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Theorem vitalilem2 25582
Description: Lemma for vitali 25586. (Contributed by Mario Carneiro, 16-Jun-2014.)
Hypotheses
Ref Expression
vitali.1 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
vitali.2 𝑆 = ((0[,]1) / )
vitali.3 (𝜑𝐹 Fn 𝑆)
vitali.4 (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
vitali.5 (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
vitali.6 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})
vitali.7 (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
Assertion
Ref Expression
vitalilem2 (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚) ∧ 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-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.
StepHypRef Expression
1 vitali.3 . . . 4 (𝜑𝐹 Fn 𝑆)
2 vitali.4 . . . . 5 (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
3 vitali.2 . . . . . . . . 9 𝑆 = ((0[,]1) / )
4 neeq1 2992 . . . . . . . . 9 ([𝑣] = 𝑧 → ([𝑣] ≠ ∅ ↔ 𝑧 ≠ ∅))
5 vitali.1 . . . . . . . . . . . . . 14 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
65vitalilem1 25581 . . . . . . . . . . . . 13 Er (0[,]1)
7 erdm 8735 . . . . . . . . . . . . 13 ( Er (0[,]1) → dom = (0[,]1))
86, 7ax-mp 5 . . . . . . . . . . . 12 dom = (0[,]1)
98eleq2i 2817 . . . . . . . . . . 11 (𝑣 ∈ dom 𝑣 ∈ (0[,]1))
10 ecdmn0 8773 . . . . . . . . . . 11 (𝑣 ∈ dom ↔ [𝑣] ≠ ∅)
119, 10bitr3i 276 . . . . . . . . . 10 (𝑣 ∈ (0[,]1) ↔ [𝑣] ≠ ∅)
1211biimpi 215 . . . . . . . . 9 (𝑣 ∈ (0[,]1) → [𝑣] ≠ ∅)
133, 4, 12ectocl 8804 . . . . . . . 8 (𝑧𝑆𝑧 ≠ ∅)
1413adantl 480 . . . . . . 7 ((𝜑𝑧𝑆) → 𝑧 ≠ ∅)
15 sseq1 4002 . . . . . . . . . 10 ([𝑤] = 𝑧 → ([𝑤] ⊆ (0[,]1) ↔ 𝑧 ⊆ (0[,]1)))
166a1i 11 . . . . . . . . . . 11 (𝑤 ∈ (0[,]1) → Er (0[,]1))
1716ecss 8772 . . . . . . . . . 10 (𝑤 ∈ (0[,]1) → [𝑤] ⊆ (0[,]1))
183, 15, 17ectocl 8804 . . . . . . . . 9 (𝑧𝑆𝑧 ⊆ (0[,]1))
1918adantl 480 . . . . . . . 8 ((𝜑𝑧𝑆) → 𝑧 ⊆ (0[,]1))
2019sseld 3975 . . . . . . 7 ((𝜑𝑧𝑆) → ((𝐹𝑧) ∈ 𝑧 → (𝐹𝑧) ∈ (0[,]1)))
2114, 20embantd 59 . . . . . 6 ((𝜑𝑧𝑆) → ((𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧) → (𝐹𝑧) ∈ (0[,]1)))
2221ralimdva 3156 . . . . 5 (𝜑 → (∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧) → ∀𝑧𝑆 (𝐹𝑧) ∈ (0[,]1)))
232, 22mpd 15 . . . 4 (𝜑 → ∀𝑧𝑆 (𝐹𝑧) ∈ (0[,]1))
24 ffnfv 7128 . . . 4 (𝐹:𝑆⟶(0[,]1) ↔ (𝐹 Fn 𝑆 ∧ ∀𝑧𝑆 (𝐹𝑧) ∈ (0[,]1)))
251, 23, 24sylanbrc 581 . . 3 (𝜑𝐹:𝑆⟶(0[,]1))
2625frnd 6731 . 2 (𝜑 → ran 𝐹 ⊆ (0[,]1))
27 vitali.5 . . . . . . . 8 (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
2827adantr 479 . . . . . . 7 ((𝜑𝑣 ∈ (0[,]1)) → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
29 f1ocnv 6850 . . . . . . 7 (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:(ℚ ∩ (-1[,]1))–1-1-onto→ℕ)
30 f1of 6838 . . . . . . 7 (𝐺:(ℚ ∩ (-1[,]1))–1-1-onto→ℕ → 𝐺:(ℚ ∩ (-1[,]1))⟶ℕ)
3128, 29, 303syl 18 . . . . . 6 ((𝜑𝑣 ∈ (0[,]1)) → 𝐺:(ℚ ∩ (-1[,]1))⟶ℕ)
