Theorem gausslemma2dlem0i 15701

Description: Auxiliary lemma 9 for gausslemma2d 15713. (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 9442	. . . 4 |- 2 e. ZZ
2		gausslemma2dlem0.p	. . . . 5 |- ( ph -> P e. ( Prime \ { 2 } ) )
3		id 19	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> P e. ( Prime \ { 2 } ) )
4	3	gausslemma2dlem0a 15693	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> P e. NN )
5	4	nnzd 9536	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
6	2, 5	syl 14	. . . 4 |- ( ph -> P e. ZZ )
7		lgscl1 15667	. . . 4 |- ( ( 2 e. ZZ /\ P e. ZZ ) -> ( 2 /L P ) e. { -u 1 , 0 , 1 } )
8	1, 6, 7	sylancr 414	. . 3 |- ( ph -> ( 2 /L P ) e. { -u 1 , 0 , 1 } )
9		eltpg 3691	. . . 4 |- ( ( 2 /L P ) e. { -u 1 , 0 , 1 } -> ( ( 2 /L P ) e. { -u 1 , 0 , 1 } <-> ( ( 2 /L P ) = -u 1 \/ ( 2 /L P ) = 0 \/ ( 2 /L P ) = 1 ) ) )
10	8, 9	syl 14	. . 3 |- ( ph -> ( ( 2 /L P ) e. { -u 1 , 0 , 1 } <-> ( ( 2 /L P ) = -u 1 \/ ( 2 /L P ) = 0 \/ ( 2 /L P ) = 1 ) ) )
11	8, 10	mpbid 147	. 2 |- ( ph -> ( ( 2 /L P ) = -u 1 \/ ( 2 /L P ) = 0 \/ ( 2 /L P ) = 1 ) )
12		gausslemma2dlem0.m	. . . . . . . . 9 |- M = ( |_ ` ( P / 4 ) )
13		gausslemma2dlem0.h	. . . . . . . . 9 |- H = ( ( P - 1 ) / 2 )
14		gausslemma2dlem0.n	. . . . . . . . 9 |- N = ( H - M )
15	2, 12, 13, 14	gausslemma2dlem0h 15700	. . . . . . . 8 |- ( ph -> N e. NN0 )
16	15	nn0zd 9535	. . . . . . 7 |- ( ph -> N e. ZZ )
17		m1expcl2 10750	. . . . . . 7 |- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )
18	16, 17	syl 14	. . . . . 6 |- ( ph -> ( -u 1 ^ N ) e. { -u 1 , 1 } )
19		elprg 3666	. . . . . . 7 |- ( ( -u 1 ^ N ) e. { -u 1 , 1 } -> ( ( -u 1 ^ N ) e. { -u 1 , 1 } <-> ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) ) )
20	18, 19	syl 14	. . . . . 6 |- ( ph -> ( ( -u 1 ^ N ) e. { -u 1 , 1 } <-> ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) ) )
21	18, 20	mpbid 147	. . . . 5 |- ( ph -> ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) )
22		eqcom 2211	. . . . . . . 8 |- ( ( -u 1 ^ N ) = -u 1 <-> -u 1 = ( -u 1 ^ N ) )
23	22	biimpi 120	. . . . . . 7 |- ( ( -u 1 ^ N ) = -u 1 -> -u 1 = ( -u 1 ^ N ) )
24	23	2a1d 23	. . . . . 6 |- ( ( -u 1 ^ N ) = -u 1 -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) )
25	2	gausslemma2dlem0a 15693	. . . . . . . . . . 11 |- ( ph -> P e. NN )
26		nnq 9796	. . . . . . . . . . 11 |- ( P e. NN -> P e. QQ )
27	25, 26	syl 14	. . . . . . . . . 10 |- ( ph -> P e. QQ )
28	2	eldifad 3188	. . . . . . . . . . 11 |- ( ph -> P e. Prime )
29		prmgt1 12620	. . . . . . . . . . 11 |- ( P e. Prime -> 1 < P )
30	28, 29	syl 14	. . . . . . . . . 10 |- ( ph -> 1 < P )
31		q1mod 10545	. . . . . . . . . 10 |- ( ( P e. QQ /\ 1 < P ) -> ( 1 mod P ) = 1 )
32	27, 30, 31	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( 1 mod P ) = 1 )
33	32	eqeq2d 2221	. . . . . . . 8 |- ( ph -> ( ( -u 1 mod P ) = ( 1 mod P ) <-> ( -u 1 mod P ) = 1 ) )
34		oddprmge3 12623	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 3 ) )
35		m1modge3gt1 10560	. . . . . . . . . 10 |- ( P e. ( ZZ>= ` 3 ) -> 1 < ( -u 1 mod P ) )
