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Theorem vitalilem2 25663

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
vitalilem2	⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)))

Distinct variable groups: 𝑚,𝑛,𝑠,𝑥,𝑦,𝑧,𝐺 𝜑,𝑚,𝑛,𝑥,𝑧 𝑧,𝑆 𝑇,𝑚,𝑥 𝑚,𝐹,𝑛,𝑠,𝑥,𝑦,𝑧 ∼ ,𝑚,𝑛,𝑠,𝑥,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑦,𝑠) 𝑆(𝑥,𝑦,𝑚,𝑛,𝑠) 𝑇(𝑦,𝑧,𝑛,𝑠)

Proof of Theorem vitalilem2 Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vitali.3	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝑆)
2		vitali.4	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
3		vitali.2	. . . . . . . . 9 ⊢ 𝑆 = ((0[,]1) / ∼ )
4		neeq1 3009	. . . . . . . . 9 ⊢ ([𝑣] ∼ = 𝑧 → ([𝑣] ∼ ≠ ∅ ↔ 𝑧 ≠ ∅))
5		vitali.1	. . . . . . . . . . . . . 14 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}
6	5	vitalilem1 25662	. . . . . . . . . . . . 13 ⊢ ∼ Er (0[,]1)
7		erdm 8773	. . . . . . . . . . . . 13 ⊢ ( ∼ Er (0[,]1) → dom ∼ = (0[,]1))
8	6, 7	ax-mp 5	. . . . . . . . . . . 12 ⊢ dom ∼ = (0[,]1)
9	8	eleq2i 2836	. . . . . . . . . . 11 ⊢ (𝑣 ∈ dom ∼ ↔ 𝑣 ∈ (0[,]1))
10		ecdmn0 8812	. . . . . . . . . . 11 ⊢ (𝑣 ∈ dom ∼ ↔ [𝑣] ∼ ≠ ∅)
11	9, 10	bitr3i 277	. . . . . . . . . 10 ⊢ (𝑣 ∈ (0[,]1) ↔ [𝑣] ∼ ≠ ∅)
12	11	biimpi 216	. . . . . . . . 9 ⊢ (𝑣 ∈ (0[,]1) → [𝑣] ∼ ≠ ∅)
13	3, 4, 12	ectocl 8843	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ≠ ∅)
14	13	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ≠ ∅)
15		sseq1 4034	. . . . . . . . . 10 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ (0[,]1) ↔ 𝑧 ⊆ (0[,]1)))
16	6	a1i 11	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (0[,]1) → ∼ Er (0[,]1))
17	16	ecss 8811	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,]1) → [𝑤] ∼ ⊆ (0[,]1))
18	3, 15, 17	ectocl 8843	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ⊆ (0[,]1))
19	18	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ⊆ (0[,]1))
20	19	sseld 4007	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑧) ∈ 𝑧 → (𝐹‘𝑧) ∈ (0[,]1)))
21	14, 20	embantd 59	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) → (𝐹‘𝑧) ∈ (0[,]1)))
22	21	ralimdva 3173	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ (0[,]1)))
23	2, 22	mpd 15	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ (0[,]1))
24		ffnfv 7153	. . . 4 ⊢ (𝐹:𝑆⟶(0[,]1) ↔ (𝐹 Fn 𝑆 ∧ ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ (0[,]1)))
25	1, 23, 24	sylanbrc 582	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶(0[,]1))
26	25	frnd 6755	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1))
27		vitali.5	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
28	27	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
29		f1ocnv 6874	. . . . . . 7 ⊢ (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → ^◡𝐺:(ℚ ∩ (-1[,]1))–1-1-onto→ℕ)
30		f1of 6862	. . . . . . 7 ⊢ (^◡𝐺:(ℚ ∩ (-1[,]1))–1-1-onto→ℕ → ^◡𝐺:(ℚ ∩ (-1[,]1))⟶ℕ)
31	28, 29, 30	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ^◡𝐺:(ℚ ∩ (-1[,]1))⟶ℕ)
32		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ (0[,]1))
33	32, 11	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ≠ ∅)
34		neeq1 3009	. . . . . . . . . . . 12 ⊢ (𝑧 = [𝑣] ∼ → (𝑧 ≠ ∅ ↔ [𝑣] ∼ ≠ ∅))
35		fveq2 6920	. . . . . . . . . . . . 13 ⊢ (𝑧 = [𝑣] ∼ → (𝐹‘𝑧) = (𝐹‘[𝑣] ∼ ))
36		id 22	. . . . . . . . . . . . 13 ⊢ (𝑧 = [𝑣] ∼ → 𝑧 = [𝑣] ∼ )
37	35, 36	eleq12d 2838	. . . . . . . . . . . 12 ⊢ (𝑧 = [𝑣] ∼ → ((𝐹‘𝑧) ∈ 𝑧 ↔ (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ))
38	34, 37	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑧 = [𝑣] ∼ → ((𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) ↔ ([𝑣] ∼ ≠ ∅ → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ )))
