Theorem vitalilem5 25569

Description: Lemma for vitali 25570. (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 11659	. . . 4 ⊢ 0 < 1
2		0re 11134	. . . . . 6 ⊢ 0 ∈ ℝ
3		1re 11132	. . . . . 6 ⊢ 1 ∈ ℝ
4		0le1 11660	. . . . . 6 ⊢ 0 ≤ 1
5		ovolicc 25480	. . . . . 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 12261	. . . . 5 ⊢ (1 − 0) = 1
8	6, 7	eqtri 2759	. . . 4 ⊢ (vol*‘(0[,]1)) = 1
9	1, 8	breqtrri 5125	. . 3 ⊢ 0 < (vol*‘(0[,]1))
10	8, 3	eqeltri 2832	. . . 4 ⊢ (vol*‘(0[,]1)) ∈ ℝ
11	2, 10	ltnlei 11254	. . 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 25566	. . . . 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 25565	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∼ Er (0[,]1)
23		erdm 8645	. . . . . . . . . . . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ ((0[,]1) / ∼ ))
27		elqsn0 8721	. . . . . . . . . . . . . . . . . . . . . 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 8713	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((0[,]1) / ∼ ) ⊆ 𝒫 (0[,]1))
31	14, 30	eqsstrid 3972	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 (0[,]1))
32	31	sselda 3933	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝒫 (0[,]1))
33	32	elpwid 4563	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ⊆ (0[,]1))
34	33	sseld 3932	. . . . . . . . . . . . . . . . . . . . 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 3148	. . . . . . . . . . . . . . . . . . 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 7064	. . . . . . . . . . . . . . . . . 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 6670	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1))
41		unitssre 13415	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) ⊆ ℝ
42	40, 41	sstrdi 3946	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
43		reex 11117	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
44	43	elpw2 5279	. . . . . . . . . . . . . . 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 3911	. . . . . . . . . . . . 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 6774	. . . . . . . . . . . . 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 4189	. . . . . . . . . . . . 13 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
54		qssre 12872	. . . . . . . . . . . . 13 ⊢ ℚ ⊆ ℝ
55	53, 54	sstri 3943	. . . . . . . . . . . 12 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℝ
56		fss 6678	. . . . . . . . . . . 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 7029	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
59		shftmbl 25495	. . . . . . . . . 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 7059	. . . . . . . 8 ⊢ (𝜑 → 𝑇:ℕ⟶dom vol)
62	61	ffvelcdmda 7029	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) ∈ dom vol)
63	62	ralrimiva 3128	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
64		iunmbl 25510	. . . . . 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 25488	. . . . 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 25442	. . . 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 2736	. . . . . 6 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))))
71		eqid 2736	. . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))
72		mblss 25488	. . . . . . 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 25568	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = 0)
75	74, 2	eqeltrdi 2844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) ∈ ℝ)
76	74	mpteq2dva 5191	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ 0))
77		fconstmpt 5686	. . . . . . . . . . 11 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
78		nnuz 12790	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ≥‘1)
79	78	xpeq1i 5650	. . . . . . . . . . 11 ⊢ (ℕ × {0}) = ((ℤ≥‘1) × {0})
80	77, 79	eqtr3i 2761	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ 0) = ((ℤ≥‘1) × {0})
81	76, 80	eqtrdi 2787	. . . . . . . . 9 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚))) = ((ℤ≥‘1) × {0}))
82	81	seqeq3d 13932	. . . . . . . 8 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) = seq1( + , ((ℤ≥‘1) × {0})))
83		1z 12521	. . . . . . . . 9 ⊢ 1 ∈ ℤ
84		serclim0 15500	. . . . . . . . 9 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0)
85	83, 84	ax-mp 5	. . . . . . . 8 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0
86	82, 85	eqbrtrdi 5137	. . . . . . 7 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) ⇝ 0)
87		seqex 13926	. . . . . . . 8 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑇‘𝑚)))) ∈ V
88		c0ex 11126	. . . . . . . 8 ⊢ 0 ∈ V
89	87, 88	breldm 5857	. . . . . . 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 25463	. . . . 5 ⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ Σ𝑚 ∈ ℕ (vol*‘(𝑇‘𝑚)))
92	74	sumeq2dv 15625	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘(𝑇‘𝑚)) = Σ𝑚 ∈ ℕ 0)
93	78	eqimssi 3994	. . . . . . . 8 ⊢ ℕ ⊆ (ℤ≥‘1)
94	93	orci 865	. . . . . . 7 ⊢ (ℕ ⊆ (ℤ≥‘1) ∨ ℕ ∈ Fin)
95		sumz 15645	. . . . . . 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 2787	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘(𝑇‘𝑚)) = 0)
98	91, 97	breqtrd 5124	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 0)
99		ovolge0 25438	. . . . 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 25435	. . . . . 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 11179	. . . . 5 ⊢ 0 ∈ ℝ*
104		xrletri3 13068	. . . . 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 5124	. 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 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  {crab 3399   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  ∪ ciun 4946   class class class wbr 5098  {copab 5160   ↦ cmpt 5179   × cxp 5622  dom cdm 5624  ran crn 5625   Fn wfn 6487  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358   Er wer 8632   / cqs 8634  Fincfn 8883  ℝcr 11025  0cc0 11026  1c1 11027   + caddc 11029  ℝ^*cxr 11165   < clt 11166   ≤ cle 11167   − cmin 11364  -cneg 11365  ℕcn 12145  2c2 12200  ℤcz 12488  ℤ_≥cuz 12751  ℚcq 12861  [,]cicc 13264  seqcseq 13924   ⇝ cli 15407  Σcsu 15609  vol*covol 25419  volcvol 25420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cc 10345  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-disj 5066  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-ec 8637  df-qs 8641  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9314  df-sup 9345  df-inf 9346  df-oi 9415  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-z 12489  df-uz 12752  df-q 12862  df-rp 12906  df-xneg 13026  df-xadd 13027  df-xmul 13028  df-ioo 13265  df-ico 13267  df-icc 13268  df-fz 13424  df-fzo 13571  df-fl 13712  df-seq 13925  df-exp 13985  df-hash 14254  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-clim 15411  df-rlim 15412  df-sum 15610  df-rest 17342  df-topgen 17363  df-psmet 21301  df-xmet 21302  df-met 21303  df-bl 21304  df-mopn 21305  df-top 22838  df-topon 22855  df-bases 22890  df-cmp 23331  df-ovol 25421  df-vol 25422

This theorem is referenced by:  vitali  25570