Theorem vitalilem2 24204

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
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 3078	. . . . . . . . 9 ⊢ ([𝑣] ∼ = 𝑧 → ([𝑣] ∼ ≠ ∅ ↔ 𝑧 ≠ ∅))
5		vitali.1	. . . . . . . . . . . . . 14 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}
6	5	vitalilem1 24203	. . . . . . . . . . . . 13 ⊢ ∼ Er (0[,]1)
7		erdm 8293	. . . . . . . . . . . . 13 ⊢ ( ∼ Er (0[,]1) → dom ∼ = (0[,]1))
8	6, 7	ax-mp 5	. . . . . . . . . . . 12 ⊢ dom ∼ = (0[,]1)
9	8	eleq2i 2904	. . . . . . . . . . 11 ⊢ (𝑣 ∈ dom ∼ ↔ 𝑣 ∈ (0[,]1))
10		ecdmn0 8330	. . . . . . . . . . 11 ⊢ (𝑣 ∈ dom ∼ ↔ [𝑣] ∼ ≠ ∅)
11	9, 10	bitr3i 279	. . . . . . . . . 10 ⊢ (𝑣 ∈ (0[,]1) ↔ [𝑣] ∼ ≠ ∅)
12	11	biimpi 218	. . . . . . . . 9 ⊢ (𝑣 ∈ (0[,]1) → [𝑣] ∼ ≠ ∅)
13	3, 4, 12	ectocl 8359	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ≠ ∅)
14	13	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ≠ ∅)
15		sseq1 3991	. . . . . . . . . 10 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ (0[,]1) ↔ 𝑧 ⊆ (0[,]1)))
16	6	a1i 11	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (0[,]1) → ∼ Er (0[,]1))
17	16	ecss 8329	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,]1) → [𝑤] ∼ ⊆ (0[,]1))
18	3, 15, 17	ectocl 8359	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ⊆ (0[,]1))
19	18	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ⊆ (0[,]1))
20	19	sseld 3965	. . . . . . 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 3177	. . . . 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 6876	. . . 4 ⊢ (𝐹:𝑆⟶(0[,]1) ↔ (𝐹 Fn 𝑆 ∧ ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ (0[,]1)))
25	1, 23, 24	sylanbrc 585	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶(0[,]1))
26	25	frnd 6515	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1))
27		vitali.5	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
28	27	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
29		f1ocnv 6621	. . . . . . 7 ⊢ (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → ◡𝐺:(ℚ ∩ (-1[,]1))–1-1-onto→ℕ)
30		f1of 6609	. . . . . . 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 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ (0[,]1))
33	32, 11	sylib 220	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ≠ ∅)
34		neeq1 3078	. . . . . . . . . . . 12 ⊢ (𝑧 = [𝑣] ∼ → (𝑧 ≠ ∅ ↔ [𝑣] ∼ ≠ ∅))
35		fveq2 6664	. . . . . . . . . . . . 13 ⊢ (𝑧 = [𝑣] ∼ → (𝐹‘𝑧) = (𝐹‘[𝑣] ∼ ))
36		id 22	. . . . . . . . . . . . 13 ⊢ (𝑧 = [𝑣] ∼ → 𝑧 = [𝑣] ∼ )
37	35, 36	eleq12d 2907	. . . . . . . . . . . 12 ⊢ (𝑧 = [𝑣] ∼ → ((𝐹‘𝑧) ∈ 𝑧 ↔ (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ))
38	34, 37	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑧 = [𝑣] ∼ → ((𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) ↔ ([𝑣] ∼ ≠ ∅ → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ )))
39	2	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
40		ovex 7183	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) ∈ V
41		erex 8307	. . . . . . . . . . . . . . 15 ⊢ ( ∼ Er (0[,]1) → ((0[,]1) ∈ V → ∼ ∈ V))
42	6, 40, 41	mp2 9	. . . . . . . . . . . . . 14 ⊢ ∼ ∈ V
43	42	ecelqsi 8347	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (0[,]1) → [𝑣] ∼ ∈ ((0[,]1) / ∼ ))
44	43	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ∈ ((0[,]1) / ∼ ))
45	44, 3	eleqtrrdi 2924	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ∈ 𝑆)
46	38, 39, 45	rspcdva 3624	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ([𝑣] ∼ ≠ ∅ → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ))
47	33, 46	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ )
48		fvex 6677	. . . . . . . . . . 11 ⊢ (𝐹‘[𝑣] ∼ ) ∈ V
49		vex 3497	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
50	48, 49	elec 8327	. . . . . . . . . 10 ⊢ ((𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ↔ 𝑣 ∼ (𝐹‘[𝑣] ∼ ))
51		oveq12 7159	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = (𝐹‘[𝑣] ∼ )) → (𝑥 − 𝑦) = (𝑣 − (𝐹‘[𝑣] ∼ )))
52	51	eleq1d 2897	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = (𝐹‘[𝑣] ∼ )) → ((𝑥 − 𝑦) ∈ ℚ ↔ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ))
