Theorem gausslemma2dlem0i 16393

Description: Auxiliary lemma 9 for gausslemma2d 16405. (Contributed by AV, 14-Jul-2021.)

Hypotheses

Ref	Expression
gausslemma2dlem0.p	⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
gausslemma2dlem0.m	⊢ 𝑀 = (⌊‘(𝑃 / 4))
gausslemma2dlem0.h	⊢ 𝐻 = ((𝑃 − 1) / 2)
gausslemma2dlem0.n	⊢ 𝑁 = (𝐻 − 𝑀)

Assertion

Ref	Expression
gausslemma2dlem0i	⊢ (𝜑 → (((2 /L 𝑃) mod 𝑃) = (( -1↑𝑁) mod 𝑃) → (2 /L 𝑃) = ( -1↑𝑁)))

Proof of Theorem gausslemma2dlem0i

Step	Hyp	Ref	Expression
1		2z 9684	. . . 4 ⊢ 2 ∈ ℤ
2		gausslemma2dlem0.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
3		id 19	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℙ ∖ {2}))
4	3	gausslemma2dlem0a 16385	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ)
5	4	nnzd 9779	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ)
6	2, 5	syl 14	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℤ)
7		lgscl1 16359	. . . 4 ⊢ ((2 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (2 /L 𝑃) ∈ { -1, 0, 1})
8	1, 6, 7	sylancr 418	. . 3 ⊢ (𝜑 → (2 /L 𝑃) ∈ { -1, 0, 1})
9		eltpg 3754	. . . 4 ⊢ ((2 /L 𝑃) ∈ { -1, 0, 1} → ((2 /L 𝑃) ∈ { -1, 0, 1} ↔ ((2 /L 𝑃) = -1 ∨ (2 /L 𝑃) = 0 ∨ (2 /L 𝑃) = 1)))
10	8, 9	syl 14	. . 3 ⊢ (𝜑 → ((2 /L 𝑃) ∈ { -1, 0, 1} ↔ ((2 /L 𝑃) = -1 ∨ (2 /L 𝑃) = 0 ∨ (2 /L 𝑃) = 1)))
11	8, 10	mpbid 147	. 2 ⊢ (𝜑 → ((2 /L 𝑃) = -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 16392	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
16	15	nn0zd 9778	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
17		m1expcl2 11020	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → ( -1↑𝑁) ∈ { -1, 1})
18	16, 17	syl 14	. . . . . 6 ⊢ (𝜑 → ( -1↑𝑁) ∈ { -1, 1})
19		elprg 3729	. . . . . . 7 ⊢ (( -1↑𝑁) ∈ { -1, 1} → (( -1↑𝑁) ∈ { -1, 1} ↔ (( -1↑𝑁) = -1 ∨ ( -1↑𝑁) = 1)))
20	18, 19	syl 14	. . . . . 6 ⊢ (𝜑 → (( -1↑𝑁) ∈ { -1, 1} ↔ (( -1↑𝑁) = -1 ∨ ( -1↑𝑁) = 1)))
21	18, 20	mpbid 147	. . . . 5 ⊢ (𝜑 → (( -1↑𝑁) = -1 ∨ ( -1↑𝑁) = 1))
22		eqcom 2240	. . . . . . . 8 ⊢ (( -1↑𝑁) = -1 ↔ -1 = ( -1↑𝑁))
23	22	biimpi 120	. . . . . . 7 ⊢ (( -1↑𝑁) = -1 → -1 = ( -1↑𝑁))
24	23	2a1d 23	. . . . . 6 ⊢ (( -1↑𝑁) = -1 → (𝜑 → (( -1 mod 𝑃) = (( -1↑𝑁) mod 𝑃) → -1 = ( -1↑𝑁))))
25	2	gausslemma2dlem0a 16385	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℕ)
