Theorem gausslemma2dlem0i 15779

Description: Auxiliary lemma 9 for gausslemma2d 15791. (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 9500	. . . 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 15771	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> P e. NN )
5	4	nnzd 9594	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
6	2, 5	syl 14	. . . 4 |- ( ph -> P e. ZZ )
7		lgscl1 15745	. . . 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 3712	. . . 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 15778	. . . . . . . 8 |- ( ph -> N e. NN0 )
16	15	nn0zd 9593	. . . . . . 7 |- ( ph -> N e. ZZ )
17		m1expcl2 10816	. . . . . . 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 3687	. . . . . . 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 2231	. . . . . . . 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 15771	. . . . . . . . . . 11 |- ( ph -> P e. NN )
26		nnq 9860	. . . . . . . . . . 11 |- ( P e. NN -> P e. QQ )
27	25, 26	syl 14	. . . . . . . . . 10 |- ( ph -> P e. QQ )
28	2	eldifad 3209	. . . . . . . . . . 11 |- ( ph -> P e. Prime )
29		prmgt1 12697	. . . . . . . . . . 11 |- ( P e. Prime -> 1 < P )
30	28, 29	syl 14	. . . . . . . . . 10 |- ( ph -> 1 < P )
31		q1mod 10611	. . . . . . . . . 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 2241	. . . . . . . 8 |- ( ph -> ( ( -u 1 mod P ) = ( 1 mod P ) <-> ( -u 1 mod P ) = 1 ) )
34		oddprmge3 12700	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 3 ) )
35		m1modge3gt1 10626	. . . . . . . . . 10 |- ( P e. ( ZZ>= ` 3 ) -> 1 < ( -u 1 mod P ) )
36		breq2 4090	. . . . . . . . . . 11 |- ( ( -u 1 mod P ) = 1 -> ( 1 < ( -u 1 mod P ) <-> 1 < 1 ) )
37		1re 8171	. . . . . . . . . . . . 13 |- 1 e. RR
38	37	ltnri 8265	. . . . . . . . . . . 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 6020	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) mod P ) = ( 1 mod P ) )
45	44	eqeq2d 2241	. . . . . . . 8 |- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( -u 1 mod P ) = ( 1 mod P ) ) )
46		eqeq2 2239	. . . . . . . 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 721	. . . . 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 6020	. . . . . 6 |- ( ( 2 /L P ) = -u 1 -> ( ( 2 /L P ) mod P ) = ( -u 1 mod P ) )
52	51	eqeq1d 2238	. . . . 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 2236	. . . . 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 9180	. . . . . . 7 |- ( ph -> 0 < P )
57		q0mod 10610	. . . . . . 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 2238	. . . . 5 |- ( ph -> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( ( -u 1 ^ N ) mod P ) ) )
60		oveq1 6020	. . . . . . . . . . 11 |- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) mod P ) = ( -u 1 mod P ) )
61	60	eqeq2d 2241	. . . . . . . . . 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 9498	. . . . . . . . . . . . . 14 |- 1 e. ZZ
64		zq 9853	. . . . . . . . . . . . . 14 |- ( 1 e. ZZ -> 1 e. QQ )
65	63, 64	ax-mp 5	. . . . . . . . . . . . 13 |- 1 e. QQ
66		negqmod0 10586	. . . . . . . . . . . . 13 |- ( ( 1 e. QQ /\ P e. QQ /\ 0 < P ) -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) )
67	65, 27, 56, 66	mp3an2i 1376	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) )
68		eqcom 2231	. . . . . . . . . . . 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 2238	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 mod P ) = 0 <-> 1 = 0 ) )
71		1ne0 9204	. . . . . . . . . . . . 13 |- 1 =/= 0
72		eqneqall 2410	. . . . . . . . . . . . 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 2241	. . . . . . . . . 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 2231	. . . . . . . . . . . 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 721	. . . . . 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 6020	. . . . . 6 |- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) mod P ) = ( 0 mod P ) )
91	90	eqeq1d 2238	. . . . 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 2236	. . . . 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 2238	. . . . 5 |- ( ph -> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( ( -u 1 ^ N ) mod P ) ) )
96		eqcom 2231	. . . . . . . . 9 |- ( 1 = ( -u 1 mod P ) <-> ( -u 1 mod P ) = 1 )
97		eqcom 2231	. . . . . . . . 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 2241	. . . . . . . . 9 |- ( ( -u 1 ^ N ) = -u 1 -> ( 1 = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( -u 1 mod P ) ) )
100		eqeq2 2239	. . . . . . . . 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 2231	. . . . . . . . 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 721	. . . . . 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 6020	. . . . . 6 |- ( ( 2 /L P ) = 1 -> ( ( 2 /L P ) mod P ) = ( 1 mod P ) )
110	109	eqeq1d 2238	. . . . 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 2236	. . . . 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 1337	. 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 713   \/ w3o 1001   = wceq 1395   e. wcel 2200   =/= wne 2400   \ cdif 3195   {csn 3667   {cpr 3668   {ctp 3669   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   0cc0 8025   1c1 8026   < clt 8207   - cmin 8343   -ucneg 8344   / cdiv 8845   NNcn 9136   2c2 9187   3c3 9188   4c4 9189   ZZcz 9472   ZZ>=cuz 9748   QQcq 9846   |_cfl 10521   mod cmo 10577   ^cexp 10793   Primecprime 12672   /Lclgs 15719

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-tp 3675  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-2o 6578  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-sup 7177  df-inf 7178  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-fz 10237  df-fzo 10371  df-fl 10523  df-mod 10578  df-seqfrec 10703  df-exp 10794  df-ihash 11031  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-clim 11833  df-proddc 12105  df-dvds 12342  df-gcd 12518  df-prm 12673  df-phi 12776  df-pc 12851  df-lgs 15720

This theorem is referenced by:  gausslemma2d  15791