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Theorem vitalilem3 24774
Description: Lemma for vitali 24777. (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
vitalilem3 (𝜑Disj 𝑚 ∈ ℕ (𝑇𝑚))
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 vitalilem3
Dummy variables 𝑘 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprlr 777 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ (𝑇𝑚))
2 simprll 776 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑚 ∈ ℕ)
3 fveq2 6774 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
43oveq2d 7291 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑚)))
54eleq1d 2823 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹))
65rabbidv 3414 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
7 vitali.6 . . . . . . . . . . . . . 14 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})
8 reex 10962 . . . . . . . . . . . . . . 15 ℝ ∈ V
98rabex 5256 . . . . . . . . . . . . . 14 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ∈ V
106, 7, 9fvmpt 6875 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
112, 10syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
121, 11eleqtrd 2841 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
13 oveq1 7282 . . . . . . . . . . . . 13 (𝑠 = 𝑤 → (𝑠 − (𝐺𝑚)) = (𝑤 − (𝐺𝑚)))
1413eleq1d 2823 . . . . . . . . . . . 12 (𝑠 = 𝑤 → ((𝑠 − (𝐺𝑚)) ∈ ran 𝐹 ↔ (𝑤 − (𝐺𝑚)) ∈ ran 𝐹))
1514elrab 3624 . . . . . . . . . . 11 (𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ↔ (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑚)) ∈ ran 𝐹))
1612, 15sylib 217 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑚)) ∈ ran 𝐹))
1716simpld 495 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ ℝ)
1817recnd 11003 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ ℂ)
19 vitali.5 . . . . . . . . . . . . 13 (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
20 f1of 6716 . . . . . . . . . . . . 13 (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
2119, 20syl 17 . . . . . . . . . . . 12 (𝜑𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
22 inss1 4162 . . . . . . . . . . . 12 (ℚ ∩ (-1[,]1)) ⊆ ℚ
23 fss 6617 . . . . . . . . . . . 12 ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℚ) → 𝐺:ℕ⟶ℚ)
2421, 22, 23sylancl 586 . . . . . . . . . . 11 (𝜑𝐺:ℕ⟶ℚ)
2524adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝐺:ℕ⟶ℚ)
2625, 2ffvelrnd 6962 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑚) ∈ ℚ)
27 qcn 12703 . . . . . . . . 9 ((𝐺𝑚) ∈ ℚ → (𝐺𝑚) ∈ ℂ)
2826, 27syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑚) ∈ ℂ)
29 simprrl 778 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑘 ∈ ℕ)
3025, 29ffvelrnd 6962 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑘) ∈ ℚ)
31 qcn 12703 . . . . . . . . 9 ((𝐺𝑘) ∈ ℚ → (𝐺𝑘) ∈ ℂ)
3230, 31syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑘) ∈ ℂ)
33 vitali.1 . . . . . . . . . . . . 13 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
3433vitalilem1 24772 . . . . . . . . . . . 12 Er (0[,]1)
3534a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → Er (0[,]1))
36 vitali.2 . . . . . . . . . . . . . . . 16 𝑆 = ((0[,]1) / )
37 vitali.3 . . . . . . . . . . . . . . . 16 (𝜑𝐹 Fn 𝑆)
38 vitali.4 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
39 vitali.7 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
4033, 36, 37, 38, 19, 7, 39vitalilem2 24773 . . . . . . . . . . . . . . 15 (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚) ∧ 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2)))
4140simp1d 1141 . . . . . . . . . . . . . 14 (𝜑 → ran 𝐹 ⊆ (0[,]1))