36		breq2 4066	. . . . . . . . . . 11 |- ( ( -u 1 mod P ) = 1 -> ( 1 < ( -u 1 mod P ) <-> 1 < 1 ) )
37		1re 8113	. . . . . . . . . . . . 13 |- 1 e. RR
38	37	ltnri 8207	. . . . . . . . . . . 12 |- -. 1 < 1
39	38	pm2.21i 649	. . . . . . . . . . 11 |- ( 1 < 1 -> -u 1 = 1 )
40	36, 39	biimtrdi 163	. . . . . . . . . 10 |- ( ( -u 1 mod P ) = 1 -> ( 1 < ( -u 1 mod P ) -> -u 1 = 1 ) )
41	35, 40	syl5com 29	. . . . . . . . 9 |- ( P e. ( ZZ>= ` 3 ) -> ( ( -u 1 mod P ) = 1 -> -u 1 = 1 ) )
42	2, 34, 41	3syl 17	. . . . . . . 8 |- ( ph -> ( ( -u 1 mod P ) = 1 -> -u 1 = 1 ) )
43	33, 42	sylbid 150	. . . . . . 7 |- ( ph -> ( ( -u 1 mod P ) = ( 1 mod P ) -> -u 1 = 1 ) )
44		oveq1 5981	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) mod P ) = ( 1 mod P ) )
45	44	eqeq2d 2221	. . . . . . . 8 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( -u 1 mod P ) = ( 1 mod P ) ) )
46		eqeq2 2219	. . . . . . . 8 |- ( ( -u 1 ^ N ) = 1 -> ( -u 1 = ( -u 1 ^ N ) <-> -u 1 = 1 ) )
47	45, 46	imbi12d 234	. . . . . . 7 |- ( ( -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 ) ) )
48	43, 47	imbitrrid 156	. . . . . 6 |- ( ( -u 1 ^ N ) = 1 -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) )
49	24, 48	jaoi 720	. . . . 5 |- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) )
50	21, 49	mpcom 36	. . . 4 |- ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) )
51		oveq1 5981	. . . . . 6 |- ( ( 2 /L P ) = -u 1 -> ( ( 2 /L P ) mod P ) = ( -u 1 mod P ) )
52	51	eqeq1d 2218	. . . . 5 |- ( ( 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 2216	. . . . 5 |- ( ( 2 /L P ) = -u 1 -> ( ( 2 /L P ) = ( -u 1 ^ N ) <-> -u 1 = ( -u 1 ^ N ) ) )
54	52, 53	imbi12d 234	. . . 4 |- ( ( 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	imbitrrid 156	. . 3 |- ( ( 2 /L P ) = -u 1 -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) )
56	25	nngt0d 9122	. . . . . . 7 |- ( ph -> 0 < P )
57		q0mod 10544	. . . . . . 7 |- ( ( P e. QQ /\ 0 < P ) -> ( 0 mod P ) = 0 )
58	27, 56, 57	syl2anc 411	. . . . . 6 |- ( ph -> ( 0 mod P ) = 0 )
59	58	eqeq1d 2218	. . . . 5 |- ( ph -> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( ( -u 1 ^ N ) mod P ) ) )
60		oveq1 5981	. . . . . . . . . . 11 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) mod P ) = ( -u 1 mod P ) )
61	60	eqeq2d 2221	. . . . . . . . . 10 |- ( ( -u 1 ^ N ) = -u 1 -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( -u 1 mod P ) ) )
62	61	adantr 276	. . . . . . . . 9 |- ( ( ( -u 1 ^ N ) = -u 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( -u 1 mod P ) ) )
63		1z 9440	. . . . . . . . . . . . . 14 |- 1 e. ZZ
64		zq 9789	. . . . . . . . . . . . . 14 |- ( 1 e. ZZ -> 1 e. QQ )
65	63, 64	ax-mp 5	. . . . . . . . . . . . 13 |- 1 e. QQ
66		negqmod0 10520	. . . . . . . . . . . . 13 |- ( ( 1 e. QQ /\ P e. QQ /\ 0 < P ) -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) )
67	65, 27, 56, 66	mp3an2i 1357	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) )
68		eqcom 2211	. . . . . . . . . . . 12 |- ( ( -u 1 mod P ) = 0 <-> 0 = ( -u 1 mod P ) )
