Theorem gausslemma2dlem0i 25089

Description: Auxiliary lemma 9 for gausslemma2d 25099. (Contributed by AV, 14-Jul-2021.)

Hypotheses

Ref	Expression
gausslemma2dlem0.p	|- ( ph -> P e. ( Prime \ { 2 } ) )
gausslemma2dlem0.m	|- M = ( |_ ` ( P / 4 ) )
gausslemma2dlem0.h	|- H = ( ( P - 1 ) / 2 )
gausslemma2dlem0.n	|- N = ( H - M )

Assertion

Ref	Expression
gausslemma2dlem0i	|- ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) )

Proof of Theorem gausslemma2dlem0i

Step	Hyp	Ref	Expression
1		2z 11409	. . 3 |- 2 e. ZZ
2		gausslemma2dlem0.p	. . . 4 |- ( ph -> P e. ( Prime \ { 2 } ) )
3		id 22	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> P e. ( Prime \ { 2 } ) )
4	3	gausslemma2dlem0a 25081	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> P e. NN )
5	4	nnzd 11481	. . . 4 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
6	2, 5	syl 17	. . 3 |- ( ph -> P e. ZZ )
7		lgscl1 25045	. . 3 |- ( ( 2 e. ZZ /\ P e. ZZ ) -> ( 2 /L P ) e. { -u 1 , 0 , 1 } )
8	1, 6, 7	sylancr 695	. 2 |- ( ph -> ( 2 /L P ) e. { -u 1 , 0 , 1 } )
9		ovex 6678	. . . 4 |- ( 2 /L P ) e. _V
10	9	eltp 4230	. . 3 |- ( ( 2 /L P ) e. { -u 1 , 0 , 1 } <-> ( ( 2 /L P ) = -u 1 \/ ( 2 /L P ) = 0 \/ ( 2 /L P ) = 1 ) )
11		gausslemma2dlem0.m	. . . . . . . . 9 |- M = ( |_ ` ( P / 4 ) )
12		gausslemma2dlem0.h	. . . . . . . . 9 |- H = ( ( P - 1 ) / 2 )
13		gausslemma2dlem0.n	. . . . . . . . 9 |- N = ( H - M )
14	2, 11, 12, 13	gausslemma2dlem0h 25088	. . . . . . . 8 |- ( ph -> N e. NN0 )
15	14	nn0zd 11480	. . . . . . 7 |- ( ph -> N e. ZZ )
16		m1expcl2 12882	. . . . . . 7 |- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )
17	15, 16	syl 17	. . . . . 6 |- ( ph -> ( -u 1 ^ N ) e. { -u 1 , 1 } )
18		ovex 6678	. . . . . . . 8 |- ( -u 1 ^ N ) e. _V
19	18	elpr 4198	. . . . . . 7 |- ( ( -u 1 ^ N ) e. { -u 1 , 1 } <-> ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) )
20		eqcom 2629	. . . . . . . . . 10 |- ( ( -u 1 ^ N ) = -u 1 <-> -u 1 = ( -u 1 ^ N ) )
21	20	biimpi 206	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = -u 1 -> -u 1 = ( -u 1 ^ N ) )
22	21	2a1d 26	. . . . . . . 8 |- ( ( -u 1 ^ N ) = -u 1 -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) )
23		eldifi 3732	. . . . . . . . . . . . 13 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
24		prmnn 15388	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> P e. NN )
25	24	nnred 11035	. . . . . . . . . . . . . 14 |- ( P e. Prime -> P e. RR )
26		prmgt1 15409	. . . . . . . . . . . . . 14 |- ( P e. Prime -> 1 < P )
27	25, 26	jca 554	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( P e. RR /\ 1 < P ) )
28	23, 27	syl 17	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( P e. RR /\ 1 < P ) )
29		1mod 12702	. . . . . . . . . . . 12 |- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 )
30	2, 28, 29	3syl 18	. . . . . . . . . . 11 |- ( ph -> ( 1 mod P ) = 1 )
31	30	eqeq2d 2632	. . . . . . . . . 10 |- ( ph -> ( ( -u 1 mod P ) = ( 1 mod P ) <-> ( -u 1 mod P ) = 1 ) )
32		oddprmge3 15412	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 3 ) )
33		m1modge3gt1 12717	. . . . . . . . . . . 12 |- ( P e. ( ZZ>= ` 3 ) -> 1 < ( -u 1 mod P ) )
