Theorem gausslemma2dlem0i 15815

Description: Auxiliary lemma 9 for gausslemma2d 15827. (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 9512	. . . 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 15807	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> P e. NN )
5	4	nnzd 9606	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
6	2, 5	syl 14	. . . 4 |- ( ph -> P e. ZZ )
7		lgscl1 15781	. . . 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 3715	. . . 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 15814	. . . . . . . 8 |- ( ph -> N e. NN0 )
16	15	nn0zd 9605	. . . . . . 7 |- ( ph -> N e. ZZ )
17		m1expcl2 10829	. . . . . . 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 3690	. . . . . . 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 2232	. . . . . . . 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 15807	. . . . . . . . . . 11 |- ( ph -> P e. NN )
26		nnq 9872	. . . . . . . . . . 11 |- ( P e. NN -> P e. QQ )
27	25, 26	syl 14	. . . . . . . . . 10 |- ( ph -> P e. QQ )
28	2	eldifad 3210	. . . . . . . . . . 11 |- ( ph -> P e. Prime )
29		prmgt1 12727	. . . . . . . . . . 11 |- ( P e. Prime -> 1 < P )
30	28, 29	syl 14	. . . . . . . . . 10 |- ( ph -> 1 < P )
31		q1mod 10624	. . . . . . . . . 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 2242	. . . . . . . 8 |- ( ph -> ( ( -u 1 mod P ) = ( 1 mod P ) <-> ( -u 1 mod P ) = 1 ) )
34		oddprmge3 12730	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 3 ) )
35		m1modge3gt1 10639	. . . . . . . . . 10 |- ( P e. ( ZZ>= ` 3 ) -> 1 < ( -u 1 mod P ) )
36		breq2 4093	. . . . . . . . . . 11 |- ( ( -u 1 mod P ) = 1 -> ( 1 < ( -u 1 mod P ) <-> 1 < 1 ) )
37		1re 8183	. . . . . . . . . . . . 13 |- 1 e. RR
38	37	ltnri 8277	. . . . . . . . . . . 12 |- -. 1 < 1
39	38	pm2.21i 651	. . . . . . . . . . 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 6030	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) mod P ) = ( 1 mod P ) )
45	44	eqeq2d 2242	. . . . . . . 8 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( -u 1 mod P ) = ( 1 mod P ) ) )
46		eqeq2 2240	. . . . . . . 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 723	. . . . 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 6030	. . . . . 6 |- ( ( 2 /L P ) = -u 1 -> ( ( 2 /L P ) mod P ) = ( -u 1 mod P ) )
52	51	eqeq1d 2239	. . . . 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 2237	. . . . 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 9192	. . . . . . 7 |- ( ph -> 0 < P )
57		q0mod 10623	. . . . . . 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 2239	. . . . 5 |- ( ph -> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( ( -u 1 ^ N ) mod P ) ) )
60		oveq1 6030	. . . . . . . . . . 11 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) mod P ) = ( -u 1 mod P ) )
61	60	eqeq2d 2242	. . . . . . . . . 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 9510	. . . . . . . . . . . . . 14 |- 1 e. ZZ
64		zq 9865	. . . . . . . . . . . . . 14 |- ( 1 e. ZZ -> 1 e. QQ )
65	63, 64	ax-mp 5	. . . . . . . . . . . . 13 |- 1 e. QQ
66		negqmod0 10599	. . . . . . . . . . . . 13 |- ( ( 1 e. QQ /\ P e. QQ /\ 0 < P ) -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) )
67	65, 27, 56, 66	mp3an2i 1378	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) )
68		eqcom 2232	. . . . . . . . . . . 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 2239	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 mod P ) = 0 <-> 1 = 0 ) )
71		1ne0 9216	. . . . . . . . . . . . 13 |- 1 =/= 0
72		eqneqall 2411	. . . . . . . . . . . . 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 2242	. . . . . . . . . 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 2232	. . . . . . . . . . . 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 723	. . . . . 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 6030	. . . . . 6 |- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) mod P ) = ( 0 mod P ) )
91	90	eqeq1d 2239	. . . . 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 2237	. . . . 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 2239	. . . . 5 |- ( ph -> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( ( -u 1 ^ N ) mod P ) ) )
96		eqcom 2232	. . . . . . . . 9 |- ( 1 = ( -u 1 mod P ) <-> ( -u 1 mod P ) = 1 )
97		eqcom 2232	. . . . . . . . 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 2242	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = -u 1 -> ( 1 = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( -u 1 mod P ) ) )
100		eqeq2 2240	. . . . . . . . 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 2232	. . . . . . . . 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 723	. . . . . 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 6030	. . . . . 6 |- ( ( 2 /L P ) = 1 -> ( ( 2 /L P ) mod P ) = ( 1 mod P ) )
110	109	eqeq1d 2239	. . . . 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 2237	. . . . 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 1339	. 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 715   \/ w3o 1003   = wceq 1397   e. wcel 2201   =/= wne 2401   \ cdif 3196   {csn 3670   {cpr 3671   {ctp 3672   class class class wbr 4089   ` cfv 5328  (class class class)co 6023   0cc0 8037   1c1 8038   < clt 8219   - cmin 8355   -ucneg 8356   / cdiv 8857   NNcn 9148   2c2 9199   3c3 9200   4c4 9201   ZZcz 9484   ZZ>=cuz 9760   QQcq 9858   |_cfl 10534   mod cmo 10590   ^cexp 10806   Primecprime 12702   /Lclgs 15755

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155  ax-arch 8156  ax-caucvg 8157

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-tp 3678  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-isom 5337  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-irdg 6541  df-frec 6562  df-1o 6587  df-2o 6588  df-oadd 6591  df-er 6707  df-en 6915  df-dom 6916  df-fin 6917  df-sup 7188  df-inf 7189  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-n0 9408  df-z 9485  df-uz 9761  df-q 9859  df-rp 9894  df-fz 10249  df-fzo 10383  df-fl 10536  df-mod 10591  df-seqfrec 10716  df-exp 10807  df-ihash 11044  df-cj 11425  df-re 11426  df-im 11427  df-rsqrt 11581  df-abs 11582  df-clim 11862  df-proddc 12135  df-dvds 12372  df-gcd 12548  df-prm 12703  df-phi 12806  df-pc 12881  df-lgs 15756

This theorem is referenced by:  gausslemma2d  15827