53	52, 5	brab2a 5638	. . . . . . . . . 10 ⊢ (𝑣 ∼ (𝐹‘[𝑣] ∼ ) ↔ ((𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ∼ ) ∈ (0[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ))
54	50, 53	bitri 277	. . . . . . . . 9 ⊢ ((𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ↔ ((𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ∼ ) ∈ (0[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ))
55	47, 54	sylib 220	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ∼ ) ∈ (0[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ))
56	55	simprd 498	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ)
57		elicc01 12848	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (0[,]1) ↔ (𝑣 ∈ ℝ ∧ 0 ≤ 𝑣 ∧ 𝑣 ≤ 1))
58	32, 57	sylib 220	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 ∈ ℝ ∧ 0 ≤ 𝑣 ∧ 𝑣 ≤ 1))
59	58	simp1d 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ ℝ)
60	55	simpld 497	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ∼ ) ∈ (0[,]1)))
61	60	simprd 498	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ (0[,]1))
62		elicc01 12848	. . . . . . . . . . 11 ⊢ ((𝐹‘[𝑣] ∼ ) ∈ (0[,]1) ↔ ((𝐹‘[𝑣] ∼ ) ∈ ℝ ∧ 0 ≤ (𝐹‘[𝑣] ∼ ) ∧ (𝐹‘[𝑣] ∼ ) ≤ 1))
63	61, 62	sylib 220	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ∼ ) ∈ ℝ ∧ 0 ≤ (𝐹‘[𝑣] ∼ ) ∧ (𝐹‘[𝑣] ∼ ) ≤ 1))
64	63	simp1d 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ ℝ)
65	59, 64	resubcld 11062	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℝ)
66	64, 59	resubcld 11062	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ∼ ) − 𝑣) ∈ ℝ)
67		1red 10636	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 1 ∈ ℝ)
68	58	simp2d 1139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 0 ≤ 𝑣)
69	64, 59	subge02d 11226	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (0 ≤ 𝑣 ↔ ((𝐹‘[𝑣] ∼ ) − 𝑣) ≤ (𝐹‘[𝑣] ∼ )))
70	68, 69	mpbid 234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ∼ ) − 𝑣) ≤ (𝐹‘[𝑣] ∼ ))
71	63	simp3d 1140	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ≤ 1)
72	66, 64, 67, 70, 71	letrd 10791	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ∼ ) − 𝑣) ≤ 1)
73	66, 67	lenegd 11213	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (((𝐹‘[𝑣] ∼ ) − 𝑣) ≤ 1 ↔ -1 ≤ -((𝐹‘[𝑣] ∼ ) − 𝑣)))
74	72, 73	mpbid 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → -1 ≤ -((𝐹‘[𝑣] ∼ ) − 𝑣))
75	64	recnd 10663	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ ℂ)
76	59	recnd 10663	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ ℂ)
77	75, 76	negsubdi2d 11007	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → -((𝐹‘[𝑣] ∼ ) − 𝑣) = (𝑣 − (𝐹‘[𝑣] ∼ )))
78	74, 77	breqtrd 5084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → -1 ≤ (𝑣 − (𝐹‘[𝑣] ∼ )))
79	63	simp2d 1139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 0 ≤ (𝐹‘[𝑣] ∼ ))
80	59, 64	subge02d 11226	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (0 ≤ (𝐹‘[𝑣] ∼ ) ↔ (𝑣 − (𝐹‘[𝑣] ∼ )) ≤ 𝑣))
81	79, 80	mpbid 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ≤ 𝑣)
82	58	simp3d 1140	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ≤ 1)
83	65, 59, 67, 81, 82	letrd 10791	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ≤ 1)
84		neg1rr 11746	. . . . . . . . 9 ⊢ -1 ∈ ℝ
85		1re 10635	. . . . . . . . 9 ⊢ 1 ∈ ℝ
86	84, 85	elicc2i 12796	. . . . . . . 8 ⊢ ((𝑣 − (𝐹‘[𝑣] ∼ )) ∈ (-1[,]1) ↔ ((𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℝ ∧ -1 ≤ (𝑣 − (𝐹‘[𝑣] ∼ )) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ≤ 1))
87	65, 78, 83, 86	syl3anbrc 1339	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ (-1[,]1))
88	56, 87	elind 4170	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ (ℚ ∩ (-1[,]1)))
89	31, 88	ffvelrnd 6846	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) ∈ ℕ)
90		oveq1 7157	. . . . . . . 8 ⊢ (𝑠 = 𝑣 → (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) = (𝑣 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))))
91	90	eleq1d 2897	. . . . . . 7 ⊢ (𝑠 = 𝑣 → ((𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹 ↔ (𝑣 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹))
92		f1ocnvfv2 7028	. . . . . . . . . . 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 683	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))) = (𝑣 − (𝐹‘[𝑣] ∼ )))