32 simpr 483 . . . . . . . . . . 11 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 ∈ (0[,]1))
3332, 11sylib 217 . . . . . . . . . 10 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] ≠ ∅)
34 neeq1 2992 . . . . . . . . . . . 12 (𝑧 = [𝑣] → (𝑧 ≠ ∅ ↔ [𝑣] ≠ ∅))
35 fveq2 6896 . . . . . . . . . . . . 13 (𝑧 = [𝑣] → (𝐹𝑧) = (𝐹‘[𝑣] ))
36 id 22 . . . . . . . . . . . . 13 (𝑧 = [𝑣] 𝑧 = [𝑣] )
3735, 36eleq12d 2819 . . . . . . . . . . . 12 (𝑧 = [𝑣] → ((𝐹𝑧) ∈ 𝑧 ↔ (𝐹‘[𝑣] ) ∈ [𝑣] ))
3834, 37imbi12d 343 . . . . . . . . . . 11 (𝑧 = [𝑣] → ((𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧) ↔ ([𝑣] ≠ ∅ → (𝐹‘[𝑣] ) ∈ [𝑣] )))
392adantr 479 . . . . . . . . . . 11 ((𝜑𝑣 ∈ (0[,]1)) → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
40 ovex 7452 . . . . . . . . . . . . . . 15 (0[,]1) ∈ V
41 erex 8749 . . . . . . . . . . . . . . 15 ( Er (0[,]1) → ((0[,]1) ∈ V → ∈ V))
426, 40, 41mp2 9 . . . . . . . . . . . . . 14 ∈ V
4342ecelqsi 8792 . . . . . . . . . . . . 13 (𝑣 ∈ (0[,]1) → [𝑣] ∈ ((0[,]1) / ))
4443adantl 480 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] ∈ ((0[,]1) / ))
4544, 3eleqtrrdi 2836 . . . . . . . . . . 11 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] 𝑆)
4638, 39, 45rspcdva 3607 . . . . . . . . . 10 ((𝜑𝑣 ∈ (0[,]1)) → ([𝑣] ≠ ∅ → (𝐹‘[𝑣] ) ∈ [𝑣] ))
4733, 46mpd 15 . . . . . . . . 9 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ) ∈ [𝑣] )
48 fvex 6909 . . . . . . . . . . 11 (𝐹‘[𝑣] ) ∈ V
49 vex 3465 . . . . . . . . . . 11 𝑣 ∈ V
5048, 49elec 8770 . . . . . . . . . 10 ((𝐹‘[𝑣] ) ∈ [𝑣] 𝑣 (𝐹‘[𝑣] ))
51 oveq12 7428 . . . . . . . . . . . 12 ((𝑥 = 𝑣𝑦 = (𝐹‘[𝑣] )) → (𝑥𝑦) = (𝑣 − (𝐹‘[𝑣] )))
5251eleq1d 2810 . . . . . . . . . . 11 ((𝑥 = 𝑣𝑦 = (𝐹‘[𝑣] )) → ((𝑥𝑦) ∈ ℚ ↔ (𝑣 − (𝐹‘[𝑣] )) ∈ ℚ))
5352, 5brab2a 5771 . . . . . . . . . 10 (𝑣 (𝐹‘[𝑣] ) ↔ ((𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ) ∈ (0[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] )) ∈ ℚ))
5450, 53bitri 274 . . . . . . . . 9 ((𝐹‘[𝑣] ) ∈ [𝑣] ↔ ((𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ) ∈ (0[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] )) ∈ ℚ))
5547, 54sylib 217 . . . . . . . 8 ((𝜑𝑣 ∈ (0[,]1)) → ((𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ) ∈ (0[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] )) ∈ ℚ))
5655simprd 494 . . . . . . 7 ((𝜑𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] )) ∈ ℚ)
57 elicc01 13478 . . . . . . . . . . 11 (𝑣 ∈ (0[,]1) ↔ (𝑣 ∈ ℝ ∧ 0 ≤ 𝑣𝑣 ≤ 1))
5832, 57sylib 217 . . . . . . . . . 10 ((𝜑𝑣 ∈ (0[,]1)) → (𝑣 ∈ ℝ ∧ 0 ≤ 𝑣𝑣 ≤ 1))
5958simp1d 1139 . . . . . . . . 9 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 ∈ ℝ)
6055simpld 493 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ (0[,]1)) → (𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ) ∈ (0[,]1)))
6160simprd 494 . . . . . . . . . . 11 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ) ∈ (0[,]1))
