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Theorem vitalilem3 23618
Description: Lemma for vitali 23621. (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 787 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ (𝑇𝑚))
2 simprll 786 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑚 ∈ ℕ)
3 fveq2 6348 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
43oveq2d 6828 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑚)))
54eleq1d 2838 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹))
65rabbidv 3343 . . . . . . . . . . . . . 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 10250 . . . . . . . . . . . . . . 15 ℝ ∈ V
98rabex 4960 . . . . . . . . . . . . . 14 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ∈ V
106, 7, 9fvmpt 6441 . . . . . . . . . . . . 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 2855 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
13 oveq1 6819 . . . . . . . . . . . . 13 (𝑠 = 𝑤 → (𝑠 − (𝐺𝑚)) = (𝑤 − (𝐺𝑚)))
1413eleq1d 2838 . . . . . . . . . . . 12 (𝑠 = 𝑤 → ((𝑠 − (𝐺𝑚)) ∈ ran 𝐹 ↔ (𝑤 − (𝐺𝑚)) ∈ ran 𝐹))
1514elrab 3521 . . . . . . . . . . 11 (𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ↔ (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑚)) ∈ ran 𝐹))
1612, 15sylib 209 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑚)) ∈ ran 𝐹))
1716simpld 483 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ ℝ)
1817recnd 10291 . . . . . . . 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 6293 . . . . . . . . . . . . 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 3988 . . . . . . . . . . . 12 (ℚ ∩ (-1[,]1)) ⊆ ℚ
23 fss 6209 . . . . . . . . . . . 12 ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℚ) → 𝐺:ℕ⟶ℚ)
2421, 22, 23sylancl 575 . . . . . . . . . . 11 (𝜑𝐺:ℕ⟶ℚ)
2524adantr 467 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝐺:ℕ⟶ℚ)
2625, 2ffvelrnd 6520 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑚) ∈ ℚ)
27 qcn 12027 . . . . . . . . 9 ((𝐺𝑚) ∈ ℚ → (𝐺𝑚) ∈ ℂ)
2826, 27syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑚) ∈ ℂ)
29 simprrl 788 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑘 ∈ ℕ)
3025, 29ffvelrnd 6520 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑘) ∈ ℚ)
31 qcn 12027 . . . . . . . . 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 23616 . . . . . . . . . . . 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 23617 . . . . . . . . . . . . . . . 16 (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚) ∧ 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2)))
4140simp1d 1163 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ⊆ (0[,]1))
4241adantr 467 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ran 𝐹 ⊆ (0[,]1))
4316simprd 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) ∈ ran 𝐹)
4442, 43sseldd 3759 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) ∈ (0[,]1))
45 simprrr 789 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ (𝑇𝑘))
46 fveq2 6348 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
4746oveq2d 6828 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑘)))
4847eleq1d 2838 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑘 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹))
4948rabbidv 3343 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
508rabex 4960 . . . . . . . . . . . . . . . . . . 19 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹} ∈ V
5149, 7, 50fvmpt 6441 . . . . . . . . . . . . . . . . . 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 2855 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹})
54 oveq1 6819 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑤 → (𝑠 − (𝐺𝑘)) = (𝑤 − (𝐺𝑘)))
5554eleq1d 2838 . . . . . . . . . . . . . . . . 17 (𝑠 = 𝑤 → ((𝑠 − (𝐺𝑘)) ∈ ran 𝐹 ↔ (𝑤 − (𝐺𝑘)) ∈ ran 𝐹))
5655elrab 3521 . . . . . . . . . . . . . . . 16 (𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑘)) ∈ ran 𝐹} ↔ (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑘)) ∈ ran 𝐹))
5753, 56sylib 209 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺𝑘)) ∈ ran 𝐹))
5857simprd 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑘)) ∈ ran 𝐹)
5942, 58sseldd 3759 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑘)) ∈ (0[,]1))
6044, 59jca 502 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝑤 − (𝐺𝑚)) ∈ (0[,]1) ∧ (𝑤 − (𝐺𝑘)) ∈ (0[,]1)))
6118, 28, 32nnncan1d 10649 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) = ((𝐺𝑘) − (𝐺𝑚)))
62 qsubcl 12032 . . . . . . . . . . . . . 14 (((𝐺𝑘) ∈ ℚ ∧ (𝐺𝑚) ∈ ℚ) → ((𝐺𝑘) − (𝐺𝑚)) ∈ ℚ)
6330, 26, 62syl2anc 574 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝐺𝑘) − (𝐺𝑚)) ∈ ℚ)
6461, 63eqeltrd 2853 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) ∈ ℚ)
65 oveq12 6821 . . . . . . . . . . . . . 14 ((𝑥 = (𝑤 − (𝐺𝑚)) ∧ 𝑦 = (𝑤 − (𝐺𝑘))) → (𝑥𝑦) = ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))))
6665eleq1d 2838 . . . . . . . . . . . . 13 ((𝑥 = (𝑤 − (𝐺𝑚)) ∧ 𝑦 = (𝑤 − (𝐺𝑘))) → ((𝑥𝑦) ∈ ℚ ↔ ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) ∈ ℚ))
6766, 33brab2a 5346 . . . . . . . . . . . 12 ((𝑤 − (𝐺𝑚)) (𝑤 − (𝐺𝑘)) ↔ (((𝑤 − (𝐺𝑚)) ∈ (0[,]1) ∧ (𝑤 − (𝐺𝑘)) ∈ (0[,]1)) ∧ ((𝑤 − (𝐺𝑚)) − (𝑤 − (𝐺𝑘))) ∈ ℚ))
6860, 64, 67sylanbrc 573 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) (𝑤 − (𝐺𝑘)))
6935, 68erthi 7966 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → [(𝑤 − (𝐺𝑚))] = [(𝑤 − (𝐺𝑘))] )
7069fveq2d 6352 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝐹‘[(𝑤 − (𝐺𝑘))] ))
71 eceq1 7955 . . . . . . . . . . . 12 (𝑧 = (𝑤 − (𝐺𝑚)) → [𝑧] = [(𝑤 − (𝐺𝑚))] )
7271fveq2d 6352 . . . . . . . . . . 11 (𝑧 = (𝑤 − (𝐺𝑚)) → (𝐹‘[𝑧] ) = (𝐹‘[(𝑤 − (𝐺𝑚))] ))
73 id 22 . . . . . . . . . . 11 (𝑧 = (𝑤 − (𝐺𝑚)) → 𝑧 = (𝑤 − (𝐺𝑚)))
7472, 73eqeq12d 2789 . . . . . . . . . 10 (𝑧 = (𝑤 − (𝐺𝑚)) → ((𝐹‘[𝑧] ) = 𝑧 ↔ (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝑤 − (𝐺𝑚))))
75 fveq2 6348 . . . . . . . . . . . . . . . . 17 ([𝑣] = 𝑤 → (𝐹‘[𝑣] ) = (𝐹𝑤))
7675eceq1d 7956 . . . . . . . . . . . . . . . 16 ([𝑣] = 𝑤 → [(𝐹‘[𝑣] )] = [(𝐹𝑤)] )
7776fveq2d 6352 . . . . . . . . . . . . . . 15 ([𝑣] = 𝑤 → (𝐹‘[(𝐹‘[𝑣] )] ) = (𝐹‘[(𝐹𝑤)] ))
7877, 75eqeq12d 2789 . . . . . . . . . . . . . 14 ([𝑣] = 𝑤 → ((𝐹‘[(𝐹‘[𝑣] )] ) = (𝐹‘[𝑣] ) ↔ (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
7934a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 ∈ (0[,]1)) → Er (0[,]1))
80 simpr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 ∈ (0[,]1))
81 erdm 7927 . . . . . . . . . . . . . . . . . . . . . . 23 ( Er (0[,]1) → dom = (0[,]1))
8234, 81ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 dom = (0[,]1)
8382eleq2i 2845 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ dom 𝑣 ∈ (0[,]1))
84 ecdmn0 7962 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ dom ↔ [𝑣] ≠ ∅)
8583, 84bitr3i 267 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ (0[,]1) ↔ [𝑣] ≠ ∅)
8680, 85sylib 209 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] ≠ ∅)
87 neeq1 3008 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = [𝑣] → (𝑧 ≠ ∅ ↔ [𝑣] ≠ ∅))