94	93	oveq2d 7166	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) = (𝑣 − (𝑣 − (𝐹‘[𝑣] ∼ ))))
95	76, 75	nncand 10996	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝑣 − (𝐹‘[𝑣] ∼ ))) = (𝐹‘[𝑣] ∼ ))
96	94, 95	eqtrd 2856	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) = (𝐹‘[𝑣] ∼ ))
97		fnfvelrn 6842	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝑆 ∧ [𝑣] ∼ ∈ 𝑆) → (𝐹‘[𝑣] ∼ ) ∈ ran 𝐹)
98	1, 45, 97	syl2an2r 683	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ ran 𝐹)
99	96, 98	eqeltrd 2913	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹)
100	91, 59, 99	elrabd 3681	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹})
101		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑛 = (◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → (𝐺‘𝑛) = (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))))
102	101	oveq2d 7166	. . . . . . . . . 10 ⊢ (𝑛 = (◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))))
103	102	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑛 = (◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹))
104	103	rabbidv 3480	. . . . . . . 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 10622	. . . . . . . . 9 ⊢ ℝ ∈ V
107	106	rabex 5227	. . . . . . . 8 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹} ∈ V
108	104, 105, 107	fvmpt 6762	. . . . . . 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 2916	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ (𝑇‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))))
111		fveq2 6664	. . . . . 6 ⊢ (𝑚 = (◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → (𝑇‘𝑚) = (𝑇‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))))
112	111	eliuni 4917	. . . . 5 ⊢ (((◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) ∈ ℕ ∧ 𝑣 ∈ (𝑇‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) → 𝑣 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚))
113	89, 110, 112	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚))
114	113	ex 415	. . 3 ⊢ (𝜑 → (𝑣 ∈ (0[,]1) → 𝑣 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
115	114	ssrdv 3972	. 2 ⊢ (𝜑 → (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚))
116		eliun 4915	. . . 4 ⊢ (𝑥 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑇‘𝑚))
117		fveq2 6664	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐺‘𝑛) = (𝐺‘𝑚))
118	117	oveq2d 7166	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘𝑚)))
119	118	eleq1d 2897	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹))
120	119	rabbidv 3480	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
121	106	rabex 5227	. . . . . . . . . . . 12 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ∈ V
122	120, 105, 121	fvmpt 6762	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
123	122	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
124	123	eleq2d 2898	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ (𝑇‘𝑚) ↔ 𝑥 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹}))
125	124	biimpa 479	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
126		oveq1 7157	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠 − (𝐺‘𝑚)) = (𝑥 − (𝐺‘𝑚)))
127	126	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑠 = 𝑥 → ((𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹 ↔ (𝑥 − (𝐺‘𝑚)) ∈ ran 𝐹))
128	127	elrab 3679	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ↔ (𝑥 ∈ ℝ ∧ (𝑥 − (𝐺‘𝑚)) ∈ ran 𝐹))
129	125, 128	sylib 220	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝑥 ∈ ℝ ∧ (𝑥 − (𝐺‘𝑚)) ∈ ran 𝐹))
130	129	simpld 497	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ∈ ℝ)
131	84	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → -1 ∈ ℝ)
132		iccssre 12812	. . . . . . . . . 10 ⊢ ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → (-1[,]1) ⊆ ℝ)
133	84, 85, 132	mp2an 690	. . . . . . . . 9 ⊢ (-1[,]1) ⊆ ℝ
134		f1of 6609	. . . . . . . . . . . 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	ffvelrnda 6845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (ℚ ∩ (-1[,]1)))
137	136	elin2d 4175	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (-1[,]1))