62 elicc01 13478 . . . . . . . . . . 11 ((𝐹‘[𝑣] ) ∈ (0[,]1) ↔ ((𝐹‘[𝑣] ) ∈ ℝ ∧ 0 ≤ (𝐹‘[𝑣] ) ∧ (𝐹‘[𝑣] ) ≤ 1))
6361, 62sylib 217 . . . . . . . . . 10 ((𝜑𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ) ∈ ℝ ∧ 0 ≤ (𝐹‘[𝑣] ) ∧ (𝐹‘[𝑣] ) ≤ 1))
6463simp1d 1139 . . . . . . . . 9 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ) ∈ ℝ)
6559, 64resubcld 11674 . . . . . . . 8 ((𝜑𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] )) ∈ ℝ)
6664, 59resubcld 11674 . . . . . . . . . . 11 ((𝜑𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ) − 𝑣) ∈ ℝ)
67 1red 11247 . . . . . . . . . . 11 ((𝜑𝑣 ∈ (0[,]1)) → 1 ∈ ℝ)
6858simp2d 1140 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ (0[,]1)) → 0 ≤ 𝑣)
6964, 59subge02d 11838 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ (0[,]1)) → (0 ≤ 𝑣 ↔ ((𝐹‘[𝑣] ) − 𝑣) ≤ (𝐹‘[𝑣] )))
7068, 69mpbid 231 . . . . . . . . . . 11 ((𝜑𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ) − 𝑣) ≤ (𝐹‘[𝑣] ))
7163simp3d 1141 . . . . . . . . . . 11 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ) ≤ 1)
7266, 64, 67, 70, 71letrd 11403 . . . . . . . . . 10 ((𝜑𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ) − 𝑣) ≤ 1)
7366, 67lenegd 11825 . . . . . . . . . 10 ((𝜑𝑣 ∈ (0[,]1)) → (((𝐹‘[𝑣] ) − 𝑣) ≤ 1 ↔ -1 ≤ -((𝐹‘[𝑣] ) − 𝑣)))
7472, 73mpbid 231 . . . . . . . . 9 ((𝜑𝑣 ∈ (0[,]1)) → -1 ≤ -((𝐹‘[𝑣] ) − 𝑣))
7564recnd 11274 . . . . . . . . . 10 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ) ∈ ℂ)
7659recnd 11274 . . . . . . . . . 10 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 ∈ ℂ)
7775, 76negsubdi2d 11619 . . . . . . . . 9 ((𝜑𝑣 ∈ (0[,]1)) → -((𝐹‘[𝑣] ) − 𝑣) = (𝑣 − (𝐹‘[𝑣] )))
7874, 77breqtrd 5175 . . . . . . . 8 ((𝜑𝑣 ∈ (0[,]1)) → -1 ≤ (𝑣 − (𝐹‘[𝑣] )))
7963simp2d 1140 . . . . . . . . . 10 ((𝜑𝑣 ∈ (0[,]1)) → 0 ≤ (𝐹‘[𝑣] ))
8059, 64subge02d 11838 . . . . . . . . . 10 ((𝜑𝑣 ∈ (0[,]1)) → (0 ≤ (𝐹‘[𝑣] ) ↔ (𝑣 − (𝐹‘[𝑣] )) ≤ 𝑣))
8179, 80mpbid 231 . . . . . . . . 9 ((𝜑𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] )) ≤ 𝑣)
8258simp3d 1141 . . . . . . . . 9 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 ≤ 1)
8365, 59, 67, 81, 82letrd 11403 . . . . . . . 8 ((𝜑𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] )) ≤ 1)
84 neg1rr 12360 . . . . . . . . 9 -1 ∈ ℝ
85 1re 11246 . . . . . . . . 9 1 ∈ ℝ
8684, 85elicc2i 13425 . . . . . . . 8 ((𝑣 − (𝐹‘[𝑣] )) ∈ (-1[,]1) ↔ ((𝑣 − (𝐹‘[𝑣] )) ∈ ℝ ∧ -1 ≤ (𝑣 − (𝐹‘[𝑣] )) ∧ (𝑣 − (𝐹‘[𝑣] )) ≤ 1))
8765, 78, 83, 86syl3anbrc 1340 . . . . . . 7 ((𝜑𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] )) ∈ (-1[,]1))
8856, 87elind 4192 . . . . . 6 ((𝜑𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] )) ∈ (ℚ ∩ (-1[,]1)))
8931, 88ffvelcdmd 7094 . . . . 5 ((𝜑𝑣 ∈ (0[,]1)) → (𝐺‘(𝑣 − (𝐹‘[𝑣] ))) ∈ ℕ)
90 oveq1 7426 . . . . . . . 8 (𝑠 = 𝑣 → (𝑠 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) = (𝑣 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))))