4241adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ran 𝐹 ⊆ (0[,]1))
4316simprd 496 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) ∈ ran 𝐹)
4442, 43sseldd 3922 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) ∈ (0[,]1))
45 simprrr 779 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ (𝑇𝑘))
46 fveq2 6774 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
4746oveq2d 7291 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑘 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑘)))
4847eleq1d 2823 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹))
4948rabbidv 3414 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
508rabex 5256 . . . . . . . . . . . . . . . . . 18 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹} ∈ V
5149, 7, 50fvmpt 6875 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → (𝑇𝑘) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
5229, 51syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑇𝑘) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
5345, 52eleqtrd 2841 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
54 oveq1 7282 . . . . . . . . . . . . . . . . 17 (𝑠 = 𝑤 → (𝑠 − (𝐺𝑘)) = (𝑤 − (𝐺𝑘)))
5554eleq1d 2823 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑤 → ((𝑠 − (𝐺𝑘)) ∈ ran 𝐹 ↔ (𝑤 − (𝐺𝑘)) ∈ ran 𝐹))
5655elrab 3624 . . . . . . . . . . . . . . 15 (𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹} ↔ (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑘)) ∈ ran 𝐹))
5753, 56sylib 217 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑘)) ∈ ran 𝐹))
5857simprd 496 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑘)) ∈ ran 𝐹)
5942, 58sseldd 3922 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑘)) ∈ (0[,]1))
6018, 28, 32nnncan1d 11366 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) = ((𝐺𝑘) − (𝐺𝑚)))
61 qsubcl 12708 . . . . . . . . . . . . . 14 (((𝐺𝑘) ∈ ℚ ∧ (𝐺𝑚) ∈ ℚ) → ((𝐺𝑘) − (𝐺𝑚)) ∈ ℚ)
6230, 26, 61syl2anc 584 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝐺𝑘) − (𝐺𝑚)) ∈ ℚ)
6360, 62eqeltrd 2839 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) ∈ ℚ)
64 oveq12 7284 . . . . . . . . . . . . . 14 ((𝑥 = (𝑤 − (𝐺𝑚)) ∧ 𝑦 = (𝑤 − (𝐺𝑘))) → (𝑥𝑦) = ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))))
6564eleq1d 2823 . . . . . . . . . . . . 13 ((𝑥 = (𝑤 − (𝐺𝑚)) ∧ 𝑦 = (𝑤 − (𝐺𝑘))) → ((𝑥𝑦) ∈ ℚ ↔ ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) ∈ ℚ))
6665, 33brab2a 5680 . . . . . . . . . . . 12 ((𝑤 − (𝐺𝑚)) (𝑤 − (𝐺𝑘)) ↔ (((𝑤 − (𝐺𝑚)) ∈ (0[,]1) ∧ (𝑤 − (𝐺𝑘)) ∈ (0[,]1)) ∧ ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) ∈ ℚ))
6744, 59, 63, 66syl21anbrc 1343 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) (𝑤 − (𝐺𝑘)))
6835, 67erthi 8549 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → [(𝑤 − (𝐺𝑚))] = [(𝑤 − (𝐺𝑘))] )
6968fveq2d 6778 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝐹‘[(𝑤 − (𝐺𝑘))] ))
70 eceq1 8536 . . . . . . . . . . . 12 (𝑧 = (𝑤 − (𝐺𝑚)) → [𝑧] = [(𝑤 − (𝐺𝑚))] )
7170fveq2d 6778 . . . . . . . . . . 11 (𝑧 = (𝑤 − (𝐺𝑚)) → (𝐹‘[𝑧] ) = (𝐹‘[(𝑤 − (𝐺𝑚))] ))
72 id 22 . . . . . . . . . . 11 (𝑧 = (𝑤 − (𝐺𝑚)) → 𝑧 = (𝑤 − (𝐺𝑚)))
7371, 72eqeq12d 2754 . . . . . . . . . 10 (𝑧 = (𝑤 − (𝐺𝑚)) → ((𝐹‘[𝑧] ) = 𝑧 ↔ (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝑤 − (𝐺𝑚))))
74 fveq2 6774 . . . . . . . . . . . . . . . . 17 ([𝑣] = 𝑤 → (𝐹‘[𝑣] ) = (𝐹𝑤))
7574eceq1d 8537 . . . . . . . . . . . . . . . 16 ([𝑣] = 𝑤 → [(𝐹‘[𝑣] )] = [(𝐹𝑤)] )
7675fveq2d 6778 . . . . . . . . . . . . . . 15 ([𝑣] = 𝑤 → (𝐹‘[(𝐹‘[𝑣] )] ) = (𝐹‘[(𝐹𝑤)] ))