138	133, 137	sseldi 3964	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ)
139	138	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝐺‘𝑚) ∈ ℝ)
140	137	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝐺‘𝑚) ∈ (-1[,]1))
141	84, 85	elicc2i 12796	. . . . . . . . 9 ⊢ ((𝐺‘𝑚) ∈ (-1[,]1) ↔ ((𝐺‘𝑚) ∈ ℝ ∧ -1 ≤ (𝐺‘𝑚) ∧ (𝐺‘𝑚) ≤ 1))
142	140, 141	sylib 220	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝐺‘𝑚) ∈ ℝ ∧ -1 ≤ (𝐺‘𝑚) ∧ (𝐺‘𝑚) ≤ 1))
143	142	simp2d 1139	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → -1 ≤ (𝐺‘𝑚))
144	26	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ran 𝐹 ⊆ (0[,]1))
145	129	simprd 498	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝑥 − (𝐺‘𝑚)) ∈ ran 𝐹)
146	144, 145	sseldd 3967	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝑥 − (𝐺‘𝑚)) ∈ (0[,]1))
147		elicc01 12848	. . . . . . . . . 10 ⊢ ((𝑥 − (𝐺‘𝑚)) ∈ (0[,]1) ↔ ((𝑥 − (𝐺‘𝑚)) ∈ ℝ ∧ 0 ≤ (𝑥 − (𝐺‘𝑚)) ∧ (𝑥 − (𝐺‘𝑚)) ≤ 1))
148	146, 147	sylib 220	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝑥 − (𝐺‘𝑚)) ∈ ℝ ∧ 0 ≤ (𝑥 − (𝐺‘𝑚)) ∧ (𝑥 − (𝐺‘𝑚)) ≤ 1))
149	148	simp2d 1139	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 0 ≤ (𝑥 − (𝐺‘𝑚)))
150	130, 139	subge0d 11224	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (0 ≤ (𝑥 − (𝐺‘𝑚)) ↔ (𝐺‘𝑚) ≤ 𝑥))
151	149, 150	mpbid 234	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝐺‘𝑚) ≤ 𝑥)
152	131, 139, 130, 143, 151	letrd 10791	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → -1 ≤ 𝑥)
153		peano2re 10807	. . . . . . . 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 11705	. . . . . . . 8 ⊢ 2 ∈ ℝ
156	155	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 2 ∈ ℝ)
157	148	simp3d 1140	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝑥 − (𝐺‘𝑚)) ≤ 1)
158		1red 10636	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 1 ∈ ℝ)
159	130, 139, 158	lesubadd2d 11233	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝑥 − (𝐺‘𝑚)) ≤ 1 ↔ 𝑥 ≤ ((𝐺‘𝑚) + 1)))
160	157, 159	mpbid 234	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ≤ ((𝐺‘𝑚) + 1))
161	142	simp3d 1140	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝐺‘𝑚) ≤ 1)
162	139, 158, 158, 161	leadd1dd 11248	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝐺‘𝑚) + 1) ≤ (1 + 1))
163		df-2 11694	. . . . . . . 8 ⊢ 2 = (1 + 1)
164	162, 163	breqtrrdi 5100	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝐺‘𝑚) + 1) ≤ 2)
165	130, 154, 156, 160, 164	letrd 10791	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ≤ 2)
166	84, 155	elicc2i 12796	. . . . . 6 ⊢ (𝑥 ∈ (-1[,]2) ↔ (𝑥 ∈ ℝ ∧ -1 ≤ 𝑥 ∧ 𝑥 ≤ 2))
167	130, 152, 165, 166	syl3anbrc 1339	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ∈ (-1[,]2))
168	167	rexlimdva2 3287	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝑥 ∈ (𝑇‘𝑚) → 𝑥 ∈ (-1[,]2)))
169	116, 168	syl5bi 244	. . 3 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) → 𝑥 ∈ (-1[,]2)))
170	169	ssrdv 3972	. 2 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2))
171	26, 115, 170	3jca 1124	1 ⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494   ∖ cdif 3932   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  ∪ ciun 4911   class class class wbr 5058  {copab 5120   ↦ cmpt 5138  ^◡ccnv 5548  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  [cec 8281   / cqs 8282  ℝcr 10530  0cc0 10531  1c1 10532   + caddc 10534   ≤ cle 10670   − cmin 10864  -cneg 10865  ℕcn 11632  2c2 11686  ℚcq 12342  [,]cicc 12735  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-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  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

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  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-iun 4913  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-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-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-ec 8285  df-qs 8289  df-en 8504  df-dom 8505  df-sdom 8506  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-n0 11892  df-z 11976  df-q 12343  df-icc 12739

This theorem is referenced by:  vitalilem3  24205  vitalilem4  24206  vitalilem5  24207