69	67, 68	bitrdi 196	. . . . . . . . . . 11 |- ( ph -> ( ( 1 mod P ) = 0 <-> 0 = ( -u 1 mod P ) ) )
70	32	eqeq1d 2218	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 mod P ) = 0 <-> 1 = 0 ) )
71		1ne0 9146	. . . . . . . . . . . . 13 |- 1 =/= 0
72		eqneqall 2390	. . . . . . . . . . . . 13 |- ( 1 = 0 -> ( 1 =/= 0 -> 0 = ( -u 1 ^ N ) ) )
73	71, 72	mpi 15	. . . . . . . . . . . 12 |- ( 1 = 0 -> 0 = ( -u 1 ^ N ) )
74	70, 73	biimtrdi 163	. . . . . . . . . . 11 |- ( ph -> ( ( 1 mod P ) = 0 -> 0 = ( -u 1 ^ N ) ) )
75	69, 74	sylbird 170	. . . . . . . . . 10 |- ( ph -> ( 0 = ( -u 1 mod P ) -> 0 = ( -u 1 ^ N ) ) )
76	75	adantl 277	. . . . . . . . 9 |- ( ( ( -u 1 ^ N ) = -u 1 /\ ph ) -> ( 0 = ( -u 1 mod P ) -> 0 = ( -u 1 ^ N ) ) )
77	62, 76	sylbid 150	. . . . . . . 8 |- ( ( ( -u 1 ^ N ) = -u 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) )
78	77	ex 115	. . . . . . 7 |- ( ( -u 1 ^ N ) = -u 1 -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) )
79	44	eqeq2d 2221	. . . . . . . . . 10 |- ( ( -u 1 ^ N ) = 1 -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( 1 mod P ) ) )
80	79	adantr 276	. . . . . . . . 9 |- ( ( ( -u 1 ^ N ) = 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( 1 mod P ) ) )
81		eqcom 2211	. . . . . . . . . . . 12 |- ( 0 = ( 1 mod P ) <-> ( 1 mod P ) = 0 )
82	81, 70	bitrid 192	. . . . . . . . . . 11 |- ( ph -> ( 0 = ( 1 mod P ) <-> 1 = 0 ) )
83	82, 73	biimtrdi 163	. . . . . . . . . 10 |- ( ph -> ( 0 = ( 1 mod P ) -> 0 = ( -u 1 ^ N ) ) )
84	83	adantl 277	. . . . . . . . 9 |- ( ( ( -u 1 ^ N ) = 1 /\ ph ) -> ( 0 = ( 1 mod P ) -> 0 = ( -u 1 ^ N ) ) )
85	80, 84	sylbid 150	. . . . . . . 8 |- ( ( ( -u 1 ^ N ) = 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) )
86	85	ex 115	. . . . . . 7 |- ( ( -u 1 ^ N ) = 1 -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) )
87	78, 86	jaoi 720	. . . . . 6 |- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) )
88	21, 87	mpcom 36	. . . . 5 |- ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) )
89	59, 88	sylbid 150	. . . 4 |- ( ph -> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) )
90		oveq1 5981	. . . . . 6 |- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) mod P ) = ( 0 mod P ) )
91	90	eqeq1d 2218	. . . . 5 |- ( ( 2 /L P ) = 0 -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) ) )
92		eqeq1 2216	. . . . 5 |- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) = ( -u 1 ^ N ) <-> 0 = ( -u 1 ^ N ) ) )
93	91, 92	imbi12d 234	. . . 4 |- ( ( 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 ) ) ) )
94	89, 93	imbitrrid 156	. . 3 |- ( ( 2 /L P ) = 0 -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) )
95	32	eqeq1d 2218	. . . . 5 |- ( ph -> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( ( -u 1 ^ N ) mod P ) ) )
96		eqcom 2211	. . . . . . . . 9 |- ( 1 = ( -u 1 mod P ) <-> ( -u 1 mod P ) = 1 )
97		eqcom 2211	. . . . . . . . 9 |- ( 1 = -u 1 <-> -u 1 = 1 )
98	42, 96, 97	3imtr4g 205	. . . . . . . 8 |- ( ph -> ( 1 = ( -u 1 mod P ) -> 1 = -u 1 ) )
99	60	eqeq2d 2221	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = -u 1 -> ( 1 = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( -u 1 mod P ) ) )