34		breq2 4657	. . . . . . . . . . . . . 14 |- ( ( -u 1 mod P ) = 1 -> ( 1 < ( -u 1 mod P ) <-> 1 < 1 ) )
35		1re 10039	. . . . . . . . . . . . . . . 16 |- 1 e. RR
36	35	ltnri 10146	. . . . . . . . . . . . . . 15 |- -. 1 < 1
37	36	pm2.21i 116	. . . . . . . . . . . . . 14 |- ( 1 < 1 -> -u 1 = 1 )
38	34, 37	syl6bi 243	. . . . . . . . . . . . 13 |- ( ( -u 1 mod P ) = 1 -> ( 1 < ( -u 1 mod P ) -> -u 1 = 1 ) )
39	38	com12 32	. . . . . . . . . . . 12 |- ( 1 < ( -u 1 mod P ) -> ( ( -u 1 mod P ) = 1 -> -u 1 = 1 ) )
40	33, 39	syl 17	. . . . . . . . . . 11 |- ( P e. ( ZZ>= ` 3 ) -> ( ( -u 1 mod P ) = 1 -> -u 1 = 1 ) )
41	2, 32, 40	3syl 18	. . . . . . . . . 10 |- ( ph -> ( ( -u 1 mod P ) = 1 -> -u 1 = 1 ) )
42	31, 41	sylbid 230	. . . . . . . . 9 |- ( ph -> ( ( -u 1 mod P ) = ( 1 mod P ) -> -u 1 = 1 ) )
43		oveq1 6657	. . . . . . . . . . 11 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) mod P ) = ( 1 mod P ) )
44	43	eqeq2d 2632	. . . . . . . . . 10 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( -u 1 mod P ) = ( 1 mod P ) ) )
45		eqeq2 2633	. . . . . . . . . 10 |- ( ( -u 1 ^ N ) = 1 -> ( -u 1 = ( -u 1 ^ N ) <-> -u 1 = 1 ) )
46	44, 45	imbi12d 334	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = 1 -> ( ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) <-> ( ( -u 1 mod P ) = ( 1 mod P ) -> -u 1 = 1 ) ) )
47	42, 46	syl5ibr 236	. . . . . . . 8 |- ( ( -u 1 ^ N ) = 1 -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) )
48	22, 47	jaoi 394	. . . . . . 7 |- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) )
49	19, 48	sylbi 207	. . . . . 6 |- ( ( -u 1 ^ N ) e. { -u 1 , 1 } -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) )
50	17, 49	mpcom 38	. . . . 5 |- ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) )
51		oveq1 6657	. . . . . . 7 |- ( ( 2 /L P ) = -u 1 -> ( ( 2 /L P ) mod P ) = ( -u 1 mod P ) )
52	51	eqeq1d 2624	. . . . . 6 |- ( ( 2 /L P ) = -u 1 -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) ) )
53		eqeq1 2626	. . . . . 6 |- ( ( 2 /L P ) = -u 1 -> ( ( 2 /L P ) = ( -u 1 ^ N ) <-> -u 1 = ( -u 1 ^ N ) ) )
54	52, 53	imbi12d 334	. . . . 5 |- ( ( 2 /L P ) = -u 1 -> ( ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) <-> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) )
55	50, 54	syl5ibr 236	. . . 4 |- ( ( 2 /L P ) = -u 1 -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) )
56	2	gausslemma2dlem0a 25081	. . . . . . . . 9 |- ( ph -> P e. NN )
57	56	nnrpd 11870	. . . . . . . 8 |- ( ph -> P e. RR+ )
58		0mod 12701	. . . . . . . 8 |- ( P e. RR+ -> ( 0 mod P ) = 0 )
59	57, 58	syl 17	. . . . . . 7 |- ( ph -> ( 0 mod P ) = 0 )
60	59	eqeq1d 2624	. . . . . 6 |- ( ph -> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( ( -u 1 ^ N ) mod P ) ) )
61		oveq1 6657	. . . . . . . . . . . . 13 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) mod P ) = ( -u 1 mod P ) )
62	61	eqeq2d 2632	. . . . . . . . . . . 12 |- ( ( -u 1 ^ N ) = -u 1 -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( -u 1 mod P ) ) )
63	62	adantr 481	. . . . . . . . . . 11 |- ( ( ( -u 1 ^ N ) = -u 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( -u 1 mod P ) ) )