88 fveq2 6348 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = [𝑣] → (𝐹𝑧) = (𝐹‘[𝑣] ))
89 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = [𝑣] 𝑧 = [𝑣] )
9088, 89eleq12d 2847 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = [𝑣] → ((𝐹𝑧) ∈ 𝑧 ↔ (𝐹‘[𝑣] ) ∈ [𝑣] ))
9187, 90imbi12d 334 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = [𝑣] → ((𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧) ↔ ([𝑣] ≠ ∅ → (𝐹‘[𝑣] ) ∈ [𝑣] )))
9238adantr 467 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑣 ∈ (0[,]1)) → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
93 ovex 6844 . . . . . . . . . . . . . . . . . . . . . . . 24 (0[,]1) ∈ V
94 erex 7941 . . . . . . . . . . . . . . . . . . . . . . . 24 ( Er (0[,]1) → ((0[,]1) ∈ V → ∈ V))
9534, 93, 94mp2 9 . . . . . . . . . . . . . . . . . . . . . . 23 ∈ V
9695ecelqsi 7976 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ∈ (0[,]1) → [𝑣] ∈ ((0[,]1) / ))
9796, 36syl6eleqr 2864 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ (0[,]1) → [𝑣] 𝑆)
9897adantl 468 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] 𝑆)
9991, 92, 98rspcdva 3471 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 ∈ (0[,]1)) → ([𝑣] ≠ ∅ → (𝐹‘[𝑣] ) ∈ [𝑣] ))
10086, 99mpd 15 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ) ∈ [𝑣] )
101 fvex 6359 . . . . . . . . . . . . . . . . . . 19 (𝐹‘[𝑣] ) ∈ V
102 vex 3358 . . . . . . . . . . . . . . . . . . 19 𝑣 ∈ V
103101, 102elec 7959 . . . . . . . . . . . . . . . . . 18 ((𝐹‘[𝑣] ) ∈ [𝑣] 𝑣 (𝐹‘[𝑣] ))
104100, 103sylib 209 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 ∈ (0[,]1)) → 𝑣 (𝐹‘[𝑣] ))
10579, 104erthi 7966 . . . . . . . . . . . . . . . 16 ((𝜑𝑣 ∈ (0[,]1)) → [𝑣] = [(𝐹‘[𝑣] )] )
106105eqcomd 2780 . . . . . . . . . . . . . . 15 ((𝜑𝑣 ∈ (0[,]1)) → [(𝐹‘[𝑣] )] = [𝑣] )
107106fveq2d 6352 . . . . . . . . . . . . . 14 ((𝜑𝑣 ∈ (0[,]1)) → (𝐹‘[(𝐹‘[𝑣] )] ) = (𝐹‘[𝑣] ))
10836, 78, 107ectocld 7987 . . . . . . . . . . . . 13 ((𝜑𝑤𝑆) → (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤))
109108ralrimiva 3118 . . . . . . . . . . . 12 (𝜑 → ∀𝑤𝑆 (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤))
110 eceq1 7955 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑤) → [𝑧] = [(𝐹𝑤)] )
111110fveq2d 6352 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑤) → (𝐹‘[𝑧] ) = (𝐹‘[(𝐹𝑤)] ))
112 id 22 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑤) → 𝑧 = (𝐹𝑤))
113111, 112eqeq12d 2789 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑤) → ((𝐹‘[𝑧] ) = 𝑧 ↔ (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
114113ralrn 6522 . . . . . . . . . . . . 13 (𝐹 Fn 𝑆 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧 ↔ ∀𝑤𝑆 (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
11537, 114syl 17 . . . . . . . . . . . 12 (𝜑 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧 ↔ ∀𝑤𝑆 (𝐹‘[(𝐹𝑤)] ) = (𝐹𝑤)))
116109, 115mpbird 248 . . . . . . . . . . 11 (𝜑 → ∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧)
117116adantr 467 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ) = 𝑧)
11874, 117, 43rspcdva 3471 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐹‘[(𝑤 − (𝐺𝑚))] ) = (𝑤 − (𝐺𝑚)))
119 eceq1 7955 . . . . . . . . . . . 12 (𝑧 = (𝑤 − (𝐺𝑘)) → [𝑧] = [(𝑤 − (𝐺𝑘))] )
120119fveq2d 6352 . . . . . . . . . . 11 (𝑧 = (𝑤 − (𝐺𝑘)) → (𝐹‘[𝑧] ) = (𝐹‘[(𝑤 − (𝐺𝑘))] ))
121 id 22 . . . . . . . . . . 11 (𝑧 = (𝑤 − (𝐺𝑘)) → 𝑧 = (𝑤 − (𝐺𝑘)))
122120, 121eqeq12d 2789 . . . . . . . . . 10 (𝑧 = (𝑤 − (𝐺𝑘)) → ((𝐹‘[𝑧] ) = 𝑧 ↔ (𝐹‘[(𝑤 − (𝐺𝑘))] ) = (𝑤 − (𝐺𝑘))))
