Theorem vitalilem5 24207

Description: Lemma for vitali 24208. (Contributed by Mario Carneiro, 16-Jun-2014.)

Hypotheses

Ref	Expression
vitali.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}
vitali.2	⊢ 𝑆 = ((0[,]1) / ∼ )
vitali.3	⊢ (𝜑 → 𝐹 Fn 𝑆)
vitali.4	⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
vitali.5	⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
vitali.6	⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})
vitali.7	⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))

Assertion

Ref	Expression
vitalilem5	⊢ ¬ 𝜑

Distinct variable groups:   𝑛,𝑠,𝑥,𝑦,𝑧,𝐺   𝜑,𝑛,𝑥,𝑧   𝑧,𝑆   𝑥,𝑇   𝑛,𝐹,𝑠,𝑥,𝑦,𝑧   ∼ ,𝑛,𝑠,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑦,𝑠)   𝑆(𝑥,𝑦,𝑛,𝑠)   𝑇(𝑦,𝑧,𝑛,𝑠)

Proof of Theorem vitalilem5

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0lt1 11156	. . . 4 ⊢ 0 < 1
2		0re 10637	. . . . . 6 ⊢ 0 ∈ ℝ
3		1re 10635	. . . . . 6 ⊢ 1 ∈ ℝ
4		0le1 11157	. . . . . 6 ⊢ 0 ≤ 1
5		ovolicc 24118	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≤ 1) → (vol*‘(0[,]1)) = (1 − 0))
6	2, 3, 4, 5	mp3an 1457	. . . . 5 ⊢ (vol*‘(0[,]1)) = (1 − 0)
7		1m0e1 11752	. . . . 5 ⊢ (1 − 0) = 1
8	6, 7	eqtri 2844	. . . 4 ⊢ (vol*‘(0[,]1)) = 1
9	1, 8	breqtrri 5085	. . 3 ⊢ 0 < (vol*‘(0[,]1))
10	8, 3	eqeltri 2909	. . . 4 ⊢ (vol*‘(0[,]1)) ∈ ℝ
11	2, 10	ltnlei 10755	. . 3 ⊢ (0 < (vol*‘(0[,]1)) ↔ ¬ (vol*‘(0[,]1)) ≤ 0)
12	9, 11	mpbi 232	. 2 ⊢ ¬ (vol*‘(0[,]1)) ≤ 0
13		vitali.1	. . . . . 6 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}
14		vitali.2	. . . . . 6 ⊢ 𝑆 = ((0[,]1) / ∼ )
15		vitali.3	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑆)
16		vitali.4	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
17		vitali.5	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
18		vitali.6	. . . . . 6 ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})
19		vitali.7	. . . . . 6 ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
20	13, 14, 15, 16, 17, 18, 19	vitalilem2 24204	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)))
21	20	simp2d 1139	. . . 4 ⊢ (𝜑 → (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚))
22	13	vitalilem1 24203	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∼ Er (0[,]1)
23		erdm 8293	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( ∼ Er (0[,]1) → dom ∼ = (0[,]1))
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom ∼ = (0[,]1)
25		simpr 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆)
26	25, 14	eleqtrdi 2923	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ ((0[,]1) / ∼ ))
27		elqsn0 8360	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((dom ∼ = (0[,]1) ∧ 𝑧 ∈ ((0[,]1) / ∼ )) → 𝑧 ≠ ∅)
28	24, 26, 27	sylancr 589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ≠ ∅)
29	22	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ∼ Er (0[,]1))
30	29	qsss 8352	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((0[,]1) / ∼ ) ⊆ 𝒫 (0[,]1))
31	14, 30	eqsstrid 4014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 (0[,]1))
32	31	sselda 3966	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝒫 (0[,]1))
33	32	elpwid 4552	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ⊆ (0[,]1))
34	33	sseld 3965	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑧) ∈ 𝑧 → (𝐹‘𝑧) ∈ (0[,]1)))
35	28, 34	embantd 59	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) → (𝐹‘𝑧) ∈ (0[,]1)))
36	35	ralimdva 3177	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ (0[,]1)))
37	16, 36	mpd 15	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ (0[,]1))
38		ffnfv 6876	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑆⟶(0[,]1) ↔ (𝐹 Fn 𝑆 ∧ ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ (0[,]1)))
39	15, 37, 38	sylanbrc 585	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝑆⟶(0[,]1))
40	39	frnd 6515	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1))
