ) )
50	47, 48, 49	mp3an13 1318	. . . . 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 146	. . 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 655	. 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 9219	. . . . . . . 8 |- -u 1 e. ZZ
55		gcdaddm 11913	. . . . . . . 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 1332	. . . . . . 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 8867	. . . . . . . 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 8867	. . . . . . . 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 8135	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC /\ B e. CC ) -> ( ( C + B ) - ( C - B ) ) = ( B + B ) )
60	59	3anidm23 1287	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) - ( C - B ) ) = ( B + B ) )
61		subcl 8093	. . . . . . . . . . . . 13 |- ( ( C e. CC /\ B e. CC ) -> ( C - B ) e. CC )
62	61	mulm1d 8304	. . . . . . . . . . . 12 |- ( ( C e. CC /\ B e. CC ) -> ( -u 1 x. ( C - B ) ) = -u ( C - B ) )
63	62	oveq2d 5857	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) = ( ( C + B ) + -u ( C - B ) ) )
64		addcl 7874	. . . . . . . . . . . 12 |- ( ( C e. CC /\ B e. CC ) -> ( C + B ) e. CC )
65	64, 61	negsubd 8211	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + -u ( C - B ) ) = ( ( C + B ) - ( C - B ) ) )
66	63, 65	eqtrd 2198	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) = ( ( C + B ) - ( C - B ) ) )
67		2times 8981	. . . . . . . . . . 11 |- ( B e. CC -> ( 2 x. B ) = ( B + B ) )
68	67	adantl 275	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( 2 x. B ) = ( B + B ) )
69	60, 66, 68	3eqtr4d 2208	. . . . . . . . 9 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) = ( 2 x. B ) )
70	69	oveq2d 5857	. . . . . . . 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 409	. . . . . . 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 2198	. . . . . 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 9308	. . . . . . . . 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 9240	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ B e. ZZ ) -> ( 2 x. B ) e. ZZ )
75	18, 73, 74	sylancr 411	. . . . . . . 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 11892	. . . . . . . 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 409	. . . . . . 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 113	. . . . . 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 4003	. . . . 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 9213	. . . . . . . 8 |- 1 e. ZZ
81		gcdaddm 11913	. . . . . . . 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 1332	. . . . . . 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 8136	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC /\ C e. CC ) -> ( ( C + B ) + ( C - B ) ) = ( C + C ) )
84	83	3anidm13 1286	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( C - B ) ) = ( C + C ) )
85	61	mulid2d 7913	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC ) -> ( 1 x. ( C - B ) ) = ( C - B ) )
86	85	oveq2d 5857	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( 1 x. ( C - B ) ) ) = ( ( C + B ) + ( C - B ) ) )
87		2times 8981	. . . . . . . . . . 11 |- ( C e. CC -> ( 2 x. C ) = ( C + C ) )
88	87	adantr 274	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( 2 x. C ) = ( C + C ) )
89	84, 86, 88	3eqtr4d 2208	. . . . . . . . 9 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( 1 x. ( C - B ) ) ) = ( 2 x. C ) )
90	57, 58, 89	syl2anc 409	. . . . . . . 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 5857	. . . . . . 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 2198	. . . . . 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 9308	. . . . . . . . 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 9240	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ C e. ZZ ) -> ( 2 x. C ) e. ZZ )
95	18, 93, 94	sylancr 411	. . . . . . . 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 11892	. . . . . . . 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 409	. . . . . . 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 113	. . . . . 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 4003	. . . . 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 8873	. . . . . . . . . . . . . 14 |- ( ( C e. NN /\ B e. NN ) -> ( C + B ) e. NN )
101	100	nnne0d 8898	. . . . . . . . . . . . 13 |- ( ( C e. NN /\ B e. NN ) -> ( C + B ) =/= 0 )
102	101	ancoms 266	. . . . . . . . . . . 12 |- ( ( B e. NN /\ C e. NN ) -> ( C + B ) =/= 0 )
103	102	3adant1 1005	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) =/= 0 )
104	103	3ad2ant1 1008	. . . . . . . . . 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 2356	. . . . . . . . 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 921	. . . . . . . 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 11891	. . . . . . . 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 1227	. . . . . . 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 9308	. . . . . 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 11941	. . . . . 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 1228	. . . . 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 430	. . . 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 9127	. . . . . 6 |- 2 e. NN0
114		mulgcd 11945	. . . . . 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 1332	. . . . 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 12195	. . . . . . 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 5857	. . . . . 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 9006	. . . . . 6 |- ( 2 x. 1 ) = 2
119	117, 118	eqtrdi 2214	. . . . 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 2198	. . . 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 4007	. . 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 12050	. . . 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 411	. . 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 146	. 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 715	. 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 103   <-> wb 104   \/ wo 698   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2335   class class class wbr 3981  (class class class)co 5841   CCcc 7747   0cc0 7749   1c1 7750   + caddc 7752   x. cmul 7754   - cmin 8065   -ucneg 8066   NNcn 8853   2c2 8904   NN0cn0 9110   ZZcz 9187   ^cexp 10450   || cdvds 11723   gcd cgcd 11871   Primecprime 12035

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-1o 6380  df-2o 6381  df-er 6497  df-en 6703  df-sup 6945  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-dvds 11724  df-gcd 11872  df-prm 12036

This theorem is referenced by:  pythagtriplem6  12198  pythagtriplem7  12199