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Type | Label | Description |
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Statement | ||
Theorem | voliunnfl 35101* | voliun 24158 is incompatible with the Feferman-Levy model; in that model, therefore, the Lebesgue measure as we've defined it isn't actually a measure. (Contributed by Brendan Leahy, 16-Dec-2017.) |
⊢ 𝑆 = seq1( + , 𝐺) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝑓‘𝑛))) & ⊢ ((∀𝑛 ∈ ℕ ((𝑓‘𝑛) ∈ dom vol ∧ (vol‘(𝑓‘𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝑓‘𝑛)) → (vol‘∪ 𝑛 ∈ ℕ (𝑓‘𝑛)) = sup(ran 𝑆, ℝ*, < )) ⇒ ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ) | ||
Theorem | volsupnfl 35102* | volsup 24160 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 2-Jan-2018.) |
⊢ ((𝑓:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ⊆ (𝑓‘(𝑛 + 1))) → (vol‘∪ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < )) ⇒ ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ) | ||
Theorem | mbfresfi 35103* | Measurability of a piecewise function across arbitrarily many subsets. (Contributed by Brendan Leahy, 31-Mar-2018.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝑆 ∈ Fin) & ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 (𝐹 ↾ 𝑠) ∈ MblFn) & ⊢ (𝜑 → ∪ 𝑆 = 𝐴) ⇒ ⊢ (𝜑 → 𝐹 ∈ MblFn) | ||
Theorem | mbfposadd 35104* | If the sum of two measurable functions is measurable, the sum of their nonnegative parts is measurable. (Contributed by Brendan Leahy, 2-Apr-2018.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ MblFn) | ||
Theorem | cnambfre 35105 | A real-valued, a.e. continuous function is measurable. (Contributed by Brendan Leahy, 4-Apr-2018.) |
⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → 𝐹 ∈ MblFn) | ||
Theorem | dvtanlem 35106 | Lemma for dvtan 35107- the domain of the tangent is open. (Contributed by Brendan Leahy, 8-Aug-2018.) (Proof shortened by OpenAI, 3-Jul-2020.) |
⊢ (◡cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld) | ||
Theorem | dvtan 35107 | Derivative of tangent. (Contributed by Brendan Leahy, 7-Aug-2018.) |
⊢ (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2)) | ||
Theorem | itg2addnclem 35108* | An alternate expression for the ∫2 integral that includes an arbitrarily small but strictly positive "buffer zone" wherever the simple function is nonzero. (Contributed by Brendan Leahy, 10-Oct-2017.) (Revised by Brendan Leahy, 10-Mar-2018.) |
⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⇒ ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < )) | ||
Theorem | itg2addnclem2 35109* | Lemma for itg2addnc 35111. The function described is a simple function. (Contributed by Brendan Leahy, 29-Oct-2017.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) ⇒ ⊢ (((𝜑 ∧ ℎ ∈ dom ∫1) ∧ 𝑣 ∈ ℝ+) → (𝑥 ∈ ℝ ↦ if(((((⌊‘((𝐹‘𝑥) / (𝑣 / 3))) − 1) · (𝑣 / 3)) ≤ (ℎ‘𝑥) ∧ (ℎ‘𝑥) ≠ 0), (((⌊‘((𝐹‘𝑥) / (𝑣 / 3))) − 1) · (𝑣 / 3)), (ℎ‘𝑥))) ∈ dom ∫1) | ||
Theorem | itg2addnclem3 35110* | Lemma incomprehensible in isolation split off to shorten proof of itg2addnc 35111. (Contributed by Brendan Leahy, 11-Mar-2018.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ) ⇒ ⊢ (𝜑 → (∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)) → ∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))) | ||
Theorem | itg2addnc 35111 | Alternate proof of itg2add 24363 using the "buffer zone" definition from the first lemma, in which every simple function in the set is divided into to by dividing its buffer by a third and finding the largest allowable function locked to a grid laid out in increments of the new, smaller buffer up to the original simple function. The measurability of this function follows from that of the augend, and subtracting it from the original simple function yields another simple function by i1fsub 24312, which is allowable by the fact that the grid must have a mark between one third and two thirds the original buffer. This has two advantages over the current approach: first, eliminating ax-cc 9846, and second, weakening the measurability hypothesis to only the augend. (Contributed by Brendan Leahy, 31-Oct-2017.) (Revised by Brendan Leahy, 13-Mar-2018.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ) ⇒ ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺))) | ||