7776, 74eqeq12d 2754 . . . . . . . . . . . . . 14 ([𝑣] = 𝑤 → ((𝐹‘[(𝐹‘[𝑣] )] ) = (𝐹‘[𝑣] ) ↔ (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
7834a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 ∈ (0[,]1)) → Er (0[,]1))
79 simpr 485 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 ∈ (0[,]1))
80 erdm 8508 . . . . . . . . . . . . . . . . . . . . . . 23 ( Er (0[,]1) → dom = (0[,]1))
8134, 80ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 dom = (0[,]1)
8281eleq2i 2830 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ dom 𝑣 ∈ (0[,]1))
83 ecdmn0 8545 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ dom ↔ [𝑣] ≠ ∅)
8482, 83bitr3i 276 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ (0[,]1) ↔ [𝑣] ≠ ∅)
8579, 84sylib 217 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] ≠ ∅)
86 neeq1 3006 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = [𝑣] → (𝑧 ≠ ∅ ↔ [𝑣] ≠ ∅))
87 fveq2 6774 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = [𝑣] → (𝐹𝑧) = (𝐹‘[𝑣] ))
88 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = [𝑣] 𝑧 = [𝑣] )
8987, 88eleq12d 2833 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = [𝑣] → ((𝐹𝑧) ∈ 𝑧 ↔ (𝐹‘[𝑣] ) ∈ [𝑣] ))
9086, 89imbi12d 345 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = [𝑣] → ((𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧) ↔ ([𝑣] ≠ ∅ → (𝐹‘[𝑣] ) ∈ [𝑣] )))
9138adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑣 ∈ (0[,]1)) → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
92 ovex 7308 . . . . . . . . . . . . . . . . . . . . . . . 24 (0[,]1) ∈ V
93 erex 8522 . . . . . . . . . . . . . . . . . . . . . . . 24 ( Er (0[,]1) → ((0[,]1) ∈ V → ∈ V))
9434, 92, 93mp2 9 . . . . . . . . . . . . . . . . . . . . . . 23 ∈ V
9594ecelqsi 8562 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ∈ (0[,]1) → [𝑣] ∈ ((0[,]1) / ))
9695, 36eleqtrrdi 2850 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ (0[,]1) → [𝑣] 𝑆)
9796adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] 𝑆)
9890, 91, 97rspcdva 3562 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 ∈ (0[,]1)) → ([𝑣] ≠ ∅ → (𝐹‘[𝑣] ) ∈ [𝑣] ))
9985, 98mpd 15 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ) ∈ [𝑣] )
100 fvex 6787 . . . . . . . . . . . . . . . . . . 19 (𝐹‘[𝑣] ) ∈ V
101 vex 3436 . . . . . . . . . . . . . . . . . . 19 𝑣 ∈ V
102100, 101elec 8542 . . . . . . . . . . . . . . . . . 18 ((𝐹‘[𝑣] ) ∈ [𝑣] 𝑣 (𝐹‘[𝑣] ))
10399, 102sylib 217 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 (𝐹‘[𝑣] ))
10478, 103erthi 8549 . . . . . . . . . . . . . . . 16 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] = [(𝐹‘[𝑣] )] )
105104eqcomd 2744 . . . . . . . . . . . . . . 15 ((𝜑𝑣 ∈ (0[,]1)) → [(𝐹‘[𝑣] )] = [𝑣] )
106105fveq2d 6778 . . . . . . . . . . . . . 14 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[(𝐹‘[𝑣] )] ) = (𝐹‘[𝑣] ))
10736, 77, 106ectocld 8573 . . . . . . . . . . . . 13 ((𝜑𝑤𝑆) → (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤))
108107ralrimiva 3103 . . . . . . . . . . . 12 (𝜑 → ∀𝑤𝑆 (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤))
109 eceq1 8536 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑤) → [𝑧] = [(𝐹𝑤)] )
110109fveq2d 6778 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑤) → (𝐹‘[𝑧] ) = (𝐹‘[(𝐹𝑤)] ))
111 id 22 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑤) → 𝑧 = (𝐹𝑤))
112110, 111eqeq12d 2754 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑤) → ((𝐹‘[𝑧] ) = 𝑧 ↔ (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
113112ralrn 6964 . . . . . . . . . . . . 13 (𝐹 Fn 𝑆 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧 ↔ ∀𝑤𝑆 (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
