Theorem gausslemma2dlem0i 15578

Description: Auxiliary lemma 9 for gausslemma2d 15590. (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 9407	. . . 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 15570	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> P e. NN )
5	4	nnzd 9501	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
6	2, 5	syl 14	. . . 4 |- ( ph -> P e. ZZ )
7		lgscl1 15544	. . . 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 3679	. . . 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 15577	. . . . . . . 8 |- ( ph -> N e. NN0 )
16	15	nn0zd 9500	. . . . . . 7 |- ( ph -> N e. ZZ )
17		m1expcl2 10713	. . . . . . 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 3654	. . . . . . 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 2208	. . . . . . . 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 15570	. . . . . . . . . . 11 |- ( ph -> P e. NN )
26		nnq 9761	. . . . . . . . . . 11 |- ( P e. NN -> P e. QQ )
27	25, 26	syl 14	. . . . . . . . . 10 |- ( ph -> P e. QQ )
28	2	eldifad 3178	. . . . . . . . . . 11 |- ( ph -> P e. Prime )
29		prmgt1 12498	. . . . . . . . . . 11 |- ( P e. Prime -> 1 < P )
30	28, 29	syl 14	. . . . . . . . . 10 |- ( ph -> 1 < P )
31		q1mod 10508	. . . . . . . . . 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 2218	. . . . . . . 8 |- ( ph -> ( ( -u 1 mod P ) = ( 1 mod P ) <-> ( -u 1 mod P ) = 1 ) )
34		oddprmge3 12501	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 3 ) )
35		m1modge3gt1 10523	. . . . . . . . . 10 |- ( P e. ( ZZ>= ` 3 ) -> 1 < ( -u 1 mod P ) )
36		breq2 4051	. . . . . . . . . . 11 |- ( ( -u 1 mod P ) = 1 -> ( 1 < ( -u 1 mod P ) <-> 1 < 1 ) )
37		1re 8078	. . . . . . . . . . . . 13 |- 1 e. RR
38	37	ltnri 8172	. . . . . . . . . . . 12 |- -. 1 < 1
39	38	pm2.21i 647	. . . . . . . . . . 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 5958	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) mod P ) = ( 1 mod P ) )
45	44	eqeq2d 2218	. . . . . . . 8 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( -u 1 mod P ) = ( 1 mod P ) ) )
46		eqeq2 2216	. . . . . . . 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 718	. . . . 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 5958	. . . . . 6 |- ( ( 2 /L P ) = -u 1 -> ( ( 2 /L P ) mod P ) = ( -u 1 mod P ) )
52	51	eqeq1d 2215	. . . . 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 2213	. . . . 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 9087	. . . . . . 7 |- ( ph -> 0 < P )
57		q0mod 10507	. . . . . . 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 2215	. . . . 5 |- ( ph -> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( ( -u 1 ^ N ) mod P ) ) )
60		oveq1 5958	. . . . . . . . . . 11 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) mod P ) = ( -u 1 mod P ) )
61	60	eqeq2d 2218	. . . . . . . . . 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 9405	. . . . . . . . . . . . . 14 |- 1 e. ZZ
64		zq 9754	. . . . . . . . . . . . . 14 |- ( 1 e. ZZ -> 1 e. QQ )
65	63, 64	ax-mp 5	. . . . . . . . . . . . 13 |- 1 e. QQ
66		negqmod0 10483	. . . . . . . . . . . . 13 |- ( ( 1 e. QQ /\ P e. QQ /\ 0 < P ) -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) )
67	65, 27, 56, 66	mp3an2i 1355	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) )
68		eqcom 2208	. . . . . . . . . . . 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 2215	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 mod P ) = 0 <-> 1 = 0 ) )
71		1ne0 9111	. . . . . . . . . . . . 13 |- 1 =/= 0
72		eqneqall 2387	. . . . . . . . . . . . 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 2218	. . . . . . . . . 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 2208	. . . . . . . . . . . 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 718	. . . . . 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 5958	. . . . . 6 |- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) mod P ) = ( 0 mod P ) )
91	90	eqeq1d 2215	. . . . 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 2213	. . . . 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 2215	. . . . 5 |- ( ph -> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( ( -u 1 ^ N ) mod P ) ) )
96		eqcom 2208	. . . . . . . . 9 |- ( 1 = ( -u 1 mod P ) <-> ( -u 1 mod P ) = 1 )
97		eqcom 2208	. . . . . . . . 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 2218	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = -u 1 -> ( 1 = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( -u 1 mod P ) ) )
100		eqeq2 2216	. . . . . . . . 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 2208	. . . . . . . . 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 718	. . . . . 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 5958	. . . . . 6 |- ( ( 2 /L P ) = 1 -> ( ( 2 /L P ) mod P ) = ( 1 mod P ) )
110	109	eqeq1d 2215	. . . . 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 2213	. . . . 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 1316	. 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 710   \/ w3o 980   = wceq 1373   e. wcel 2177   =/= wne 2377   \ cdif 3164   {csn 3634   {cpr 3635   {ctp 3636   class class class wbr 4047   ` cfv 5276  (class class class)co 5951   0cc0 7932   1c1 7933   < clt 8114   - cmin 8250   -ucneg 8251   / cdiv 8752   NNcn 9043   2c2 9094   3c3 9095   4c4 9096   ZZcz 9379   ZZ>=cuz 9655   QQcq 9747   |_cfl 10418   mod cmo 10474   ^cexp 10690   Primecprime 12473   /Lclgs 15518

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-xor 1396  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-tp 3642  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-frec 6484  df-1o 6509  df-2o 6510  df-oadd 6513  df-er 6627  df-en 6835  df-dom 6836  df-fin 6837  df-sup 7093  df-inf 7094  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-fz 10138  df-fzo 10272  df-fl 10420  df-mod 10475  df-seqfrec 10600  df-exp 10691  df-ihash 10928  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-clim 11634  df-proddc 11906  df-dvds 12143  df-gcd 12319  df-prm 12474  df-phi 12577  df-pc 12652  df-lgs 15519

This theorem is referenced by:  gausslemma2d  15590