Theorem | itg2gt0cn 35112* | itg2gt0 24364 holds on functions continuous on an open interval in the absence of ax-cc 9846. The fourth hypothesis is made unnecessary by the continuity hypothesis. (Contributed by Brendan Leahy, 16-Nov-2017.) |
⊢ (𝜑 → 𝑋 < 𝑌) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 0 < (𝐹‘𝑥)) & ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)𝑌)) ∈ ((𝑋(,)𝑌)–cn→ℂ)) ⇒ ⊢ (𝜑 → 0 < (∫2‘𝐹)) | ||
Theorem | ibladdnclem 35113* | Lemma for ibladdnc 35114; cf ibladdlem 24423, whose fifth hypothesis is rendered unnecessary by the weakened hypotheses of itg2addnc 35111. (Contributed by Brendan Leahy, 31-Oct-2017.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 = (𝐵 + 𝐶)) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) & ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ) & ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ) ⇒ ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ∈ ℝ) | ||
Theorem | ibladdnc 35114* | Choice-free analogue of itgadd 24428. A measurability hypothesis is necessitated by the loss of mbfadd 24265; for large classes of functions, such as continuous functions, it should be relatively easy to show. (Contributed by Brendan Leahy, 1-Nov-2017.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝐿1) | ||
Theorem | itgaddnclem1 35115* | Lemma for itgaddnc 35117; cf. itgaddlem1 24426. (Contributed by Brendan Leahy, 7-Nov-2017.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐶) ⇒ ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥)) | ||
Theorem | itgaddnclem2 35116* | Lemma for itgaddnc 35117; cf. itgaddlem2 24427. (Contributed by Brendan Leahy, 10-Nov-2017.) (Revised by Brendan Leahy, 3-Apr-2018.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥)) | ||
Theorem | itgaddnc 35117* | Choice-free analogue of itgadd 24428. (Contributed by Brendan Leahy, 11-Nov-2017.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn) ⇒ ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥)) | ||
Theorem | iblsubnc 35118* | Choice-free analogue of iblsub 24425. (Contributed by Brendan Leahy, 11-Nov-2017.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ MblFn) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ 𝐿1) | ||
Theorem | itgsubnc 35119* | Choice-free analogue of itgsub 24429. (Contributed by Brendan Leahy, 11-Nov-2017.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ MblFn) ⇒ ⊢ (𝜑 → ∫𝐴(𝐵 − 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 − ∫𝐴𝐶 d𝑥)) | ||
Theorem | iblabsnclem 35120* | Lemma for iblabsnc 35121; cf. iblabslem 24431. (Contributed by Brendan Leahy, 7-Nov-2017.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ 𝐿1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐺 ∈ MblFn ∧ (∫2‘𝐺) ∈ ℝ)) | ||
Theorem | iblabsnc 35121* | Choice-free analogue of iblabs 24432. As with ibladdnc 35114, a measurability hypothesis is needed. (Contributed by Brendan Leahy, 7-Nov-2017.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿1) | ||
Theorem | iblmulc2nc 35122* | Choice-free analogue of iblmulc2 24434. (Contributed by Brendan Leahy, 17-Nov-2017.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿1) | ||
Theorem | itgmulc2nclem1 35123* | Lemma for itgmulc2nc 35125; cf. itgmulc2lem1 24435. (Contributed by Brendan Leahy, 17-Nov-2017.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥) | ||
Theorem | itgmulc2nclem2 35124* | Lemma for itgmulc2nc 35125; cf. itgmulc2lem2 24436. (Contributed by Brendan Leahy, 19-Nov-2017.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥) | ||
Theorem | itgmulc2nc 35125* | Choice-free analogue of itgmulc2 24437. (Contributed by Brendan Leahy, 19-Nov-2017.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn) ⇒ ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥) | ||
Theorem | itgabsnc 35126* | Choice-free analogue of itgabs 24438. (Contributed by Brendan Leahy, 19-Nov-2017.) (Revised by Brendan Leahy, 19-Jun-2018.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn) & ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈ MblFn) ⇒ ⊢ (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥) | ||
