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Theorem vitalilem3 23424
Description: Lemma for vitali 23427. (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 820 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ (𝑇𝑚))
2 simprll 819 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑚 ∈ ℕ)
3 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
43oveq2d 6706 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑚)))
54eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹))
65rabbidv 3220 . . . . . . . . . . . . . 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 10065 . . . . . . . . . . . . . . 15 ℝ ∈ V
98rabex 4845 . . . . . . . . . . . . . 14 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ∈ V
106, 7, 9fvmpt 6321 . . . . . . . . . . . . 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 2732 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
13 oveq1 6697 . . . . . . . . . . . . 13 (𝑠 = 𝑤 → (𝑠 − (𝐺𝑚)) = (𝑤 − (𝐺𝑚)))
1413eleq1d 2715 . . . . . . . . . . . 12 (𝑠 = 𝑤 → ((𝑠 − (𝐺𝑚)) ∈ ran 𝐹 ↔ (𝑤 − (𝐺𝑚)) ∈ ran 𝐹))
1514elrab 3396 . . . . . . . . . . 11 (𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ↔ (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑚)) ∈ ran 𝐹))
1612, 15sylib 208 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑚)) ∈ ran 𝐹))
1716simpld 474 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ ℝ)
1817recnd 10106 . . . . . . . 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 6175 . . . . . . . . . . . . 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 3866 . . . . . . . . . . . 12 (ℚ ∩ (-1[,]1)) ⊆ ℚ
23 fss 6094 . . . . . . . . . . . 12 ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℚ) → 𝐺:ℕ⟶ℚ)
2421, 22, 23sylancl 695 . . . . . . . . . . 11 (𝜑𝐺:ℕ⟶ℚ)
2524adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝐺:ℕ⟶ℚ)
2625, 2ffvelrnd 6400 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑚) ∈ ℚ)
27 qcn 11840 . . . . . . . . 9 ((𝐺𝑚) ∈ ℚ → (𝐺𝑚) ∈ ℂ)
2826, 27syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑚) ∈ ℂ)
29 simprrl 821 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑘 ∈ ℕ)
3025, 29ffvelrnd 6400 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑘) ∈ ℚ)
31 qcn 11840 . . . . . . . . 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 23422 . . . . . . . . . . . 12 Er (0[,]1)
3534a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → Er (0[,]1))
36 vitali.2 . . . . . . . . . . . . . . . . 17 𝑆 = ((0[,]1) / )
37 vitali.3 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 Fn 𝑆)
38 vitali.4 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
39 vitali.7 . . . . . . . . . . . . . . . . 17 (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
4033, 36, 37, 38, 19, 7, 39vitalilem2 23423 . . . . . . . . . . . . . . . 16 (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚) ∧ 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2)))
4140simp1d 1093 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ⊆ (0[,]1))
4241adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ran 𝐹 ⊆ (0[,]1))
4316simprd 478 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) ∈ ran 𝐹)
4442, 43sseldd 3637 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) ∈ (0[,]1))
45 simprrr 822 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ (𝑇𝑘))
46 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
4746oveq2d 6706 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑘)))
4847eleq1d 2715 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑘 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹))