123122, 117, 58rspcdva 3471 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐹‘[(𝑤 − (𝐺𝑘))] ) = (𝑤 − (𝐺𝑘)))
12470, 118, 1233eqtr3d 2816 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝑤 − (𝐺𝑚)) = (𝑤 − (𝐺𝑘)))
12518, 28, 32, 124subcand 10656 . . . . . . 7 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → (𝐺𝑚) = (𝐺𝑘))
12619adantr 467 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
127 f1of1 6292 . . . . . . . . 9 (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)))
128126, 127syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)))
129 f1fveq 6681 . . . . . . . 8 ((𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)) ∧ (𝑚 ∈ ℕ ∧ 𝑘 ∈ ℕ)) → ((𝐺𝑚) = (𝐺𝑘) ↔ 𝑚 = 𝑘))
130128, 2, 29, 129syl12anc 854 . . . . . . 7 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → ((𝐺𝑚) = (𝐺𝑘) ↔ 𝑚 = 𝑘))
131125, 130mpbid 223 . . . . . 6 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘)))) → 𝑚 = 𝑘)
132131ex 398 . . . . 5 (𝜑 → (((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))) → 𝑚 = 𝑘))
133132alrimivv 2011 . . . 4 (𝜑 → ∀𝑚𝑘(((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))) → 𝑚 = 𝑘))
134 eleq1w 2836 . . . . . 6 (𝑚 = 𝑘 → (𝑚 ∈ ℕ ↔ 𝑘 ∈ ℕ))
135 fveq2 6348 . . . . . . 7 (𝑚 = 𝑘 → (𝑇𝑚) = (𝑇𝑘))
136135eleq2d 2839 . . . . . 6 (𝑚 = 𝑘 → (𝑤 ∈ (𝑇𝑚) ↔ 𝑤 ∈ (𝑇𝑘)))
137134, 136anbi12d 617 . . . . 5 (𝑚 = 𝑘 → ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ↔ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))))
138137mo4 2669 . . . 4 (∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ↔ ∀𝑚𝑘(((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑘))) → 𝑚 = 𝑘))
139133, 138sylibr 225 . . 3 (𝜑 → ∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)))
140139alrimiv 2010 . 2 (𝜑 → ∀𝑤∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)))
141 dfdisj2 4767 . 2 (Disj 𝑚 ∈ ℕ (𝑇𝑚) ↔ ∀𝑤∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇𝑚)))
142140, 141sylibr 225 1 (𝜑Disj 𝑚 ∈ ℕ (𝑇𝑚))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 383  wal 1632   = wceq 1634  wcel 2148  ∃*wmo 2622  wne 2946  wral 3064  {crab 3068  Vcvv 3355  cdif 3726  cin 3728  wss 3729  c0 4073  𝒫 cpw 4307   ciun 4665  Disj wdisj 4765   class class class wbr 4797  {copab 4859  cmpt 4876  dom cdm 5263  ran crn 5264   Fn wfn 6037  wf 6038  1-1wf1 6039  1-1-ontowf1o 6041  cfv 6042  (class class class)co 6812   Er wer 7914  [cec 7915   / cqs 7916  cc 10157  cr 10158  0cc0 10159  1c1 10160  cmin 10489  -cneg 10490  cn 11243  2c2 11293  cq 12013  [,]cicc 12402  volcvol 23471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-cnex 10215  ax-resscn 10216  ax-1cn 10217  ax-icn 10218  ax-addcl 10219  ax-addrcl 10220  ax-mulcl 10221  ax-mulrcl 10222  ax-mulcom 10223  ax-addass 10224  ax-mulass 10225  ax-distr 10226  ax-i2m1 10227  ax-1ne0 10228  ax-1rid 10229  ax-rnegex 10230  ax-rrecex 10231  ax-cnre 10232  ax-pre-lttri 10233  ax-pre-lttrn 10234  ax-pre-ltadd 10235  ax-pre-mulgt0 10236
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-iun 4667  df-disj 4766  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-1st 7336  df-2nd 7337  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-er 7917  df-ec 7919  df-qs 7923  df-en 8131  df-dom 8132  df-sdom 8133  df-pnf 10299  df-mnf 10300  df-xr 10301  df-ltxr 10302  df-le 10303  df-sub 10491  df-neg 10492  df-div 10908  df-nn 11244  df-2 11302  df-n0 11517  df-z 11602  df-q 12014  df-icc 12406
This theorem is referenced by:  vitalilem4  23619
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