9190eleq1d 2810 . . . . . . 7 (𝑠 = 𝑣 → ((𝑠 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) ∈ ran 𝐹 ↔ (𝑣 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) ∈ ran 𝐹))
92 f1ocnvfv2 7286 . . . . . . . . . . 11 ((𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] )) ∈ (ℚ ∩ (-1[,]1))) → (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] )))) = (𝑣 − (𝐹‘[𝑣] )))
9327, 88, 92syl2an2r 683 . . . . . . . . . 10 ((𝜑𝑣 ∈ (0[,]1)) → (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] )))) = (𝑣 − (𝐹‘[𝑣] )))
9493oveq2d 7435 . . . . . . . . 9 ((𝜑𝑣 ∈ (0[,]1)) → (𝑣 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) = (𝑣 − (𝑣 − (𝐹‘[𝑣] ))))
9576, 75nncand 11608 . . . . . . . . 9 ((𝜑𝑣 ∈ (0[,]1)) → (𝑣 − (𝑣 − (𝐹‘[𝑣] ))) = (𝐹‘[𝑣] ))
9694, 95eqtrd 2765 . . . . . . . 8 ((𝜑𝑣 ∈ (0[,]1)) → (𝑣 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) = (𝐹‘[𝑣] ))
97 fnfvelrn 7089 . . . . . . . . 9 ((𝐹 Fn 𝑆 ∧ [𝑣] 𝑆) → (𝐹‘[𝑣] ) ∈ ran 𝐹)
981, 45, 97syl2an2r 683 . . . . . . . 8 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ) ∈ ran 𝐹)
9996, 98eqeltrd 2825 . . . . . . 7 ((𝜑𝑣 ∈ (0[,]1)) → (𝑣 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) ∈ ran 𝐹)
10091, 59, 99elrabd 3681 . . . . . 6 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) ∈ ran 𝐹})
101 fveq2 6896 . . . . . . . . . . 11 (𝑛 = (𝐺‘(𝑣 − (𝐹‘[𝑣] ))) → (𝐺𝑛) = (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] )))))
102101oveq2d 7435 . . . . . . . . . 10 (𝑛 = (𝐺‘(𝑣 − (𝐹‘[𝑣] ))) → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))))
103102eleq1d 2810 . . . . . . . . 9 (𝑛 = (𝐺‘(𝑣 − (𝐹‘[𝑣] ))) → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) ∈ ran 𝐹))
104103rabbidv 3426 . . . . . . . 8 (𝑛 = (𝐺‘(𝑣 − (𝐹‘[𝑣] ))) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) ∈ ran 𝐹})
105 vitali.6 . . . . . . . 8 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})
106 reex 11231 . . . . . . . . 9 ℝ ∈ V
107106rabex 5335 . . . . . . . 8 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) ∈ ran 𝐹} ∈ V
108104, 105, 107fvmpt 7004 . . . . . . 7 ((𝐺‘(𝑣 − (𝐹‘[𝑣] ))) ∈ ℕ → (𝑇‘(𝐺‘(𝑣 − (𝐹‘[𝑣] )))) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) ∈ ran 𝐹})
10989, 108syl 17 . . . . . 6 ((𝜑𝑣 ∈ (0[,]1)) → (𝑇‘(𝐺‘(𝑣 − (𝐹‘[𝑣] )))) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) ∈ ran 𝐹})
110100, 109eleqtrrd 2828 . . . . 5 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 ∈ (𝑇‘(𝐺‘(𝑣 − (𝐹‘[𝑣] )))))
111 fveq2 6896 . . . . . 6 (𝑚 = (𝐺‘(𝑣 − (𝐹‘[𝑣] ))) → (𝑇𝑚) = (𝑇‘(𝐺‘(𝑣 − (𝐹‘[𝑣] )))))
112111eliuni 5003 . . . . 5 (((𝐺‘(𝑣 − (𝐹‘[𝑣] ))) ∈ ℕ ∧ 𝑣 ∈ (𝑇‘(𝐺‘(𝑣 − (𝐹‘[𝑣] ))))) → 𝑣 𝑚 ∈ ℕ (𝑇𝑚))
11389, 110, 112syl2anc 582 . . . 4 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 𝑚 ∈ ℕ (𝑇𝑚))
114113ex 411 . . 3 (𝜑 → (𝑣 ∈ (0[,]1) → 𝑣 𝑚 ∈ ℕ (𝑇𝑚)))
115114ssrdv 3982 . 2 (𝜑 → (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚))
116 eliun 5001 . . . 4 (𝑥 𝑚 ∈ ℕ (𝑇𝑚) ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑇𝑚))
