Theorem gausslemma2dlem0i 16059

Description: Auxiliary lemma 9 for gausslemma2d 16071. (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 9625	. . . 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 16051	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> P e. NN )
5	4	nnzd 9720	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
6	2, 5	syl 14	. . . 4 |- ( ph -> P e. ZZ )
7		lgscl1 16025	. . . 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 3739	. . . 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 16058	. . . . . . . 8 |- ( ph -> N e. NN0 )
16	15	nn0zd 9719	. . . . . . 7 |- ( ph -> N e. ZZ )
17		m1expcl2 10950	. . . . . . 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 3714	. . . . . . 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 2236	. . . . . . . 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 16051	. . . . . . . . . . 11 |- ( ph -> P e. NN )
26		nnq 9986	. . . . . . . . . . 11 |- ( P e. NN -> P e. QQ )
27	25, 26	syl 14	. . . . . . . . . 10 |- ( ph -> P e. QQ )
28	2	eldifad 3225	. . . . . . . . . . 11 |- ( ph -> P e. Prime )
29		prmgt1 12857	. . . . . . . . . . 11 |- ( P e. Prime -> 1 < P )
30	28, 29	syl 14	. . . . . . . . . 10 |- ( ph -> 1 < P )
31		q1mod 10745	. . . . . . . . . 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 2246	. . . . . . . 8 |- ( ph -> ( ( -u 1 mod P ) = ( 1 mod P ) <-> ( -u 1 mod P ) = 1 ) )
34		oddprmge3 12860	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 3 ) )
35		m1modge3gt1 10760	. . . . . . . . . 10 |- ( P e. ( ZZ>= ` 3 ) -> 1 < ( -u 1 mod P ) )
36		breq2 4118	. . . . . . . . . . 11 |- ( ( -u 1 mod P ) = 1 -> ( 1 < ( -u 1 mod P ) <-> 1 < 1 ) )
37		1re 8289	. . . . . . . . . . . . 13 |- 1 e. RR
38	37	ltnri 8382	. . . . . . . . . . . 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 6065	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) mod P ) = ( 1 mod P ) )
45	44	eqeq2d 2246	. . . . . . . 8 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( -u 1 mod P ) = ( 1 mod P ) ) )
46		eqeq2 2244	. . . . . . . 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 724	. . . . 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 6065	. . . . . 6 |- ( ( 2 /L P ) = -u 1 -> ( ( 2 /L P ) mod P ) = ( -u 1 mod P ) )
52	51	eqeq1d 2243	. . . . 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 2241	. . . . 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 9301	. . . . . . 7 |- ( ph -> 0 < P )
57		q0mod 10744	. . . . . . 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 2243	. . . . 5 |- ( ph -> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( ( -u 1 ^ N ) mod P ) ) )
60		oveq1 6065	. . . . . . . . . . 11 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) mod P ) = ( -u 1 mod P ) )
61	60	eqeq2d 2246	. . . . . . . . . 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 9623	. . . . . . . . . . . . . 14 |- 1 e. ZZ
64		zq 9979	. . . . . . . . . . . . . 14 |- ( 1 e. ZZ -> 1 e. QQ )
65	63, 64	ax-mp 5	. . . . . . . . . . . . 13 |- 1 e. QQ
66		negqmod0 10720	. . . . . . . . . . . . 13 |- ( ( 1 e. QQ /\ P e. QQ /\ 0 < P ) -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) )
67	65, 27, 56, 66	mp3an2i 1379	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) )
68		eqcom 2236	. . . . . . . . . . . 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 2243	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 mod P ) = 0 <-> 1 = 0 ) )
71		1ne0 9325	. . . . . . . . . . . . 13 |- 1 =/= 0
72		eqneqall 2424	. . . . . . . . . . . . 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 2246	. . . . . . . . . 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 2236	. . . . . . . . . . . 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 724	. . . . . 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 6065	. . . . . 6 |- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) mod P ) = ( 0 mod P ) )
91	90	eqeq1d 2243	. . . . 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 2241	. . . . 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 2243	. . . . 5 |- ( ph -> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( ( -u 1 ^ N ) mod P ) ) )
96		eqcom 2236	. . . . . . . . 9 |- ( 1 = ( -u 1 mod P ) <-> ( -u 1 mod P ) = 1 )
97		eqcom 2236	. . . . . . . . 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 2246	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = -u 1 -> ( 1 = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( -u 1 mod P ) ) )
100		eqeq2 2244	. . . . . . . . 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 2236	. . . . . . . . 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 724	. . . . . 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 6065	. . . . . 6 |- ( ( 2 /L P ) = 1 -> ( ( 2 /L P ) mod P ) = ( 1 mod P ) )
110	109	eqeq1d 2243	. . . . 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 2241	. . . . 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 1340	. 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 716   \/ w3o 1004   = wceq 1398   e. wcel 2205   =/= wne 2414   \ cdif 3211   {csn 3694   {cpr 3695   {ctp 3696   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   0cc0 8143   1c1 8144   < clt 8324   - cmin 8461   -ucneg 8462   / cdiv 8966   NNcn 9257   2c2 9308   3c3 9309   4c4 9310   ZZcz 9597   ZZ>=cuz 9874   QQcq 9972   |_cfl 10655   mod cmo 10711   ^cexp 10927   Primecprime 12832   /Lclgs 15999

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-reap 8867  df-ap 8874  df-div 8967  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-7 9321  df-8 9322  df-n0 9517  df-z 9598  df-uz 9875  df-q 9973  df-rp 10008  df-fz 10365  df-fzo 10502  df-fl 10657  df-mod 10712  df-seqfrec 10837  df-exp 10928  df-ihash 11167  df-cj 11555  df-re 11556  df-im 11557  df-rsqrt 11711  df-abs 11712  df-clim 11992  df-proddc 12265  df-dvds 12502  df-gcd 12678  df-prm 12833  df-phi 12936  df-pc 13011  df-lgs 16000

This theorem is referenced by:  gausslemma2d  16071