4948rabbidv 3220 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
508rabex 4845 . . . . . . . . . . . . . . . . . . 19 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹} ∈ V
5149, 7, 50fvmpt 6321 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → (𝑇𝑘) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
5229, 51syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑇𝑘) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
5345, 52eleqtrd 2732 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
54 oveq1 6697 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑤 → (𝑠 − (𝐺𝑘)) = (𝑤 − (𝐺𝑘)))
5554eleq1d 2715 . . . . . . . . . . . . . . . . 17 (𝑠 = 𝑤 → ((𝑠 − (𝐺𝑘)) ∈ ran 𝐹 ↔ (𝑤 − (𝐺𝑘)) ∈ ran 𝐹))
5655elrab 3396 . . . . . . . . . . . . . . . 16 (𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹} ↔ (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑘)) ∈ ran 𝐹))
5753, 56sylib 208 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑘)) ∈ ran 𝐹))
5857simprd 478 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑘)) ∈ ran 𝐹)
5942, 58sseldd 3637 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑘)) ∈ (0[,]1))
6044, 59jca 553 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝑤 − (𝐺𝑚)) ∈ (0[,]1) ∧ (𝑤 − (𝐺𝑘)) ∈ (0[,]1)))
6118, 28, 32nnncan1d 10464 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) = ((𝐺𝑘) − (𝐺𝑚)))
62 qsubcl 11845 . . . . . . . . . . . . . 14 (((𝐺𝑘) ∈ ℚ ∧ (𝐺𝑚) ∈ ℚ) → ((𝐺𝑘) − (𝐺𝑚)) ∈ ℚ)
6330, 26, 62syl2anc 694 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝐺𝑘) − (𝐺𝑚)) ∈ ℚ)
6461, 63eqeltrd 2730 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) ∈ ℚ)
65 oveq12 6699 . . . . . . . . . . . . . 14 ((𝑥 = (𝑤 − (𝐺𝑚)) ∧ 𝑦 = (𝑤 − (𝐺𝑘))) → (𝑥𝑦) = ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))))
6665eleq1d 2715 . . . . . . . . . . . . 13 ((𝑥 = (𝑤 − (𝐺𝑚)) ∧ 𝑦 = (𝑤 − (𝐺𝑘))) → ((𝑥𝑦) ∈ ℚ ↔ ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) ∈ ℚ))
6766, 33brab2a 5228 . . . . . . . . . . . 12 ((𝑤 − (𝐺𝑚)) (𝑤 − (𝐺𝑘)) ↔ (((𝑤 − (𝐺𝑚)) ∈ (0[,]1) ∧ (𝑤 − (𝐺𝑘)) ∈ (0[,]1)) ∧ ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) ∈ ℚ))
6860, 64, 67sylanbrc 699 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) (𝑤 − (𝐺𝑘)))
6935, 68erthi 7836 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → [(𝑤 − (𝐺𝑚))] = [(𝑤 − (𝐺𝑘))] )
7069fveq2d 6233 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝐹‘[(𝑤 − (𝐺𝑘))] ))
71 fveq2 6229 . . . . . . . . . . . . . . . . 17 ([𝑣] = 𝑤 → (𝐹‘[𝑣] ) = (𝐹𝑤))
7271eceq1d 7826 . . . . . . . . . . . . . . . 16 ([𝑣] = 𝑤 → [(𝐹‘[𝑣] )] = [(𝐹𝑤)] )
7372fveq2d 6233 . . . . . . . . . . . . . . 15 ([𝑣] = 𝑤 → (𝐹‘[(𝐹‘[𝑣] )] ) = (𝐹‘[(𝐹𝑤)] ))
7473, 71eqeq12d 2666 . . . . . . . . . . . . . 14 ([𝑣] = 𝑤 → ((𝐹‘[(𝐹‘[𝑣] )] ) = (𝐹‘[𝑣] ) ↔ (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
7534a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 ∈ (0[,]1)) → Er (0[,]1))
76 ovex 6718 . . . . . . . . . . . . . . . . . . . . . . 23 (0[,]1) ∈ V
77 erex 7811 . . . . . . . . . . . . . . . . . . . . . . 23 ( Er (0[,]1) → ((0[,]1) ∈ V → ∈ V))
7834, 76, 77mp2 9 . . . . . . . . . . . . . . . . . . . . . 22 ∈ V
7978ecelqsi 7846 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ (0[,]1) → [𝑣] ∈ ((0[,]1) / ))
8079, 36syl6eleqr 2741 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ (0[,]1) → [𝑣] 𝑆)
8180adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] 𝑆)
8238adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 ∈ (0[,]1)) → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
