Theorem vitalilem5 25666

Description: Lemma for vitali 25667. (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 11812	. . . 4 ⊢ 0 < 1
2		0re 11292	. . . . . 6 ⊢ 0 ∈ ℝ
3		1re 11290	. . . . . 6 ⊢ 1 ∈ ℝ
4		0le1 11813	. . . . . 6 ⊢ 0 ≤ 1
5		ovolicc 25577	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≤ 1) → (vol*‘(0[,]1)) = (1 − 0))
6	2, 3, 4, 5	mp3an 1461	. . . . 5 ⊢ (vol*‘(0[,]1)) = (1 − 0)
7		1m0e1 12414	. . . . 5 ⊢ (1 − 0) = 1
8	6, 7	eqtri 2768	. . . 4 ⊢ (vol*‘(0[,]1)) = 1
9	1, 8	breqtrri 5193	. . 3 ⊢ 0 < (vol*‘(0[,]1))
10	8, 3	eqeltri 2840	. . . 4 ⊢ (vol*‘(0[,]1)) ∈ ℝ
11	2, 10	ltnlei 11411	. . 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 25663	. . . . 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 25662	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∼ Er (0[,]1)
23		erdm 8773	. . . . . . . . . . . . . . . . . . . . . . 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 2854	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ ((0[,]1) / ∼ ))
27		elqsn0 8844	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((dom ∼ = (0[,]1) ∧ 𝑧 ∈ ((0[,]1) / ∼ )) → 𝑧 ≠ ∅)
28	24, 26, 27	sylancr 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ≠ ∅)
29	22	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ∼ Er (0[,]1))
30	29	qsss 8836	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((0[,]1) / ∼ ) ⊆ 𝒫 (0[,]1))
31	14, 30	eqsstrid 4057	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 (0[,]1))
32	31	sselda 4008	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝒫 (0[,]1))
33	32	elpwid 4631	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ⊆ (0[,]1))
34	33	sseld 4007	. . . . . . . . . . . . . . . . . . . . 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 3173	. . . . . . . . . . . . . . . . . . 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 7153	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑆⟶(0[,]1) ↔ (𝐹 Fn 𝑆 ∧ ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ (0[,]1)))
39	15, 37, 38	sylanbrc 582	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝑆⟶(0[,]1))
40	39	frnd 6755	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1))
41		unitssre 13559	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) ⊆ ℝ
42	40, 41	sstrdi 4021	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
43		reex 11275	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
44	43	elpw2 5352	. . . . . . . . . . . . . . 15 ⊢ (ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
45	42, 44	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐹 ∈ 𝒫 ℝ)
46	45	anim1i 614	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
47		eldif 3986	. . . . . . . . . . . . 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 6862	. . . . . . . . . . . . 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 4258	. . . . . . . . . . . . 13 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
54		qssre 13024	. . . . . . . . . . . . 13 ⊢ ℚ ⊆ ℝ
55	53, 54	sstri 4018	. . . . . . . . . . . 12 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℝ
56		fss 6763	. . . . . . . . . . . 12 ⊢ ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℝ) → 𝐺:ℕ⟶ℝ)
57	52, 55, 56	sylancl 585	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
58	57	ffvelcdmda 7118	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
59		shftmbl 25592	. . . . . . . . . 10 ⊢ ((ran 𝐹 ∈ dom vol ∧ (𝐺‘𝑛) ∈ ℝ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol)
60	50, 58, 59	syl2an2r 684	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol)
61	60, 18	fmptd 7148	. . . . . . . 8 ⊢ (𝜑 → 𝑇:ℕ⟶dom vol)
62	61	ffvelcdmda 7118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) ∈ dom vol)
63	62	ralrimiva 3152	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
64		iunmbl 25607	. . . . . 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 25585	. . . . 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 25539	. . . 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 583	. . 3 ⊢ (𝜑 → (vol*‘(0[,]1)) ≤ (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
70		eqid 2740	. . . . . 6 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))))
71		eqid 2740	. . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))
72		mblss 25585	. . . . . . 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 25665	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = 0)
75	74, 2	eqeltrdi 2852	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) ∈ ℝ)
76	74	mpteq2dva 5266	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ 0))
77		fconstmpt 5762	. . . . . . . . . . 11 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
78		nnuz 12946	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ≥‘1)
79	78	xpeq1i 5726	. . . . . . . . . . 11 ⊢ (ℕ × {0}) = ((ℤ≥‘1) × {0})
80	77, 79	eqtr3i 2770	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ 0) = ((ℤ≥‘1) × {0})
81	76, 80	eqtrdi 2796	. . . . . . . . 9 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))) = ((ℤ≥‘1) × {0}))
82	81	seqeq3d 14060	. . . . . . . 8 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) = seq1( + , ((ℤ≥‘1) × {0})))
83		1z 12673	. . . . . . . . 9 ⊢ 1 ∈ ℤ
84		serclim0 15623	. . . . . . . . 9 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0)
85	83, 84	ax-mp 5	. . . . . . . 8 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0
86	82, 85	eqbrtrdi 5205	. . . . . . 7 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) ⇝ 0)
87		seqex 14054	. . . . . . . 8 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) ∈ V
88		c0ex 11284	. . . . . . . 8 ⊢ 0 ∈ V
89	87, 88	breldm 5933	. . . . . . 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 25560	. . . . 5 ⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ Σ𝑚 ∈ ℕ (vol*‘(𝑇‘𝑚)))
92	74	sumeq2dv 15750	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘(𝑇‘𝑚)) = Σ𝑚 ∈ ℕ 0)
93	78	eqimssi 4069	. . . . . . . 8 ⊢ ℕ ⊆ (ℤ≥‘1)
94	93	orci 864	. . . . . . 7 ⊢ (ℕ ⊆ (ℤ≥‘1) ∨ ℕ ∈ Fin)
95		sumz 15770	. . . . . . 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 2796	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘(𝑇‘𝑚)) = 0)
98	91, 97	breqtrd 5192	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 0)
99		ovolge0 25535	. . . . 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 25532	. . . . . 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 11337	. . . . 5 ⊢ 0 ∈ ℝ*
104		xrletri3 13216	. . . . 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 585	. . . 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 712	. . 3 ⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = 0)
107	69, 106	breqtrd 5192	. 2 ⊢ (𝜑 → (vol*‘(0[,]1)) ≤ 0)
108	12, 107	mto 197	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  {crab 3443   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ∪ ciun 5015   class class class wbr 5166  {copab 5228   ↦ cmpt 5249   × cxp 5698  dom cdm 5700  ran crn 5701   Fn wfn 6568  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448   Er wer 8760   / cqs 8762  Fincfn 9003  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325   − cmin 11520  -cneg 11521  ℕcn 12293  2c2 12348  ℤcz 12639  ℤ_≥cuz 12903  ℚcq 13013  [,]cicc 13410  seqcseq 14052   ⇝ cli 15530  Σcsu 15734  vol*covol 25516  volcvol 25517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cc 10504  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-ec 8765  df-qs 8769  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fi 9480  df-sup 9511  df-inf 9512  df-oi 9579  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-rlim 15535  df-sum 15735  df-rest 17482  df-topgen 17503  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-top 22921  df-topon 22938  df-bases 22974  df-cmp 23416  df-ovol 25518  df-vol 25519

This theorem is referenced by:  vitali  25667