26		nnq 10050	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
27	25, 26	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℚ)
28	2	eldifad 3231	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℙ)
29		prmgt1 12937	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
30	28, 29	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 1 < 𝑃)
31		q1mod 10815	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℚ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
32	27, 30, 31	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → (1 mod 𝑃) = 1)
33	32	eqeq2d 2250	. . . . . . . 8 ⊢ (𝜑 → (( -1 mod 𝑃) = (1 mod 𝑃) ↔ ( -1 mod 𝑃) = 1))
34		oddprmge3 12940	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ≥‘3))
35		m1modge3gt1 10830	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℤ≥‘3) → 1 < ( -1 mod 𝑃))
36		breq2 4135	. . . . . . . . . . 11 ⊢ (( -1 mod 𝑃) = 1 → (1 < ( -1 mod 𝑃) ↔ 1 < 1))
37		1re 8332	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
38	37	ltnri 8426	. . . . . . . . . . . 12 ⊢ ¬ 1 < 1
39	38	pm2.21i 655	. . . . . . . . . . 11 ⊢ (1 < 1 → -1 = 1)
40	36, 39	biimtrdi 163	. . . . . . . . . 10 ⊢ (( -1 mod 𝑃) = 1 → (1 < ( -1 mod 𝑃) → -1 = 1))
41	35, 40	syl5com 29	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℤ≥‘3) → (( -1 mod 𝑃) = 1 → -1 = 1))
42	2, 34, 41	3syl 17	. . . . . . . 8 ⊢ (𝜑 → (( -1 mod 𝑃) = 1 → -1 = 1))
43	33, 42	sylbid 150	. . . . . . 7 ⊢ (𝜑 → (( -1 mod 𝑃) = (1 mod 𝑃) → -1 = 1))
44		oveq1 6095	. . . . . . . . 9 ⊢ (( -1↑𝑁) = 1 → (( -1↑𝑁) mod 𝑃) = (1 mod 𝑃))
45	44	eqeq2d 2250	. . . . . . . 8 ⊢ (( -1↑𝑁) = 1 → (( -1 mod 𝑃) = (( -1↑𝑁) mod 𝑃) ↔ ( -1 mod 𝑃) = (1 mod 𝑃)))
46		eqeq2 2248	. . . . . . . 8 ⊢ (( -1↑𝑁) = 1 → ( -1 = ( -1↑𝑁) ↔ -1 = 1))
47	45, 46	imbi12d 234	. . . . . . 7 ⊢ (( -1↑𝑁) = 1 → ((( -1 mod 𝑃) = (( -1↑𝑁) mod 𝑃) → -1 = ( -1↑𝑁)) ↔ (( -1 mod 𝑃) = (1 mod 𝑃) → -1 = 1)))
48	43, 47	imbitrrid 156	. . . . . 6 ⊢ (( -1↑𝑁) = 1 → (𝜑 → (( -1 mod 𝑃) = (( -1↑𝑁) mod 𝑃) → -1 = ( -1↑𝑁))))
49	24, 48	jaoi 728	. . . . 5 ⊢ ((( -1↑𝑁) = -1 ∨ ( -1↑𝑁) = 1) → (𝜑 → (( -1 mod 𝑃) = (( -1↑𝑁) mod 𝑃) → -1 = ( -1↑𝑁))))
50	21, 49	mpcom 36	. . . 4 ⊢ (𝜑 → (( -1 mod 𝑃) = (( -1↑𝑁) mod 𝑃) → -1 = ( -1↑𝑁)))
51		oveq1 6095	. . . . . 6 ⊢ ((2 /L 𝑃) = -1 → ((2 /L 𝑃) mod 𝑃) = ( -1 mod 𝑃))