39	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
40		ovex 7481	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) ∈ V
41		erex 8787	. . . . . . . . . . . . . . 15 ⊢ ( ∼ Er (0[,]1) → ((0[,]1) ∈ V → ∼ ∈ V))
42	6, 40, 41	mp2 9	. . . . . . . . . . . . . 14 ⊢ ∼ ∈ V
43	42	ecelqsi 8831	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (0[,]1) → [𝑣] ∼ ∈ ((0[,]1) / ∼ ))
44	43	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ∈ ((0[,]1) / ∼ ))
45	44, 3	eleqtrrdi 2855	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ∈ 𝑆)
46	38, 39, 45	rspcdva 3636	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ([𝑣] ∼ ≠ ∅ → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ))
47	33, 46	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ )
48		fvex 6933	. . . . . . . . . . 11 ⊢ (𝐹‘[𝑣] ∼ ) ∈ V
49		vex 3492	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
50	48, 49	elec 8809	. . . . . . . . . 10 ⊢ ((𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ↔ 𝑣 ∼ (𝐹‘[𝑣] ∼ ))
51		oveq12 7457	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = (𝐹‘[𝑣] ∼ )) → (𝑥 − 𝑦) = (𝑣 − (𝐹‘[𝑣] ∼ )))
52	51	eleq1d 2829	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = (𝐹‘[𝑣] ∼ )) → ((𝑥 − 𝑦) ∈ ℚ ↔ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ))
53	52, 5	brab2a 5793	. . . . . . . . . 10 ⊢ (𝑣 ∼ (𝐹‘[𝑣] ∼ ) ↔ ((𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ∼ ) ∈ (0[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ))
54	50, 53	bitri 275	. . . . . . . . 9 ⊢ ((𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ↔ ((𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ∼ ) ∈ (0[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ))
55	47, 54	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ∼ ) ∈ (0[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ))
56	55	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ)
57		elicc01 13526	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (0[,]1) ↔ (𝑣 ∈ ℝ ∧ 0 ≤ 𝑣 ∧ 𝑣 ≤ 1))
58	32, 57	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 ∈ ℝ ∧ 0 ≤ 𝑣 ∧ 𝑣 ≤ 1))
59	58	simp1d 1142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ ℝ)
60	55	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ∼ ) ∈ (0[,]1)))
61	60	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ (0[,]1))
62		elicc01 13526	. . . . . . . . . . 11 ⊢ ((𝐹‘[𝑣] ∼ ) ∈ (0[,]1) ↔ ((𝐹‘[𝑣] ∼ ) ∈ ℝ ∧ 0 ≤ (𝐹‘[𝑣] ∼ ) ∧ (𝐹‘[𝑣] ∼ ) ≤ 1))
63	61, 62	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ∼ ) ∈ ℝ ∧ 0 ≤ (𝐹‘[𝑣] ∼ ) ∧ (𝐹‘[𝑣] ∼ ) ≤ 1))
64	63	simp1d 1142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ ℝ)
65	59, 64	resubcld 11718	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℝ)
66	64, 59	resubcld 11718	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ∼ ) − 𝑣) ∈ ℝ)
67		1red 11291	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 1 ∈ ℝ)
68	58	simp2d 1143	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 0 ≤ 𝑣)
69	64, 59	subge02d 11882	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (0 ≤ 𝑣 ↔ ((𝐹‘[𝑣] ∼ ) − 𝑣) ≤ (𝐹‘[𝑣] ∼ )))
70	68, 69	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ∼ ) − 𝑣) ≤ (𝐹‘[𝑣] ∼ ))
71	63	simp3d 1144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ≤ 1)
72	66, 64, 67, 70, 71	letrd 11447	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ∼ ) − 𝑣) ≤ 1)
73	66, 67	lenegd 11869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (((𝐹‘[𝑣] ∼ ) − 𝑣) ≤ 1 ↔ -1 ≤ -((𝐹‘[𝑣] ∼ ) − 𝑣)))