Theorem | itggt0cn 35127* | itggt0 24447 holds for continuous functions in the absence of ax-cc 9846. (Contributed by Brendan Leahy, 16-Nov-2017.) |
⊢ (𝜑 → 𝑋 < 𝑌) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ 𝐿1) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ ((𝑋(,)𝑌)–cn→ℂ)) ⇒ ⊢ (𝜑 → 0 < ∫(𝑋(,)𝑌)𝐵 d𝑥) | ||
Theorem | ftc1cnnclem 35128* | Lemma for ftc1cnnc 35129; cf. ftc1lem4 24642. The stronger assumptions of ftc1cn 24646 are exploited to make use of weaker theorems. (Contributed by Brendan Leahy, 19-Nov-2017.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝑐 ∈ (𝐴(,)𝐵)) & ⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐))) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑦 − 𝑐)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → (abs‘(𝑋 − 𝑐)) < 𝑅) & ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → (abs‘(𝑌 − 𝑐)) < 𝑅) ⇒ ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝑐))) < 𝐸) | ||
Theorem | ftc1cnnc 35129* | Choice-free proof of ftc1cn 24646. (Contributed by Brendan Leahy, 20-Nov-2017.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) ⇒ ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹) | ||
Theorem | ftc1anclem1 35130 | Lemma for ftc1anc 35138- the absolute value of a real-valued measurable function is measurable. Would be trivial with cncombf 24262, but this proof avoids ax-cc 9846. (Contributed by Brendan Leahy, 18-Jun-2018.) |
⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐹 ∈ MblFn) → (abs ∘ 𝐹) ∈ MblFn) | ||
Theorem | ftc1anclem2 35131* | Lemma for ftc1anc 35138- restriction of an integrable function to the absolute value of its real or imaginary part. (Contributed by Brendan Leahy, 19-Jun-2018.) (Revised by Brendan Leahy, 8-Aug-2018.) |
⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1 ∧ 𝐺 ∈ {ℜ, ℑ}) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝐴, (abs‘(𝐺‘(𝐹‘𝑡))), 0))) ∈ ℝ) | ||
Theorem | ftc1anclem3 35132 | Lemma for ftc1anc 35138- the absolute value of the sum of a simple function and i times another simple function is itself a simple function. (Contributed by Brendan Leahy, 27-May-2018.) |
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (abs ∘ (𝐹 ∘f + ((ℝ × {i}) ∘f · 𝐺))) ∈ dom ∫1) | ||
Theorem | ftc1anclem4 35133* | Lemma for ftc1anc 35138. (Contributed by Brendan Leahy, 17-Jun-2018.) |
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ 𝐿1 ∧ 𝐺:ℝ⟶ℝ) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ∈ ℝ) | ||
Theorem | ftc1anclem5 35134* | Lemma for ftc1anc 35138, the existence of a simple function the integral of whose pointwise difference from the function is less than a given positive real. (Contributed by Brendan Leahy, 17-Jun-2018.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) ⇒ ⊢ ((𝜑 ∧ 𝑌 ∈ ℝ+) → ∃𝑓 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0)) − (𝑓‘𝑡))))) < 𝑌) | ||
Theorem | ftc1anclem6 35135* | Lemma for ftc1anc 35138- construction of simple functions within an arbitrary absolute distance of the given function. Similar to Lemma 565Ib of [Fremlin5] p. 218, but without Fremlin's additional step of converting the simple function into a continuous one, which is unnecessary to this lemma's use; also, two simple functions are used to allow for complex-valued 𝐹. (Contributed by Brendan Leahy, 31-May-2018.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) ⇒ ⊢ ((𝜑 ∧ 𝑌 ∈ ℝ+) → ∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < 𝑌) | ||
Theorem | ftc1anclem7 35136* | Lemma for ftc1anc 35138. (Contributed by Brendan Leahy, 13-May-2018.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) ⇒ ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)))) < ((𝑦 / 2) + (𝑦 / 2))) | ||
Theorem | ftc1anclem8 35137* | Lemma for ftc1anc 35138. (Contributed by Brendan Leahy, 29-May-2018.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) ⇒ ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) + (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) < 𝑦) | ||