117 fveq2 6896 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
118117oveq2d 7435 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑚)))
119118eleq1d 2810 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹))
120119rabbidv 3426 . . . . . . . . . . . 12 (𝑛 = 𝑚 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
121106rabex 5335 . . . . . . . . . . . 12 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ∈ V
122120, 105, 121fvmpt 7004 . . . . . . . . . . 11 (𝑚 ∈ ℕ → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
123122adantl 480 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
124123eleq2d 2811 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ (𝑇𝑚) ↔ 𝑥 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹}))
125124biimpa 475 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → 𝑥 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
126 oveq1 7426 . . . . . . . . . 10 (𝑠 = 𝑥 → (𝑠 − (𝐺𝑚)) = (𝑥 − (𝐺𝑚)))
127126eleq1d 2810 . . . . . . . . 9 (𝑠 = 𝑥 → ((𝑠 − (𝐺𝑚)) ∈ ran 𝐹 ↔ (𝑥 − (𝐺𝑚)) ∈ ran 𝐹))
128127elrab 3679 . . . . . . . 8 (𝑥 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ↔ (𝑥 ∈ ℝ ∧ (𝑥 − (𝐺𝑚)) ∈ ran 𝐹))
129125, 128sylib 217 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → (𝑥 ∈ ℝ ∧ (𝑥 − (𝐺𝑚)) ∈ ran 𝐹))
130129simpld 493 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → 𝑥 ∈ ℝ)
13184a1i 11 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → -1 ∈ ℝ)
132 iccssre 13441 . . . . . . . . . 10 ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → (-1[,]1) ⊆ ℝ)
13384, 85, 132mp2an 690 . . . . . . . . 9 (-1[,]1) ⊆ ℝ
134 f1of 6838 . . . . . . . . . . . 12 (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
13527, 134syl 17 . . . . . . . . . . 11 (𝜑𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
136135ffvelcdmda 7093 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ (ℚ ∩ (-1[,]1)))
137136elin2d 4197 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ (-1[,]1))
138133, 137sselid 3974 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ ℝ)
139138adantr 479 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → (𝐺𝑚) ∈ ℝ)
140137adantr 479 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → (𝐺𝑚) ∈ (-1[,]1))
14184, 85elicc2i 13425 . . . . . . . . 9 ((𝐺𝑚) ∈ (-1[,]1) ↔ ((𝐺𝑚) ∈ ℝ ∧ -1 ≤ (𝐺𝑚) ∧ (𝐺𝑚) ≤ 1))
142140, 141sylib 217 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → ((𝐺𝑚) ∈ ℝ ∧ -1 ≤ (𝐺𝑚) ∧ (𝐺𝑚) ≤ 1))
143142simp2d 1140 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → -1 ≤ (𝐺𝑚))
14426ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → ran 𝐹 ⊆ (0[,]1))
145129simprd 494 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → (𝑥 − (𝐺𝑚)) ∈ ran 𝐹)
146144, 145sseldd 3977 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → (𝑥 − (𝐺𝑚)) ∈ (0[,]1))
147 elicc01 13478 . . . . . . . . . 10 ((𝑥 − (𝐺𝑚)) ∈ (0[,]1) ↔ ((𝑥 − (𝐺𝑚)) ∈ ℝ ∧ 0 ≤ (𝑥 − (𝐺𝑚)) ∧ (𝑥 − (𝐺𝑚)) ≤ 1))