74	72, 73	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → -1 ≤ -((𝐹‘[𝑣] ∼ ) − 𝑣))
75	64	recnd 11318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ ℂ)
76	59	recnd 11318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ ℂ)
77	75, 76	negsubdi2d 11663	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → -((𝐹‘[𝑣] ∼ ) − 𝑣) = (𝑣 − (𝐹‘[𝑣] ∼ )))
78	74, 77	breqtrd 5192	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → -1 ≤ (𝑣 − (𝐹‘[𝑣] ∼ )))
79	63	simp2d 1143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 0 ≤ (𝐹‘[𝑣] ∼ ))
80	59, 64	subge02d 11882	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (0 ≤ (𝐹‘[𝑣] ∼ ) ↔ (𝑣 − (𝐹‘[𝑣] ∼ )) ≤ 𝑣))
81	79, 80	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ≤ 𝑣)
82	58	simp3d 1144	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ≤ 1)
83	65, 59, 67, 81, 82	letrd 11447	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ≤ 1)
84		neg1rr 12408	. . . . . . . . 9 ⊢ -1 ∈ ℝ
85		1re 11290	. . . . . . . . 9 ⊢ 1 ∈ ℝ
86	84, 85	elicc2i 13473	. . . . . . . 8 ⊢ ((𝑣 − (𝐹‘[𝑣] ∼ )) ∈ (-1[,]1) ↔ ((𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℝ ∧ -1 ≤ (𝑣 − (𝐹‘[𝑣] ∼ )) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ≤ 1))
87	65, 78, 83, 86	syl3anbrc 1343	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ (-1[,]1))
88	56, 87	elind 4223	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ (ℚ ∩ (-1[,]1)))
89	31, 88	ffvelcdmd 7119	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) ∈ ℕ)
90		oveq1 7455	. . . . . . . 8 ⊢ (𝑠 = 𝑣 → (𝑠 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) = (𝑣 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))))
91	90	eleq1d 2829	. . . . . . 7 ⊢ (𝑠 = 𝑣 → ((𝑠 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹 ↔ (𝑣 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹))
92		f1ocnvfv2 7313	. . . . . . . . . . 11 ⊢ ((𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ (ℚ ∩ (-1[,]1))) → (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))) = (𝑣 − (𝐹‘[𝑣] ∼ )))
93	27, 88, 92	syl2an2r 684	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))) = (𝑣 − (𝐹‘[𝑣] ∼ )))
94	93	oveq2d 7464	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) = (𝑣 − (𝑣 − (𝐹‘[𝑣] ∼ ))))
95	76, 75	nncand 11652	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝑣 − (𝐹‘[𝑣] ∼ ))) = (𝐹‘[𝑣] ∼ ))
96	94, 95	eqtrd 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) = (𝐹‘[𝑣] ∼ ))
97		fnfvelrn 7114	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝑆 ∧ [𝑣] ∼ ∈ 𝑆) → (𝐹‘[𝑣] ∼ ) ∈ ran 𝐹)
98	1, 45, 97	syl2an2r 684	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ ran 𝐹)
99	96, 98	eqeltrd 2844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹)
100	91, 59, 99	elrabd 3710	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹})
101		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑛 = (^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → (𝐺‘𝑛) = (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))))
102	101	oveq2d 7464	. . . . . . . . . 10 ⊢ (𝑛 = (^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))))
103	102	eleq1d 2829	. . . . . . . . 9 ⊢ (𝑛 = (^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹))
104	103	rabbidv 3451	. . . . . . . 8 ⊢ (𝑛 = (^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹})
105		vitali.6	. . . . . . . 8 ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})
106		reex 11275	. . . . . . . . 9 ⊢ ℝ ∈ V
107	106	rabex 5357	. . . . . . . 8 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹} ∈ V
108	104, 105, 107	fvmpt 7029	. . . . . . 7 ⊢ ((^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) ∈ ℕ → (𝑇‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹})