64		negmod0 12677	. . . . . . . . . . . . . . 15 |- ( ( 1 e. RR /\ P e. RR+ ) -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) )
65		eqcom 2629	. . . . . . . . . . . . . . 15 |- ( ( -u 1 mod P ) = 0 <-> 0 = ( -u 1 mod P ) )
66	64, 65	syl6bb 276	. . . . . . . . . . . . . 14 |- ( ( 1 e. RR /\ P e. RR+ ) -> ( ( 1 mod P ) = 0 <-> 0 = ( -u 1 mod P ) ) )
67	35, 57, 66	sylancr 695	. . . . . . . . . . . . 13 |- ( ph -> ( ( 1 mod P ) = 0 <-> 0 = ( -u 1 mod P ) ) )
68	30	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( ph -> ( ( 1 mod P ) = 0 <-> 1 = 0 ) )
69		ax-1ne0 10005	. . . . . . . . . . . . . . 15 |- 1 =/= 0
70		eqneqall 2805	. . . . . . . . . . . . . . 15 |- ( 1 = 0 -> ( 1 =/= 0 -> 0 = ( -u 1 ^ N ) ) )
71	69, 70	mpi 20	. . . . . . . . . . . . . 14 |- ( 1 = 0 -> 0 = ( -u 1 ^ N ) )
72	68, 71	syl6bi 243	. . . . . . . . . . . . 13 |- ( ph -> ( ( 1 mod P ) = 0 -> 0 = ( -u 1 ^ N ) ) )
73	67, 72	sylbird 250	. . . . . . . . . . . 12 |- ( ph -> ( 0 = ( -u 1 mod P ) -> 0 = ( -u 1 ^ N ) ) )
74	73	adantl 482	. . . . . . . . . . 11 |- ( ( ( -u 1 ^ N ) = -u 1 /\ ph ) -> ( 0 = ( -u 1 mod P ) -> 0 = ( -u 1 ^ N ) ) )
75	63, 74	sylbid 230	. . . . . . . . . 10 |- ( ( ( -u 1 ^ N ) = -u 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) )
76	75	ex 450	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = -u 1 -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) )
77	43	eqeq2d 2632	. . . . . . . . . . . 12 |- ( ( -u 1 ^ N ) = 1 -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( 1 mod P ) ) )
78	77	adantr 481	. . . . . . . . . . 11 |- ( ( ( -u 1 ^ N ) = 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( 1 mod P ) ) )
79		eqcom 2629	. . . . . . . . . . . . . 14 |- ( 0 = ( 1 mod P ) <-> ( 1 mod P ) = 0 )
80	79, 68	syl5bb 272	. . . . . . . . . . . . 13 |- ( ph -> ( 0 = ( 1 mod P ) <-> 1 = 0 ) )
81	80, 71	syl6bi 243	. . . . . . . . . . . 12 |- ( ph -> ( 0 = ( 1 mod P ) -> 0 = ( -u 1 ^ N ) ) )
82	81	adantl 482	. . . . . . . . . . 11 |- ( ( ( -u 1 ^ N ) = 1 /\ ph ) -> ( 0 = ( 1 mod P ) -> 0 = ( -u 1 ^ N ) ) )
83	78, 82	sylbid 230	. . . . . . . . . 10 |- ( ( ( -u 1 ^ N ) = 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) )
84	83	ex 450	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = 1 -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) )
85	76, 84	jaoi 394	. . . . . . . 8 |- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) )
86	19, 85	sylbi 207	. . . . . . 7 |- ( ( -u 1 ^ N ) e. { -u 1 , 1 } -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) )
87	17, 86	mpcom 38	. . . . . 6 |- ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) )
88	60, 87	sylbid 230	. . . . 5 |- ( ph -> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) )
89		oveq1 6657	. . . . . . 7 |- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) mod P ) = ( 0 mod P ) )
90	89	eqeq1d 2624	. . . . . 6 |- ( ( 2 /L P ) = 0 -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) ) )
91		eqeq1 2626	. . . . . 6 |- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) = ( -u 1 ^ N ) <-> 0 = ( -u 1 ^ N ) ) )