52	51	eqeq1d 2247	. . . . 5 ⊢ ((2 /L 𝑃) = -1 → (((2 /L 𝑃) mod 𝑃) = (( -1↑𝑁) mod 𝑃) ↔ ( -1 mod 𝑃) = (( -1↑𝑁) mod 𝑃)))
53		eqeq1 2245	. . . . 5 ⊢ ((2 /L 𝑃) = -1 → ((2 /L 𝑃) = ( -1↑𝑁) ↔ -1 = ( -1↑𝑁)))
54	52, 53	imbi12d 234	. . . 4 ⊢ ((2 /L 𝑃) = -1 → ((((2 /L 𝑃) mod 𝑃) = (( -1↑𝑁) mod 𝑃) → (2 /L 𝑃) = ( -1↑𝑁)) ↔ (( -1 mod 𝑃) = (( -1↑𝑁) mod 𝑃) → -1 = ( -1↑𝑁))))
55	50, 54	imbitrrid 156	. . 3 ⊢ ((2 /L 𝑃) = -1 → (𝜑 → (((2 /L 𝑃) mod 𝑃) = (( -1↑𝑁) mod 𝑃) → (2 /L 𝑃) = ( -1↑𝑁))))
56	25	nngt0d 9358	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑃)
57		q0mod 10814	. . . . . . 7 ⊢ ((𝑃 ∈ ℚ ∧ 0 < 𝑃) → (0 mod 𝑃) = 0)
58	27, 56, 57	syl2anc 415	. . . . . 6 ⊢ (𝜑 → (0 mod 𝑃) = 0)
59	58	eqeq1d 2247	. . . . 5 ⊢ (𝜑 → ((0 mod 𝑃) = (( -1↑𝑁) mod 𝑃) ↔ 0 = (( -1↑𝑁) mod 𝑃)))
60		oveq1 6095	. . . . . . . . . . 11 ⊢ (( -1↑𝑁) = -1 → (( -1↑𝑁) mod 𝑃) = ( -1 mod 𝑃))
61	60	eqeq2d 2250	. . . . . . . . . 10 ⊢ (( -1↑𝑁) = -1 → (0 = (( -1↑𝑁) mod 𝑃) ↔ 0 = ( -1 mod 𝑃)))
62	61	adantr 276	. . . . . . . . 9 ⊢ ((( -1↑𝑁) = -1 ∧ 𝜑) → (0 = (( -1↑𝑁) mod 𝑃) ↔ 0 = ( -1 mod 𝑃)))
63		1z 9682	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
64		zq 10043	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
65	63, 64	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℚ
66		negqmod0 10790	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℚ ∧ 𝑃 ∈ ℚ ∧ 0 < 𝑃) → ((1 mod 𝑃) = 0 ↔ ( -1 mod 𝑃) = 0))
67	65, 27, 56, 66	mp3an2i 1383	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1 mod 𝑃) = 0 ↔ ( -1 mod 𝑃) = 0))
68		eqcom 2240	. . . . . . . . . . . 12 ⊢ (( -1 mod 𝑃) = 0 ↔ 0 = ( -1 mod 𝑃))
69	67, 68	bitrdi 196	. . . . . . . . . . 11 ⊢ (𝜑 → ((1 mod 𝑃) = 0 ↔ 0 = ( -1 mod 𝑃)))
70	32	eqeq1d 2247	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1 mod 𝑃) = 0 ↔ 1 = 0))
71		1ne0 9382	. . . . . . . . . . . . 13 ⊢ 1 ≠ 0
72		eqneqall 2430	. . . . . . . . . . . . 13 ⊢ (1 = 0 → (1 ≠ 0 → 0 = ( -1↑𝑁)))
73	71, 72	mpi 15	. . . . . . . . . . . 12 ⊢ (1 = 0 → 0 = ( -1↑𝑁))
74	70, 73	biimtrdi 163	. . . . . . . . . . 11 ⊢ (𝜑 → ((1 mod 𝑃) = 0 → 0 = ( -1↑𝑁)))
75	69, 74	sylbird 170	. . . . . . . . . 10 ⊢ (𝜑 → (0 = ( -1 mod 𝑃) → 0 = ( -1↑𝑁)))
76	75	adantl 277	. . . . . . . . 9 ⊢ ((( -1↑𝑁) = -1 ∧ 𝜑) → (0 = ( -1 mod 𝑃) → 0 = ( -1↑𝑁)))
77	62, 76	sylbid 150	. . . . . . . 8 ⊢ ((( -1↑𝑁) = -1 ∧ 𝜑) → (0 = (( -1↑𝑁) mod 𝑃) → 0 = ( -1↑𝑁)))