109	89, 108	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑇‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹})
110	100, 109	eleqtrrd 2847	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ (𝑇‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))))
111		fveq2 6920	. . . . . 6 ⊢ (𝑚 = (^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → (𝑇‘𝑚) = (𝑇‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))))
112	111	eliuni 5021	. . . . 5 ⊢ (((^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) ∈ ℕ ∧ 𝑣 ∈ (𝑇‘(^◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) → 𝑣 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚))
113	89, 110, 112	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚))
114	113	ex 412	. . 3 ⊢ (𝜑 → (𝑣 ∈ (0[,]1) → 𝑣 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
115	114	ssrdv 4014	. 2 ⊢ (𝜑 → (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚))
116		eliun 5019	. . . 4 ⊢ (𝑥 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑇‘𝑚))
117		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐺‘𝑛) = (𝐺‘𝑚))
118	117	oveq2d 7464	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘𝑚)))
119	118	eleq1d 2829	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹))
120	119	rabbidv 3451	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
121	106	rabex 5357	. . . . . . . . . . . 12 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ∈ V
122	120, 105, 121	fvmpt 7029	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
123	122	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
124	123	eleq2d 2830	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ (𝑇‘𝑚) ↔ 𝑥 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹}))
125	124	biimpa 476	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
126		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠 − (𝐺‘𝑚)) = (𝑥 − (𝐺‘𝑚)))
127	126	eleq1d 2829	. . . . . . . . 9 ⊢ (𝑠 = 𝑥 → ((𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹 ↔ (𝑥 − (𝐺‘𝑚)) ∈ ran 𝐹))
128	127	elrab 3708	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ↔ (𝑥 ∈ ℝ ∧ (𝑥 − (𝐺‘𝑚)) ∈ ran 𝐹))
129	125, 128	sylib 218	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝑥 ∈ ℝ ∧ (𝑥 − (𝐺‘𝑚)) ∈ ran 𝐹))
130	129	simpld 494	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ∈ ℝ)
131	84	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → -1 ∈ ℝ)
132		iccssre 13489	. . . . . . . . . 10 ⊢ ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → (-1[,]1) ⊆ ℝ)
133	84, 85, 132	mp2an 691	. . . . . . . . 9 ⊢ (-1[,]1) ⊆ ℝ
134		f1of 6862	. . . . . . . . . . . 12 ⊢ (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
135	27, 134	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
136	135	ffvelcdmda 7118	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (ℚ ∩ (-1[,]1)))
137	136	elin2d 4228	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (-1[,]1))
138	133, 137	sselid 4006	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ)
139	138	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝐺‘𝑚) ∈ ℝ)
140	137	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝐺‘𝑚) ∈ (-1[,]1))
141	84, 85	elicc2i 13473	. . . . . . . . 9 ⊢ ((𝐺‘𝑚) ∈ (-1[,]1) ↔ ((𝐺‘𝑚) ∈ ℝ ∧ -1 ≤ (𝐺‘𝑚) ∧ (𝐺‘𝑚) ≤ 1))
142	140, 141	sylib 218	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝐺‘𝑚) ∈ ℝ ∧ -1 ≤ (𝐺‘𝑚) ∧ (𝐺‘𝑚) ≤ 1))
143	142	simp2d 1143	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → -1 ≤ (𝐺‘𝑚))
144	26	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ran 𝐹 ⊆ (0[,]1))