41		unitssre 12879	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) ⊆ ℝ
42	40, 41	sstrdi 3978	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
43		reex 10622	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
44	43	elpw2 5240	. . . . . . . . . . . . . . 15 ⊢ (ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
45	42, 44	sylibr 236	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐹 ∈ 𝒫 ℝ)
46	45	anim1i 616	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
47		eldif 3945	. . . . . . . . . . . . 13 ⊢ (ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol) ↔ (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
48	46, 47	sylibr 236	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
49	48	ex 415	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ ran 𝐹 ∈ dom vol → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol)))
50	19, 49	mt3d 150	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝐹 ∈ dom vol)
51		f1of 6609	. . . . . . . . . . . . 13 ⊢ (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
52	17, 51	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
53		inss1 4204	. . . . . . . . . . . . 13 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
54		qssre 12352	. . . . . . . . . . . . 13 ⊢ ℚ ⊆ ℝ
55	53, 54	sstri 3975	. . . . . . . . . . . 12 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℝ
56		fss 6521	. . . . . . . . . . . 12 ⊢ ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℝ) → 𝐺:ℕ⟶ℝ)
57	52, 55, 56	sylancl 588	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
58	57	ffvelrnda 6845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
59		shftmbl 24133	. . . . . . . . . 10 ⊢ ((ran 𝐹 ∈ dom vol ∧ (𝐺‘𝑛) ∈ ℝ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol)
60	50, 58, 59	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol)
61	60, 18	fmptd 6872	. . . . . . . 8 ⊢ (𝜑 → 𝑇:ℕ⟶dom vol)
62	61	ffvelrnda 6845	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) ∈ dom vol)
63	62	ralrimiva 3182	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
64		iunmbl 24148	. . . . . 6 ⊢ (∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
65	63, 64	syl 17	. . . . 5 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
66		mblss 24126	. . . . 5 ⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ)
67	65, 66	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ)
68		ovolss 24080	. . . 4 ⊢ (((0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ) → (vol*‘(0[,]1)) ≤ (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
69	21, 67, 68	syl2anc 586	. . 3 ⊢ (𝜑 → (vol*‘(0[,]1)) ≤ (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
70		eqid 2821	. . . . . 6 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))))
71		eqid 2821	. . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))
72		mblss 24126	. . . . . . 7 ⊢ ((𝑇‘𝑚) ∈ dom vol → (𝑇‘𝑚) ⊆ ℝ)
73	62, 72	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) ⊆ ℝ)
74	13, 14, 15, 16, 17, 18, 19	vitalilem4 24206	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = 0)
75	74, 2	eqeltrdi 2921	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) ∈ ℝ)
76	74	mpteq2dva 5153	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ 0))
77		fconstmpt 5608	. . . . . . . . . . 11 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
78		nnuz 12275	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ≥‘1)
79	78	xpeq1i 5575	. . . . . . . . . . 11 ⊢ (ℕ × {0}) = ((ℤ≥‘1) × {0})
80	77, 79	eqtr3i 2846	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ 0) = ((ℤ≥‘1) × {0})
81	76, 80	syl6eq 2872	. . . . . . . . 9 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))) = ((ℤ≥‘1) × {0}))
82	81	seqeq3d 13371	. . . . . . . 8 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) = seq1( + , ((ℤ≥‘1) × {0})))
83		1z 12006	. . . . . . . . 9 ⊢ 1 ∈ ℤ