78	77	ex 115	. . . . . . 7 ⊢ (( -1↑𝑁) = -1 → (𝜑 → (0 = (( -1↑𝑁) mod 𝑃) → 0 = ( -1↑𝑁))))
79	44	eqeq2d 2250	. . . . . . . . . 10 ⊢ (( -1↑𝑁) = 1 → (0 = (( -1↑𝑁) mod 𝑃) ↔ 0 = (1 mod 𝑃)))
80	79	adantr 276	. . . . . . . . 9 ⊢ ((( -1↑𝑁) = 1 ∧ 𝜑) → (0 = (( -1↑𝑁) mod 𝑃) ↔ 0 = (1 mod 𝑃)))
81		eqcom 2240	. . . . . . . . . . . 12 ⊢ (0 = (1 mod 𝑃) ↔ (1 mod 𝑃) = 0)
82	81, 70	bitrid 192	. . . . . . . . . . 11 ⊢ (𝜑 → (0 = (1 mod 𝑃) ↔ 1 = 0))
83	82, 73	biimtrdi 163	. . . . . . . . . 10 ⊢ (𝜑 → (0 = (1 mod 𝑃) → 0 = ( -1↑𝑁)))
84	83	adantl 277	. . . . . . . . 9 ⊢ ((( -1↑𝑁) = 1 ∧ 𝜑) → (0 = (1 mod 𝑃) → 0 = ( -1↑𝑁)))
85	80, 84	sylbid 150	. . . . . . . 8 ⊢ ((( -1↑𝑁) = 1 ∧ 𝜑) → (0 = (( -1↑𝑁) mod 𝑃) → 0 = ( -1↑𝑁)))
86	85	ex 115	. . . . . . 7 ⊢ (( -1↑𝑁) = 1 → (𝜑 → (0 = (( -1↑𝑁) mod 𝑃) → 0 = ( -1↑𝑁))))
87	78, 86	jaoi 728	. . . . . 6 ⊢ ((( -1↑𝑁) = -1 ∨ ( -1↑𝑁) = 1) → (𝜑 → (0 = (( -1↑𝑁) mod 𝑃) → 0 = ( -1↑𝑁))))
88	21, 87	mpcom 36	. . . . 5 ⊢ (𝜑 → (0 = (( -1↑𝑁) mod 𝑃) → 0 = ( -1↑𝑁)))
89	59, 88	sylbid 150	. . . 4 ⊢ (𝜑 → ((0 mod 𝑃) = (( -1↑𝑁) mod 𝑃) → 0 = ( -1↑𝑁)))
90		oveq1 6095	. . . . . 6 ⊢ ((2 /L 𝑃) = 0 → ((2 /L 𝑃) mod 𝑃) = (0 mod 𝑃))
91	90	eqeq1d 2247	. . . . 5 ⊢ ((2 /L 𝑃) = 0 → (((2 /L 𝑃) mod 𝑃) = (( -1↑𝑁) mod 𝑃) ↔ (0 mod 𝑃) = (( -1↑𝑁) mod 𝑃)))
92		eqeq1 2245	. . . . 5 ⊢ ((2 /L 𝑃) = 0 → ((2 /L 𝑃) = ( -1↑𝑁) ↔ 0 = ( -1↑𝑁)))
93	91, 92	imbi12d 234	. . . 4 ⊢ ((2 /L 𝑃) = 0 → ((((2 /L 𝑃) mod 𝑃) = (( -1↑𝑁) mod 𝑃) → (2 /L 𝑃) = ( -1↑𝑁)) ↔ ((0 mod 𝑃) = (( -1↑𝑁) mod 𝑃) → 0 = ( -1↑𝑁))))
94	89, 93	imbitrrid 156	. . 3 ⊢ ((2 /L 𝑃) = 0 → (𝜑 → (((2 /L 𝑃) mod 𝑃) = (( -1↑𝑁) mod 𝑃) → (2 /L 𝑃) = ( -1↑𝑁))))
95	32	eqeq1d 2247	. . . . 5 ⊢ (𝜑 → ((1 mod 𝑃) = (( -1↑𝑁) mod 𝑃) ↔ 1 = (( -1↑𝑁) mod 𝑃)))
96		eqcom 2240	. . . . . . . . 9 ⊢ (1 = ( -1 mod 𝑃) ↔ ( -1 mod 𝑃) = 1)
97		eqcom 2240	. . . . . . . . 9 ⊢ (1 = -1 ↔ -1 = 1)
98	42, 96, 97	3imtr4g 205	. . . . . . . 8 ⊢ (𝜑 → (1 = ( -1 mod 𝑃) → 1 = -1))
99	60	eqeq2d 2250	. . . . . . . . 9 ⊢ (( -1↑𝑁) = -1 → (1 = (( -1↑𝑁) mod 𝑃) ↔ 1 = ( -1 mod 𝑃)))
100		eqeq2 2248	. . . . . . . . 9 ⊢ (( -1↑𝑁) = -1 → (1 = ( -1↑𝑁) ↔ 1 = -1))
101	99, 100	imbi12d 234	. . . . . . . 8 ⊢ (( -1↑𝑁) = -1 → ((1 = (( -1↑𝑁) mod 𝑃) → 1 = ( -1↑𝑁)) ↔ (1 = ( -1 mod 𝑃) → 1 = -1)))
102	98, 101	imbitrrid 156	. . . . . . 7 ⊢ (( -1↑𝑁) = -1 → (𝜑 → (1 = (( -1↑𝑁) mod 𝑃) → 1 = ( -1↑𝑁))))