92	90, 91	imbi12d 334	. . . . 5 |- ( ( 2 /L P ) = 0 -> ( ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) <-> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) )
93	88, 92	syl5ibr 236	. . . 4 |- ( ( 2 /L P ) = 0 -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) )
94	30	eqeq1d 2624	. . . . . 6 |- ( ph -> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( ( -u 1 ^ N ) mod P ) ) )
95		eqcom 2629	. . . . . . . . . . 11 |- ( 1 = ( -u 1 mod P ) <-> ( -u 1 mod P ) = 1 )
96		eqcom 2629	. . . . . . . . . . 11 |- ( 1 = -u 1 <-> -u 1 = 1 )
97	41, 95, 96	3imtr4g 285	. . . . . . . . . 10 |- ( ph -> ( 1 = ( -u 1 mod P ) -> 1 = -u 1 ) )
98	61	eqeq2d 2632	. . . . . . . . . . 11 |- ( ( -u 1 ^ N ) = -u 1 -> ( 1 = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( -u 1 mod P ) ) )
99		eqeq2 2633	. . . . . . . . . . 11 |- ( ( -u 1 ^ N ) = -u 1 -> ( 1 = ( -u 1 ^ N ) <-> 1 = -u 1 ) )
100	98, 99	imbi12d 334	. . . . . . . . . 10 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) <-> ( 1 = ( -u 1 mod P ) -> 1 = -u 1 ) ) )
101	97, 100	syl5ibr 236	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = -u 1 -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) )
102		eqcom 2629	. . . . . . . . . . 11 |- ( ( -u 1 ^ N ) = 1 <-> 1 = ( -u 1 ^ N ) )
103	102	biimpi 206	. . . . . . . . . 10 |- ( ( -u 1 ^ N ) = 1 -> 1 = ( -u 1 ^ N ) )
104	103	2a1d 26	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = 1 -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) )
105	101, 104	jaoi 394	. . . . . . . 8 |- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) )
106	19, 105	sylbi 207	. . . . . . 7 |- ( ( -u 1 ^ N ) e. { -u 1 , 1 } -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) )
107	17, 106	mpcom 38	. . . . . 6 |- ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) )
108	94, 107	sylbid 230	. . . . 5 |- ( ph -> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) )
109		oveq1 6657	. . . . . . 7 |- ( ( 2 /L P ) = 1 -> ( ( 2 /L P ) mod P ) = ( 1 mod P ) )
110	109	eqeq1d 2624	. . . . . 6 |- ( ( 2 /L P ) = 1 -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) ) )
111		eqeq1 2626	. . . . . 6 |- ( ( 2 /L P ) = 1 -> ( ( 2 /L P ) = ( -u 1 ^ N ) <-> 1 = ( -u 1 ^ N ) ) )
112	110, 111	imbi12d 334	. . . . 5 |- ( ( 2 /L P ) = 1 -> ( ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) <-> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) )
113	108, 112	syl5ibr 236	. . . 4 |- ( ( 2 /L P ) = 1 -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) )
114	55, 93, 113	3jaoi 1391	. . 3 |- ( ( ( 2 /L P ) = -u 1 \/ ( 2 /L P ) = 0 \/ ( 2 /L P ) = 1 ) -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) )
115	10, 114	sylbi 207	. 2 |- ( ( 2 /L P ) e. { -u 1 , 0 , 1 } -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) )
116	8, 115	mpcom 38	1 |- ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   {cpr 4179   {ctp 4181   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   3c3 11071   4c4 11072   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   |_cfl 12591   mod cmo 12668   ^cexp 12860   Primecprime 15385   /Lclgs 25019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-pc 15542  df-lgs 25020

This theorem is referenced by:  gausslemma2d  25099