84		serclim0 14928	. . . . . . . . 9 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0)
85	83, 84	ax-mp 5	. . . . . . . 8 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0
86	82, 85	eqbrtrdi 5097	. . . . . . 7 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) ⇝ 0)
87		seqex 13365	. . . . . . . 8 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) ∈ V
88		c0ex 10629	. . . . . . . 8 ⊢ 0 ∈ V
89	87, 88	breldm 5771	. . . . . . 7 ⊢ (seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) ⇝ 0 → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) ∈ dom ⇝ )
90	86, 89	syl 17	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) ∈ dom ⇝ )
91	70, 71, 73, 75, 90	ovoliun2 24101	. . . . 5 ⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ Σ𝑚 ∈ ℕ (vol*‘(𝑇‘𝑚)))
92	74	sumeq2dv 15054	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘(𝑇‘𝑚)) = Σ𝑚 ∈ ℕ 0)
93	78	eqimssi 4024	. . . . . . . 8 ⊢ ℕ ⊆ (ℤ≥‘1)
94	93	orci 861	. . . . . . 7 ⊢ (ℕ ⊆ (ℤ≥‘1) ∨ ℕ ∈ Fin)
95		sumz 15073	. . . . . . 7 ⊢ ((ℕ ⊆ (ℤ≥‘1) ∨ ℕ ∈ Fin) → Σ𝑚 ∈ ℕ 0 = 0)
96	94, 95	ax-mp 5	. . . . . 6 ⊢ Σ𝑚 ∈ ℕ 0 = 0
97	92, 96	syl6eq 2872	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘(𝑇‘𝑚)) = 0)
98	91, 97	breqtrd 5084	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 0)
99		ovolge0 24076	. . . . 5 ⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ → 0 ≤ (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
100	67, 99	syl 17	. . . 4 ⊢ (𝜑 → 0 ≤ (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
101		ovolcl 24073	. . . . . 6 ⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈ ℝ*)
102	67, 101	syl 17	. . . . 5 ⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈ ℝ*)
103		0xr 10682	. . . . 5 ⊢ 0 ∈ ℝ*
104		xrletri3 12541	. . . . 5 ⊢ (((vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = 0 ↔ ((vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 0 ∧ 0 ≤ (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))))
105	102, 103, 104	sylancl 588	. . . 4 ⊢ (𝜑 → ((vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = 0 ↔ ((vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 0 ∧ 0 ≤ (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))))
106	98, 100, 105	mpbir2and 711	. . 3 ⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = 0)
107	69, 106	breqtrd 5084	. 2 ⊢ (𝜑 → (vol*‘(0[,]1)) ≤ 0)
108	12, 107	mto 199	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  {crab 3142   ∖ cdif 3932   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  {csn 4560  ∪ ciun 4911   class class class wbr 5058  {copab 5120   ↦ cmpt 5138   × cxp 5547  dom cdm 5549  ran crn 5550   Fn wfn 6344  ⟶wf 6345  –1-1-onto→wf1o 6348  ‘cfv 6349  (class class class)co 7150   Er wer 8280   / cqs 8282  Fincfn 8503  ℝcr 10530  0cc0 10531  1c1 10532   + caddc 10534  ℝ^*cxr 10668   < clt 10669   ≤ cle 10670   − cmin 10864  -cneg 10865  ℕcn 11632  2c2 11686  ℤcz 11975  ℤ_≥cuz 12237  ℚcq 12342  [,]cicc 12735  seqcseq 13363   ⇝ cli 14835  Σcsu 15036  vol*covol 24057  volcvol 24058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098  ax-cc 9851  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-disj 5024  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-ec 8285  df-qs 8289  df-map 8402  df-pm 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-sup 8900  df-inf 8901  df-oi 8968  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-q 12343  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-ioo 12736  df-ico 12738  df-icc 12739  df-fz 12887  df-fzo 13028  df-fl 13156  df-seq 13364  df-exp 13424  df-hash 13685  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839  df-rlim 14840  df-sum 15037  df-rest 16690  df-topgen 16711  df-psmet 20531  df-xmet 20532  df-met 20533  df-bl 20534  df-mopn 20535  df-top 21496  df-topon 21513  df-bases 21548  df-cmp 21989  df-ovol 24059  df-vol 24060

This theorem is referenced by:  vitali  24208