83 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 ∈ (0[,]1))
84 erdm 7797 . . . . . . . . . . . . . . . . . . . . . . 23 ( Er (0[,]1) → dom = (0[,]1))
8534, 84ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 dom = (0[,]1)
8685eleq2i 2722 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ dom 𝑣 ∈ (0[,]1))
87 ecdmn0 7832 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ dom ↔ [𝑣] ≠ ∅)
8886, 87bitr3i 266 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ (0[,]1) ↔ [𝑣] ≠ ∅)
8983, 88sylib 208 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] ≠ ∅)
90 neeq1 2885 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = [𝑣] → (𝑧 ≠ ∅ ↔ [𝑣] ≠ ∅))
91 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = [𝑣] → (𝐹𝑧) = (𝐹‘[𝑣] ))
92 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = [𝑣] 𝑧 = [𝑣] )
9391, 92eleq12d 2724 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = [𝑣] → ((𝐹𝑧) ∈ 𝑧 ↔ (𝐹‘[𝑣] ) ∈ [𝑣] ))
9490, 93imbi12d 333 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = [𝑣] → ((𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧) ↔ ([𝑣] ≠ ∅ → (𝐹‘[𝑣] ) ∈ [𝑣] )))
9594rspcv 3336 . . . . . . . . . . . . . . . . . . 19 ([𝑣] 𝑆 → (∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧) → ([𝑣] ≠ ∅ → (𝐹‘[𝑣] ) ∈ [𝑣] )))
9681, 82, 89, 95syl3c 66 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ) ∈ [𝑣] )
97 fvex 6239 . . . . . . . . . . . . . . . . . . 19 (𝐹‘[𝑣] ) ∈ V
98 vex 3234 . . . . . . . . . . . . . . . . . . 19 𝑣 ∈ V
9997, 98elec 7829 . . . . . . . . . . . . . . . . . 18 ((𝐹‘[𝑣] ) ∈ [𝑣] 𝑣 (𝐹‘[𝑣] ))
10096, 99sylib 208 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 (𝐹‘[𝑣] ))
10175, 100erthi 7836 . . . . . . . . . . . . . . . 16 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] = [(𝐹‘[𝑣] )] )
102101eqcomd 2657 . . . . . . . . . . . . . . 15 ((𝜑𝑣 ∈ (0[,]1)) → [(𝐹‘[𝑣] )] = [𝑣] )
103102fveq2d 6233 . . . . . . . . . . . . . 14 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[(𝐹‘[𝑣] )] ) = (𝐹‘[𝑣] ))
10436, 74, 103ectocld 7857 . . . . . . . . . . . . 13 ((𝜑𝑤𝑆) → (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤))
105104ralrimiva 2995 . . . . . . . . . . . 12 (𝜑 → ∀𝑤𝑆 (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤))
106 eceq1 7825 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑤) → [𝑧] = [(𝐹𝑤)] )
107106fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑤) → (𝐹‘[𝑧] ) = (𝐹‘[(𝐹𝑤)] ))
108 id 22 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑤) → 𝑧 = (𝐹𝑤))
109107, 108eqeq12d 2666 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑤) → ((𝐹‘[𝑧] ) = 𝑧 ↔ (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
110109ralrn 6402 . . . . . . . . . . . . 13 (𝐹 Fn 𝑆 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧 ↔ ∀𝑤𝑆 (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
11137, 110syl 17 . . . . . . . . . . . 12 (𝜑 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧 ↔ ∀𝑤𝑆 (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
112105, 111mpbird 247 . . . . . . . . . . 11 (𝜑 → ∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧)
113112adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧)
114 eceq1 7825 . . . . . . . . . . . . 13 (𝑧 = (𝑤 − (𝐺𝑚)) → [𝑧] = [(𝑤 − (𝐺𝑚))] )
115114fveq2d 6233 . . . . . . . . . . . 12 (𝑧 = (𝑤 − (𝐺𝑚)) → (𝐹‘[𝑧] ) = (𝐹‘[(𝑤 − (𝐺𝑚))] ))
116 id 22 . . . . . . . . . . . 12 (𝑧 = (𝑤 − (𝐺𝑚)) → 𝑧 = (𝑤 − (𝐺𝑚)))
117115, 116eqeq12d 2666 . . . . . . . . . . 11 (𝑧 = (𝑤 − (𝐺𝑚)) → ((𝐹‘[𝑧] ) = 𝑧 ↔ (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝑤 − (𝐺𝑚))))