148146, 147sylib 217 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → ((𝑥 − (𝐺𝑚)) ∈ ℝ ∧ 0 ≤ (𝑥 − (𝐺𝑚)) ∧ (𝑥 − (𝐺𝑚)) ≤ 1))
149148simp2d 1140 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → 0 ≤ (𝑥 − (𝐺𝑚)))
150130, 139subge0d 11836 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → (0 ≤ (𝑥 − (𝐺𝑚)) ↔ (𝐺𝑚) ≤ 𝑥))
151149, 150mpbid 231 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → (𝐺𝑚) ≤ 𝑥)
152131, 139, 130, 143, 151letrd 11403 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → -1 ≤ 𝑥)
153 peano2re 11419 . . . . . . . 8 ((𝐺𝑚) ∈ ℝ → ((𝐺𝑚) + 1) ∈ ℝ)
154139, 153syl 17 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → ((𝐺𝑚) + 1) ∈ ℝ)
155 2re 12319 . . . . . . . 8 2 ∈ ℝ
156155a1i 11 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → 2 ∈ ℝ)
157148simp3d 1141 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → (𝑥 − (𝐺𝑚)) ≤ 1)
158 1red 11247 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → 1 ∈ ℝ)
159130, 139, 158lesubadd2d 11845 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → ((𝑥 − (𝐺𝑚)) ≤ 1 ↔ 𝑥 ≤ ((𝐺𝑚) + 1)))
160157, 159mpbid 231 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → 𝑥 ≤ ((𝐺𝑚) + 1))
161142simp3d 1141 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → (𝐺𝑚) ≤ 1)
162139, 158, 158, 161leadd1dd 11860 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → ((𝐺𝑚) + 1) ≤ (1 + 1))
163 df-2 12308 . . . . . . . 8 2 = (1 + 1)
164162, 163breqtrrdi 5191 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → ((𝐺𝑚) + 1) ≤ 2)
165130, 154, 156, 160, 164letrd 11403 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → 𝑥 ≤ 2)
16684, 155elicc2i 13425 . . . . . 6 (𝑥 ∈ (-1[,]2) ↔ (𝑥 ∈ ℝ ∧ -1 ≤ 𝑥𝑥 ≤ 2))
167130, 152, 165, 166syl3anbrc 1340 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇𝑚)) → 𝑥 ∈ (-1[,]2))
168167rexlimdva2 3146 . . . 4 (𝜑 → (∃𝑚 ∈ ℕ 𝑥 ∈ (𝑇𝑚) → 𝑥 ∈ (-1[,]2)))
169116, 168biimtrid 241 . . 3 (𝜑 → (𝑥 𝑚 ∈ ℕ (𝑇𝑚) → 𝑥 ∈ (-1[,]2)))
170169ssrdv 3982 . 2 (𝜑 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2))
17126, 115, 1703jca 1125 1 (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚) ∧ 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  cdif 3941  cin 3943  wss 3944  c0 4322  𝒫 cpw 4604   ciun 4997   class class class wbr 5149  {copab 5211  cmpt 5232  ccnv 5677  dom cdm 5678  ran crn 5679   Fn wfn 6544  wf 6545  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419   Er wer 8722  [cec 8723   / cqs 8724  cr 11139  0cc0 11140  1c1 11141   + caddc 11143  cle 11281  cmin 11476  -cneg 11477  cn 12245  2c2 12300  cq 12965  [,]cicc 13362  volcvol 25436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-ec 8727  df-qs 8731  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-n0 12506  df-z 12592  df-q 12966  df-icc 13366
This theorem is referenced by:  vitalilem3  25583  vitalilem4  25584  vitalilem5  25585
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