Theorem pythagtriplem4 12251

Description: Lemma for pythagtrip 12266. 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 1026	. . 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 9261	. . . . . . . . . . . . 13 |- ( C e. NN -> C e. ZZ )
3		nnz 9261	. . . . . . . . . . . . 13 |- ( B e. NN -> B e. ZZ )
4		zsubcl 9283	. . . . . . . . . . . . 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 1015	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C - B ) e. ZZ )
7	6	3ad2ant1 1018	. . . . . . . . . 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 1029	. . . . . . . . . . . 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 1028	. . . . . . . . . . . 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 8956	. . . . . . . . . . 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 9363	. . . . . . . . . 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 11947	. . . . . . . . . 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 4003	. . . . . . . . 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 9270	. . . . . . . 8 |- 2 e. ZZ
19		simpl13 1074	. . . . . . . . . 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 9363	. . . . . . . . 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 1073	. . . . . . . . . 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 9363	. . . . . . . . 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 9368	. . . . . . . 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 9369	. . . . . . . 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 11822	. . . . . . . 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 1342	. . . . . . 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 8922	. . . . . . 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 8922	. . . . . . 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 10612	. . . . . . 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 4028	. . . . 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 1001	. . . . . . 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 5884	. . . . . 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 1072	. . . . . . . . 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 10660	. . . . . . . 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 8922	. . . . . . 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 10660	. . . . . . . 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 8922	. . . . . . 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 8259	. . . . . 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 2212	. . . . 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 4026	. . . 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 9261	. . . . . . . 8 |- ( A e. NN -> A e. ZZ )
44	43	3ad2ant1 1018	. . . . . . 7 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> A e. ZZ )
45	44	3ad2ant1 1018	. . . . . 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 12110	. . . . . 6 |- 2 e. Prime
48		2nn 9069	. . . . . 6 |- 2 e. NN
49		prmdvdsexp 12131	. . . . . 6 |- ( ( 2 e. Prime /\ A e. ZZ /\ 2 e. NN ) -> ( 2 || ( A ^ 2 ) <-> 2 || A ) )
50	47, 48, 49	mp3an13 1328	. . . . 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 665	. 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 9274	. . . . . . . 8 |- -u 1 e. ZZ
55		gcdaddm 11968	. . . . . . . 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 1342	. . . . . . 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 8922	. . . . . . . 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 8922	. . . . . . . 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 8188	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC /\ B e. CC ) -> ( ( C + B ) - ( C - B ) ) = ( B + B ) )
60	59	3anidm23 1297	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) - ( C - B ) ) = ( B + B ) )
61		subcl 8146	. . . . . . . . . . . . 13 |- ( ( C e. CC /\ B e. CC ) -> ( C - B ) e. CC )
62	61	mulm1d 8357	. . . . . . . . . . . 12 |- ( ( C e. CC /\ B e. CC ) -> ( -u 1 x. ( C - B ) ) = -u ( C - B ) )
63	62	oveq2d 5885	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) = ( ( C + B ) + -u ( C - B ) ) )
64		addcl 7927	. . . . . . . . . . . 12 |- ( ( C e. CC /\ B e. CC ) -> ( C + B ) e. CC )
65	64, 61	negsubd 8264	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + -u ( C - B ) ) = ( ( C + B ) - ( C - B ) ) )
66	63, 65	eqtrd 2210	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) = ( ( C + B ) - ( C - B ) ) )
67		2times 9036	. . . . . . . . . . 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 2220	. . . . . . . . 9 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( -u 1 x. ( C - B ) ) ) = ( 2 x. B ) )
70	69	oveq2d 5885	. . . . . . . 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 2210	. . . . . 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 9363	. . . . . . . . 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 9295	. . . . . . . . 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 11947	. . . . . . . 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 4022	. . . . 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 9268	. . . . . . . 8 |- 1 e. ZZ
81		gcdaddm 11968	. . . . . . . 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 1342	. . . . . . 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 8189	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC /\ C e. CC ) -> ( ( C + B ) + ( C - B ) ) = ( C + C ) )
84	83	3anidm13 1296	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( C - B ) ) = ( C + C ) )
85	61	mulid2d 7966	. . . . . . . . . . 11 |- ( ( C e. CC /\ B e. CC ) -> ( 1 x. ( C - B ) ) = ( C - B ) )
86	85	oveq2d 5885	. . . . . . . . . 10 |- ( ( C e. CC /\ B e. CC ) -> ( ( C + B ) + ( 1 x. ( C - B ) ) ) = ( ( C + B ) + ( C - B ) ) )
87		2times 9036	. . . . . . . . . . 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 2220	. . . . . . . . 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 5885	. . . . . . 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 2210	. . . . . 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 9363	. . . . . . . . 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 9295	. . . . . . . . 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 11947	. . . . . . . 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 4022	. . . . 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 8928	. . . . . . . . . . . . . 14 |- ( ( C e. NN /\ B e. NN ) -> ( C + B ) e. NN )
101	100	nnne0d 8953	. . . . . . . . . . . . 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 1015	. . . . . . . . . . 11 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( C + B ) =/= 0 )
104	103	3ad2ant1 1018	. . . . . . . . . 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 2368	. . . . . . . . 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 931	. . . . . . . 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 11946	. . . . . . . 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 1237	. . . . . . 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 9363	. . . . . 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 11996	. . . . . 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 1238	. . . . 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 9182	. . . . . 6 |- 2 e. NN0
114		mulgcd 12000	. . . . . 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 1342	. . . . 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 12250	. . . . . . 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 5885	. . . . . 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 9061	. . . . . 6 |- ( 2 x. 1 ) = 2
119	117, 118	eqtrdi 2226	. . . . 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 2210	. . . 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 4026	. . 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 12105	. . . 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 725	. 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 708   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4000  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   - cmin 8118   -ucneg 8119   NNcn 8908   2c2 8959   NN0cn0 9165   ZZcz 9242   ^cexp 10505   || cdvds 11778   gcd cgcd 11926   Primecprime 12090

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-1o 6411  df-2o 6412  df-er 6529  df-en 6735  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927  df-prm 12091

This theorem is referenced by:  pythagtriplem6  12253  pythagtriplem7  12254