Theorem vitalilem5 25511

Description: Lemma for vitali 25512. (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 11642	. . . 4 ⊢ 0 < 1
2		0re 11117	. . . . . 6 ⊢ 0 ∈ ℝ
3		1re 11115	. . . . . 6 ⊢ 1 ∈ ℝ
4		0le1 11643	. . . . . 6 ⊢ 0 ≤ 1
5		ovolicc 25422	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≤ 1) → (vol*‘(0[,]1)) = (1 − 0))
6	2, 3, 4, 5	mp3an 1463	. . . . 5 ⊢ (vol*‘(0[,]1)) = (1 − 0)
7		1m0e1 12244	. . . . 5 ⊢ (1 − 0) = 1
8	6, 7	eqtri 2752	. . . 4 ⊢ (vol*‘(0[,]1)) = 1
9	1, 8	breqtrri 5119	. . 3 ⊢ 0 < (vol*‘(0[,]1))
10	8, 3	eqeltri 2824	. . . 4 ⊢ (vol*‘(0[,]1)) ∈ ℝ
11	2, 10	ltnlei 11237	. . 3 ⊢ (0 < (vol*‘(0[,]1)) ↔ ¬ (vol*‘(0[,]1)) ≤ 0)
12	9, 11	mpbi 230	. 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 25508	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)))
21	20	simp2d 1143	. . . 4 ⊢ (𝜑 → (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚))
22	13	vitalilem1 25507	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∼ Er (0[,]1)
23		erdm 8635	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( ∼ Er (0[,]1) → dom ∼ = (0[,]1))
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom ∼ = (0[,]1)
25		simpr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆)
26	25, 14	eleqtrdi 2838	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ ((0[,]1) / ∼ ))
27		elqsn0 8711	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((dom ∼ = (0[,]1) ∧ 𝑧 ∈ ((0[,]1) / ∼ )) → 𝑧 ≠ ∅)
28	24, 26, 27	sylancr 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ≠ ∅)
29	22	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ∼ Er (0[,]1))
30	29	qsss 8703	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((0[,]1) / ∼ ) ⊆ 𝒫 (0[,]1))
31	14, 30	eqsstrid 3974	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 (0[,]1))
32	31	sselda 3935	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝒫 (0[,]1))
33	32	elpwid 4560	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ⊆ (0[,]1))
34	33	sseld 3934	. . . . . . . . . . . . . . . . . . . . 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 3141	. . . . . . . . . . . . . . . . . . 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 7053	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑆⟶(0[,]1) ↔ (𝐹 Fn 𝑆 ∧ ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ (0[,]1)))
39	15, 37, 38	sylanbrc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝑆⟶(0[,]1))
40	39	frnd 6660	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1))
41		unitssre 13402	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) ⊆ ℝ
42	40, 41	sstrdi 3948	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
43		reex 11100	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
44	43	elpw2 5273	. . . . . . . . . . . . . . 15 ⊢ (ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
45	42, 44	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐹 ∈ 𝒫 ℝ)
46	45	anim1i 615	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
47		eldif 3913	. . . . . . . . . . . . 13 ⊢ (ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol) ↔ (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
48	46, 47	sylibr 234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
49	48	ex 412	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ ran 𝐹 ∈ dom vol → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol)))
50	19, 49	mt3d 148	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝐹 ∈ dom vol)
51		f1of 6764	. . . . . . . . . . . . 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 4188	. . . . . . . . . . . . 13 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
54		qssre 12860	. . . . . . . . . . . . 13 ⊢ ℚ ⊆ ℝ
55	53, 54	sstri 3945	. . . . . . . . . . . 12 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℝ
56		fss 6668	. . . . . . . . . . . 12 ⊢ ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℝ) → 𝐺:ℕ⟶ℝ)
57	52, 55, 56	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
58	57	ffvelcdmda 7018	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
59		shftmbl 25437	. . . . . . . . . 10 ⊢ ((ran 𝐹 ∈ dom vol ∧ (𝐺‘𝑛) ∈ ℝ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol)
60	50, 58, 59	syl2an2r 685	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol)
61	60, 18	fmptd 7048	. . . . . . . 8 ⊢ (𝜑 → 𝑇:ℕ⟶dom vol)
62	61	ffvelcdmda 7018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) ∈ dom vol)
63	62	ralrimiva 3121	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