103		eqcom 2240	. . . . . . . . 9 ⊢ (( -1↑𝑁) = 1 ↔ 1 = ( -1↑𝑁))
104	103	biimpi 120	. . . . . . . 8 ⊢ (( -1↑𝑁) = 1 → 1 = ( -1↑𝑁))
105	104	2a1d 23	. . . . . . 7 ⊢ (( -1↑𝑁) = 1 → (𝜑 → (1 = (( -1↑𝑁) mod 𝑃) → 1 = ( -1↑𝑁))))
106	102, 105	jaoi 728	. . . . . 6 ⊢ ((( -1↑𝑁) = -1 ∨ ( -1↑𝑁) = 1) → (𝜑 → (1 = (( -1↑𝑁) mod 𝑃) → 1 = ( -1↑𝑁))))
107	21, 106	mpcom 36	. . . . 5 ⊢ (𝜑 → (1 = (( -1↑𝑁) mod 𝑃) → 1 = ( -1↑𝑁)))
108	95, 107	sylbid 150	. . . 4 ⊢ (𝜑 → ((1 mod 𝑃) = (( -1↑𝑁) mod 𝑃) → 1 = ( -1↑𝑁)))
109		oveq1 6095	. . . . . 6 ⊢ ((2 /L 𝑃) = 1 → ((2 /L 𝑃) mod 𝑃) = (1 mod 𝑃))
110	109	eqeq1d 2247	. . . . 5 ⊢ ((2 /L 𝑃) = 1 → (((2 /L 𝑃) mod 𝑃) = (( -1↑𝑁) mod 𝑃) ↔ (1 mod 𝑃) = (( -1↑𝑁) mod 𝑃)))
111		eqeq1 2245	. . . . 5 ⊢ ((2 /L 𝑃) = 1 → ((2 /L 𝑃) = ( -1↑𝑁) ↔ 1 = ( -1↑𝑁)))
112	110, 111	imbi12d 234	. . . 4 ⊢ ((2 /L 𝑃) = 1 → ((((2 /L 𝑃) mod 𝑃) = (( -1↑𝑁) mod 𝑃) → (2 /L 𝑃) = ( -1↑𝑁)) ↔ ((1 mod 𝑃) = (( -1↑𝑁) mod 𝑃) → 1 = ( -1↑𝑁))))
113	108, 112	imbitrrid 156	. . 3 ⊢ ((2 /L 𝑃) = 1 → (𝜑 → (((2 /L 𝑃) mod 𝑃) = (( -1↑𝑁) mod 𝑃) → (2 /L 𝑃) = ( -1↑𝑁))))
114	55, 94, 113	3jaoi 1344	. 2 ⊢ (((2 /L 𝑃) = -1 ∨ (2 /L 𝑃) = 0 ∨ (2 /L 𝑃) = 1) → (𝜑 → (((2 /L 𝑃) mod 𝑃) = (( -1↑𝑁) mod 𝑃) → (2 /L 𝑃) = ( -1↑𝑁))))
115	11, 114	mpcom 36	1 ⊢ (𝜑 → (((2 /L 𝑃) mod 𝑃) = (( -1↑𝑁) mod 𝑃) → (2 /L 𝑃) = ( -1↑𝑁)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   ∨ w3o 1008   = wceq 1402   ∈ wcel 2209   ≠ wne 2420   ∖ cdif 3217  {csn 3709  {cpr 3710  {ctp 3711   class class class wbr 4131  ‘cfv 5380  (class class class)co 6088  0cc0 8186  1c1 8187   < clt 8367   − cmin 8505   -cneg 8506   / cdiv 9011  ℕcn 9314  2c2 9365  3c3 9366  4c4 9367  ℤcz 9656  ℤ_≥cuz 9938  ℚcq 10036  ⌊cfl 10721   mod cmo 10781  ↑cexp 10997  ℙcprime 12911   /_L clgs 16333

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-en 7028  df-dom 7029  df-fin 7030  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-5 9376  df-6 9377  df-7 9378  df-8 9379  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-fz 10430  df-fzo 10568  df-fl 10723  df-mod 10782  df-seqfrec 10907  df-exp 10998  df-ihash 11238  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-clim 12071  df-proddc 12344  df-dvds 12581  df-gcd 12757  df-prm 12912  df-phi 13019  df-pc 13094  df-lgs 16334

This theorem is used by:  gausslemma2d  16405