100		eqeq2 2219	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = -u 1 -> ( 1 = ( -u 1 ^ N ) <-> 1 = -u 1 ) )
101	99, 100	imbi12d 234	. . . . . . . 8 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) <-> ( 1 = ( -u 1 mod P ) -> 1 = -u 1 ) ) )
102	98, 101	imbitrrid 156	. . . . . . 7 |- ( ( -u 1 ^ N ) = -u 1 -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) )
103		eqcom 2211	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = 1 <-> 1 = ( -u 1 ^ N ) )
104	103	biimpi 120	. . . . . . . 8 |- ( ( -u 1 ^ N ) = 1 -> 1 = ( -u 1 ^ N ) )
105	104	2a1d 23	. . . . . . 7 |- ( ( -u 1 ^ N ) = 1 -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) )
106	102, 105	jaoi 720	. . . . . 6 |- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) )
107	21, 106	mpcom 36	. . . . 5 |- ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) )
108	95, 107	sylbid 150	. . . 4 |- ( ph -> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) )
109		oveq1 5981	. . . . . 6 |- ( ( 2 /L P ) = 1 -> ( ( 2 /L P ) mod P ) = ( 1 mod P ) )
110	109	eqeq1d 2218	. . . . 5 |- ( ( 2 /L P ) = 1 -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) ) )
111		eqeq1 2216	. . . . 5 |- ( ( 2 /L P ) = 1 -> ( ( 2 /L P ) = ( -u 1 ^ N ) <-> 1 = ( -u 1 ^ N ) ) )
112	110, 111	imbi12d 234	. . . 4 |- ( ( 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	imbitrrid 156	. . 3 |- ( ( 2 /L P ) = 1 -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) )
114	55, 94, 113	3jaoi 1318	. 2 |- ( ( ( 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	11, 114	mpcom 36	1 |- ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 712   \/ w3o 982   = wceq 1375   e. wcel 2180   =/= wne 2380   \ cdif 3174   {csn 3646   {cpr 3647   {ctp 3648   class class class wbr 4062   ` cfv 5294  (class class class)co 5974   0cc0 7967   1c1 7968   < clt 8149   - cmin 8285   -ucneg 8286   / cdiv 8787   NNcn 9078   2c2 9129   3c3 9130   4c4 9131   ZZcz 9414   ZZ>=cuz 9690   QQcq 9782   |_cfl 10455   mod cmo 10511   ^cexp 10727   Primecprime 12595   /Lclgs 15641

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086  ax-caucvg 8087

This theorem depends on definitions:  df-bi 117  df-stab 835  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-xor 1398  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-tp 3654  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-isom 5303  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-frec 6507  df-1o 6532  df-2o 6533  df-oadd 6536  df-er 6650  df-en 6858  df-dom 6859  df-fin 6860  df-sup 7119  df-inf 7120  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-n0 9338  df-z 9415  df-uz 9691  df-q 9783  df-rp 9818  df-fz 10173  df-fzo 10307  df-fl 10457  df-mod 10512  df-seqfrec 10637  df-exp 10728  df-ihash 10965  df-cj 11319  df-re 11320  df-im 11321  df-rsqrt 11475  df-abs 11476  df-clim 11756  df-proddc 12028  df-dvds 12265  df-gcd 12441  df-prm 12596  df-phi 12699  df-pc 12774  df-lgs 15642

This theorem is referenced by:  gausslemma2d  15713