118117rspcv 3336 . . . . . . . . . 10 ((𝑤 − (𝐺𝑚)) ∈ ran 𝐹 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧 → (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝑤 − (𝐺𝑚))))
11943, 113, 118sylc 65 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝑤 − (𝐺𝑚)))
120 eceq1 7825 . . . . . . . . . . . . 13 (𝑧 = (𝑤 − (𝐺𝑘)) → [𝑧] = [(𝑤 − (𝐺𝑘))] )
121120fveq2d 6233 . . . . . . . . . . . 12 (𝑧 = (𝑤 − (𝐺𝑘)) → (𝐹‘[𝑧] ) = (𝐹‘[(𝑤 − (𝐺𝑘))] ))
122 id 22 . . . . . . . . . . . 12 (𝑧 = (𝑤 − (𝐺𝑘)) → 𝑧 = (𝑤 − (𝐺𝑘)))
123121, 122eqeq12d 2666 . . . . . . . . . . 11 (𝑧 = (𝑤 − (𝐺𝑘)) → ((𝐹‘[𝑧] ) = 𝑧 ↔ (𝐹‘[(𝑤 − (𝐺𝑘))] ) = (𝑤 − (𝐺𝑘))))
124123rspcv 3336 . . . . . . . . . 10 ((𝑤 − (𝐺𝑘)) ∈ ran 𝐹 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧 → (𝐹‘[(𝑤 − (𝐺𝑘))] ) = (𝑤 − (𝐺𝑘))))
12558, 113, 124sylc 65 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐹‘[(𝑤 − (𝐺𝑘))] ) = (𝑤 − (𝐺𝑘)))
12670, 119, 1253eqtr3d 2693 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) = (𝑤 − (𝐺𝑘)))
12718, 28, 32, 126subcand 10471 . . . . . . 7 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑚) = (𝐺𝑘))
12819adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
129 f1of1 6174 . . . . . . . . 9 (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)))
130128, 129syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)))
131 f1fveq 6559 . . . . . . . 8 ((𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)) ∧ (𝑚 ∈ ℕ ∧ 𝑘 ∈ ℕ)) → ((𝐺𝑚) = (𝐺𝑘) ↔ 𝑚 = 𝑘))
132130, 2, 29, 131syl12anc 1364 . . . . . . 7 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝐺𝑚) = (𝐺𝑘) ↔ 𝑚 = 𝑘))
133127, 132mpbid 222 . . . . . 6 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑚 = 𝑘)
134133ex 449 . . . . 5 (𝜑 → (((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))) → 𝑚 = 𝑘))
135134alrimivv 1896 . . . 4 (𝜑 → ∀𝑚𝑘(((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))) → 𝑚 = 𝑘))
136 eleq1 2718 . . . . . 6 (𝑚 = 𝑘 → (𝑚 ∈ ℕ ↔ 𝑘 ∈ ℕ))
137 fveq2 6229 . . . . . . 7 (𝑚 = 𝑘 → (𝑇𝑚) = (𝑇𝑘))
138137eleq2d 2716 . . . . . 6 (𝑚 = 𝑘 → (𝑤 ∈ (𝑇𝑚) ↔ 𝑤 ∈ (𝑇𝑘)))
139136, 138anbi12d 747 . . . . 5 (𝑚 = 𝑘 → ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ↔ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))))
140139mo4 2546 . . . 4 (∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ↔ ∀𝑚𝑘(((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))) → 𝑚 = 𝑘))
141135, 140sylibr 224 . . 3 (𝜑 → ∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)))
142141alrimiv 1895 . 2 (𝜑 → ∀𝑤∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)))
143 dfdisj2 4654 . 2 (Disj 𝑚 ∈ ℕ (𝑇𝑚) ↔ ∀𝑤∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)))
144142, 143sylibr 224 1 (𝜑Disj 𝑚 ∈ ℕ (𝑇𝑚))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wcel 2030  ∃*wmo 2499  wne 2823  wral 2941  {crab 2945  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   ciun 4552  Disj wdisj 4652   class class class wbr 4685  {copab 4745  cmpt 4762  dom cdm 5143  ran crn 5144   Fn wfn 5921  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690   Er wer 7784  [cec 7785   / cqs 7786  cc 9972  cr 9973  0cc0 9974  1c1 9975  cmin 10304  -cneg 10305  cn 11058  2c2 11108  cq 11826  [,]cicc 12216  volcvol 23278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-ec 7789  df-qs 7793  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-q 11827  df-icc 12220
This theorem is referenced by:  vitalilem4  23425
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