64		iunmbl 25452	. . . . . 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 25430	. . . . 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 25384	. . . 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 584	. . 3 ⊢ (𝜑 → (vol*‘(0[,]1)) ≤ (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
70		eqid 2729	. . . . . 6 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))))
71		eqid 2729	. . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))
72		mblss 25430	. . . . . . 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 25510	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = 0)
75	74, 2	eqeltrdi 2836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) ∈ ℝ)
76	74	mpteq2dva 5185	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ 0))
77		fconstmpt 5681	. . . . . . . . . . 11 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
78		nnuz 12778	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ≥‘1)
79	78	xpeq1i 5645	. . . . . . . . . . 11 ⊢ (ℕ × {0}) = ((ℤ≥‘1) × {0})
80	77, 79	eqtr3i 2754	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ 0) = ((ℤ≥‘1) × {0})
81	76, 80	eqtrdi 2780	. . . . . . . . 9 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))) = ((ℤ≥‘1) × {0}))
82	81	seqeq3d 13916	. . . . . . . 8 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) = seq1( + , ((ℤ≥‘1) × {0})))
83		1z 12505	. . . . . . . . 9 ⊢ 1 ∈ ℤ
84		serclim0 15484	. . . . . . . . 9 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0)
85	83, 84	ax-mp 5	. . . . . . . 8 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0
86	82, 85	eqbrtrdi 5131	. . . . . . 7 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) ⇝ 0)
87		seqex 13910	. . . . . . . 8 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) ∈ V
88		c0ex 11109	. . . . . . . 8 ⊢ 0 ∈ V
89	87, 88	breldm 5851	. . . . . . 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 25405	. . . . 5 ⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ Σ𝑚 ∈ ℕ (vol*‘(𝑇‘𝑚)))
92	74	sumeq2dv 15609	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘(𝑇‘𝑚)) = Σ𝑚 ∈ ℕ 0)
93	78	eqimssi 3996	. . . . . . . 8 ⊢ ℕ ⊆ (ℤ≥‘1)
94	93	orci 865	. . . . . . 7 ⊢ (ℕ ⊆ (ℤ≥‘1) ∨ ℕ ∈ Fin)
95		sumz 15629	. . . . . . 7 ⊢ ((ℕ ⊆ (ℤ≥‘1) ∨ ℕ ∈ Fin) → Σ𝑚 ∈ ℕ 0 = 0)
96	94, 95	ax-mp 5	. . . . . 6 ⊢ Σ𝑚 ∈ ℕ 0 = 0
97	92, 96	eqtrdi 2780	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘(𝑇‘𝑚)) = 0)
98	91, 97	breqtrd 5118	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 0)
99		ovolge0 25380	. . . . 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 25377	. . . . . 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 11162	. . . . 5 ⊢ 0 ∈ ℝ*
104		xrletri3 13056	. . . . 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 586	. . . 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 713	. . 3 ⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = 0)
107	69, 106	breqtrd 5118	. 2 ⊢ (𝜑 → (vol*‘(0[,]1)) ≤ 0)
108	12, 107	mto 197	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3394   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  {csn 4577  ∪ ciun 4941   class class class wbr 5092  {copab 5154   ↦ cmpt 5173   × cxp 5617  dom cdm 5619  ran crn 5620   Fn wfn 6477  ⟶wf 6478  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349   Er wer 8622   / cqs 8624  Fincfn 8872  ℝcr 11008  0cc0 11009  1c1 11010   + caddc 11012  ℝ^*cxr 11148   < clt 11149   ≤ cle 11150   − cmin 11347  -cneg 11348  ℕcn 12128  2c2 12183  ℤcz 12471  ℤ_≥cuz 12735  ℚcq 12849  [,]cicc 13251  seqcseq 13908   ⇝ cli 15391  Σcsu 15593  vol*covol 25361  volcvol 25362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cc 10329  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-disj 5060  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-ec 8627  df-qs 8631  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fi 9301  df-sup 9332  df-inf 9333  df-oi 9402  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-q 12850  df-rp 12894  df-xneg 13014  df-xadd 13015  df-xmul 13016  df-ioo 13252  df-ico 13254  df-icc 13255  df-fz 13411  df-fzo 13558  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-rest 17326  df-topgen 17347  df-psmet 21253  df-xmet 21254  df-met 21255  df-bl 21256  df-mopn 21257  df-top 22779  df-topon 22796  df-bases 22831  df-cmp 23272  df-ovol 25363  df-vol 25364

This theorem is referenced by:  vitali  25512