11437, 113syl 17 . . . . . . . . . . . 12 (𝜑 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧 ↔ ∀𝑤𝑆 (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
115108, 114mpbird 256 . . . . . . . . . . 11 (𝜑 → ∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧)
116115adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧)
11773, 116, 43rspcdva 3562 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝑤 − (𝐺𝑚)))
118 eceq1 8536 . . . . . . . . . . . 12 (𝑧 = (𝑤 − (𝐺𝑘)) → [𝑧] = [(𝑤 − (𝐺𝑘))] )
119118fveq2d 6778 . . . . . . . . . . 11 (𝑧 = (𝑤 − (𝐺𝑘)) → (𝐹‘[𝑧] ) = (𝐹‘[(𝑤 − (𝐺𝑘))] ))
120 id 22 . . . . . . . . . . 11 (𝑧 = (𝑤 − (𝐺𝑘)) → 𝑧 = (𝑤 − (𝐺𝑘)))
121119, 120eqeq12d 2754 . . . . . . . . . 10 (𝑧 = (𝑤 − (𝐺𝑘)) → ((𝐹‘[𝑧] ) = 𝑧 ↔ (𝐹‘[(𝑤 − (𝐺𝑘))] ) = (𝑤 − (𝐺𝑘))))
122121, 116, 58rspcdva 3562 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐹‘[(𝑤 − (𝐺𝑘))] ) = (𝑤 − (𝐺𝑘)))
12369, 117, 1223eqtr3d 2786 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) = (𝑤 − (𝐺𝑘)))
12418, 28, 32, 123subcand 11373 . . . . . . 7 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑚) = (𝐺𝑘))
12519adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
126 f1of1 6715 . . . . . . . . 9 (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)))
127125, 126syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)))
128 f1fveq 7135 . . . . . . . 8 ((𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)) ∧ (𝑚 ∈ ℕ ∧ 𝑘 ∈ ℕ)) → ((𝐺𝑚) = (𝐺𝑘) ↔ 𝑚 = 𝑘))
129127, 2, 29, 128syl12anc 834 . . . . . . 7 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝐺𝑚) = (𝐺𝑘) ↔ 𝑚 = 𝑘))
130124, 129mpbid 231 . . . . . 6 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑚 = 𝑘)
131130ex 413 . . . . 5 (𝜑 → (((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))) → 𝑚 = 𝑘))
132131alrimivv 1931 . . . 4 (𝜑 → ∀𝑚𝑘(((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))) → 𝑚 = 𝑘))
133 eleq1w 2821 . . . . . 6 (𝑚 = 𝑘 → (𝑚 ∈ ℕ ↔ 𝑘 ∈ ℕ))
134 fveq2 6774 . . . . . . 7 (𝑚 = 𝑘 → (𝑇𝑚) = (𝑇𝑘))
135134eleq2d 2824 . . . . . 6 (𝑚 = 𝑘 → (𝑤 ∈ (𝑇𝑚) ↔ 𝑤 ∈ (𝑇𝑘)))
136133, 135anbi12d 631 . . . . 5 (𝑚 = 𝑘 → ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ↔ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))))
137136mo4 2566 . . . 4 (∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ↔ ∀𝑚𝑘(((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))) → 𝑚 = 𝑘))
138132, 137sylibr 233 . . 3 (𝜑 → ∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)))
139138alrimiv 1930 . 2 (𝜑 → ∀𝑤∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)))
140 dfdisj2 5041 . 2 (Disj 𝑚 ∈ ℕ (𝑇𝑚) ↔ ∀𝑤∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)))
141139, 140sylibr 233 1 (𝜑Disj 𝑚 ∈ ℕ (𝑇𝑚))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1537   = wceq 1539  wcel 2106  ∃*wmo 2538  wne 2943  wral 3064  {crab 3068  Vcvv 3432  cdif 3884  cin 3886  wss 3887  c0 4256  𝒫 cpw 4533   ciun 4924  Disj wdisj 5039   class class class wbr 5074  {copab 5136  cmpt 5157  dom cdm 5589  ran crn 5590   Fn wfn 6428  wf 6429  1-1wf1 6430  1-1-ontowf1o 6432  cfv 6433  (class class class)co 7275   Er wer 8495  [cec 8496   / cqs 8497  cc 10869  cr 10870  0cc0 10871  1c1 10872  cmin 11205  -cneg 11206  cn 11973  2c2 12028  cq 12688  [,]cicc 13082  volcvol 24627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-disj 5040  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-ec 8500  df-qs 8504  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-q 12689  df-icc 13086
This theorem is referenced by:  vitalilem4  24775
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