145	129	simprd 495	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝑥 − (𝐺‘𝑚)) ∈ ran 𝐹)
146	144, 145	sseldd 4009	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝑥 − (𝐺‘𝑚)) ∈ (0[,]1))
147		elicc01 13526	. . . . . . . . . 10 ⊢ ((𝑥 − (𝐺‘𝑚)) ∈ (0[,]1) ↔ ((𝑥 − (𝐺‘𝑚)) ∈ ℝ ∧ 0 ≤ (𝑥 − (𝐺‘𝑚)) ∧ (𝑥 − (𝐺‘𝑚)) ≤ 1))
148	146, 147	sylib 218	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝑥 − (𝐺‘𝑚)) ∈ ℝ ∧ 0 ≤ (𝑥 − (𝐺‘𝑚)) ∧ (𝑥 − (𝐺‘𝑚)) ≤ 1))
149	148	simp2d 1143	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 0 ≤ (𝑥 − (𝐺‘𝑚)))
150	130, 139	subge0d 11880	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (0 ≤ (𝑥 − (𝐺‘𝑚)) ↔ (𝐺‘𝑚) ≤ 𝑥))
151	149, 150	mpbid 232	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝐺‘𝑚) ≤ 𝑥)
152	131, 139, 130, 143, 151	letrd 11447	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → -1 ≤ 𝑥)
153		peano2re 11463	. . . . . . . 8 ⊢ ((𝐺‘𝑚) ∈ ℝ → ((𝐺‘𝑚) + 1) ∈ ℝ)
154	139, 153	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝐺‘𝑚) + 1) ∈ ℝ)
155		2re 12367	. . . . . . . 8 ⊢ 2 ∈ ℝ
156	155	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 2 ∈ ℝ)
157	148	simp3d 1144	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝑥 − (𝐺‘𝑚)) ≤ 1)
158		1red 11291	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 1 ∈ ℝ)
159	130, 139, 158	lesubadd2d 11889	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝑥 − (𝐺‘𝑚)) ≤ 1 ↔ 𝑥 ≤ ((𝐺‘𝑚) + 1)))
160	157, 159	mpbid 232	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ≤ ((𝐺‘𝑚) + 1))
161	142	simp3d 1144	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝐺‘𝑚) ≤ 1)
162	139, 158, 158, 161	leadd1dd 11904	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝐺‘𝑚) + 1) ≤ (1 + 1))
163		df-2 12356	. . . . . . . 8 ⊢ 2 = (1 + 1)
164	162, 163	breqtrrdi 5208	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝐺‘𝑚) + 1) ≤ 2)
165	130, 154, 156, 160, 164	letrd 11447	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ≤ 2)
166	84, 155	elicc2i 13473	. . . . . 6 ⊢ (𝑥 ∈ (-1[,]2) ↔ (𝑥 ∈ ℝ ∧ -1 ≤ 𝑥 ∧ 𝑥 ≤ 2))
167	130, 152, 165, 166	syl3anbrc 1343	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ∈ (-1[,]2))
168	167	rexlimdva2 3163	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝑥 ∈ (𝑇‘𝑚) → 𝑥 ∈ (-1[,]2)))
169	116, 168	biimtrid 242	. . 3 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) → 𝑥 ∈ (-1[,]2)))
170	169	ssrdv 4014	. 2 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2))
171	26, 115, 170	3jca 1128	1 ⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 Vcvv 3488 ∖ cdif 3973 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 𝒫 cpw 4622 ∪ ciun 5015 class class class wbr 5166 {copab 5228 ↦ cmpt 5249 ^◡ccnv 5699 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 [cec 8761 / cqs 8762 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 ≤ cle 11325 − cmin 11520 -cneg 11521 ℕcn 12293 2c2 12348 ℚcq 13013 [,]cicc 13410 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-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 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

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-iun 5017 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-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-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-ec 8765 df-qs 8769 df-en 9004 df-dom 9005 df-sdom 9006 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-n0 12554 df-z 12640 df-q 13014 df-icc 13414

This theorem is referenced by: vitalilem3 25664 vitalilem4 25665 vitalilem5 25666

W3C validator