Theorem | ftc1anc 35138* | ftc1a 24640 holds for functions that obey the triangle inequality in the absence of ax-cc 9846. Theorem 565Ma of [Fremlin5] p. 220. (Contributed by Brendan Leahy, 11-May-2018.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) & ⊢ (𝜑 → ∀𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))(abs‘∫𝑠(𝐹‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘(𝐹‘𝑡)), 0)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ)) | ||
Theorem | ftc2nc 35139* | Choice-free proof of ftc2 24647. (Contributed by Brendan Leahy, 19-Jun-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ)) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿1) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) ⇒ ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴))) | ||
Theorem | asindmre 35140 | Real part of domain of differentiability of arcsine. (Contributed by Brendan Leahy, 19-Dec-2018.) |
⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ⇒ ⊢ (𝐷 ∩ ℝ) = (-1(,)1) | ||
Theorem | dvasin 35141* | Derivative of arcsine. (Contributed by Brendan Leahy, 18-Dec-2018.) |
⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ⇒ ⊢ (ℂ D (arcsin ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (1 / (√‘(1 − (𝑥↑2))))) | ||
Theorem | dvacos 35142* | Derivative of arccosine. (Contributed by Brendan Leahy, 18-Dec-2018.) |
⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ⇒ ⊢ (ℂ D (arccos ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (-1 / (√‘(1 − (𝑥↑2))))) | ||
Theorem | dvreasin 35143 | Real derivative of arcsine. (Contributed by Brendan Leahy, 3-Aug-2017.) (Proof shortened by Brendan Leahy, 18-Dec-2018.) |
⊢ (ℝ D (arcsin ↾ (-1(,)1))) = (𝑥 ∈ (-1(,)1) ↦ (1 / (√‘(1 − (𝑥↑2))))) | ||
Theorem | dvreacos 35144 | Real derivative of arccosine. (Contributed by Brendan Leahy, 3-Aug-2017.) (Proof shortened by Brendan Leahy, 18-Dec-2018.) |
⊢ (ℝ D (arccos ↾ (-1(,)1))) = (𝑥 ∈ (-1(,)1) ↦ (-1 / (√‘(1 − (𝑥↑2))))) | ||
Theorem | areacirclem1 35145* | Antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.) |
⊢ (𝑅 ∈ ℝ+ → (ℝ D (𝑡 ∈ (-𝑅(,)𝑅) ↦ ((𝑅↑2) · ((arcsin‘(𝑡 / 𝑅)) + ((𝑡 / 𝑅) · (√‘(1 − ((𝑡 / 𝑅)↑2)))))))) = (𝑡 ∈ (-𝑅(,)𝑅) ↦ (2 · (√‘((𝑅↑2) − (𝑡↑2)))))) | ||
Theorem | areacirclem2 35146* | Endpoint-inclusive continuity of Cartesian ordinate of circle. (Contributed by Brendan Leahy, 29-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.) |
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ (√‘((𝑅↑2) − (𝑡↑2)))) ∈ ((-𝑅[,]𝑅)–cn→ℂ)) | ||
Theorem | areacirclem3 35147* | Integrability of cross-section of circle. (Contributed by Brendan Leahy, 26-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.) |
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ (2 · (√‘((𝑅↑2) − (𝑡↑2))))) ∈ 𝐿1) | ||
Theorem | areacirclem4 35148* | Endpoint-inclusive continuity of antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.) |
⊢ (𝑅 ∈ ℝ+ → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) · ((arcsin‘(𝑡 / 𝑅)) + ((𝑡 / 𝑅) · (√‘(1 − ((𝑡 / 𝑅)↑2))))))) ∈ ((-𝑅[,]𝑅)–cn→ℂ)) | ||
Theorem | areacirclem5 35149* | Finding the cross-section of a circle. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.) (Revised by Brendan Leahy, 11-Jul-2018.) |
⊢ 𝑆 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))} ⇒ ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑆 “ {𝑡}) = if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅)) | ||
Theorem | areacirc 35150* | The area of a circle of radius 𝑅 is π · 𝑅↑2. This is Metamath 100 proof #9. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.) (Revised by Brendan Leahy, 11-Jul-2018.) |
⊢ 𝑆 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))} ⇒ ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (area‘𝑆) = (π · (𝑅↑2))) | ||
Theorem | unirep 35151* | Define a quantity whose definition involves a choice of representative, but which is uniquely determined regardless of the choice. (Contributed by Jeff Madsen, 1-Jun-2011.) |
⊢ (𝑦 = 𝐷 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐷 → 𝐵 = 𝐶) & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑧 → 𝐵 = 𝐹) & ⊢ 𝐵 ∈ V ⇒ ⊢ ((∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝜑 ∧ 𝜒) → 𝐵 = 𝐹) ∧ (𝐷 ∈ 𝐴 ∧ 𝜓)) → (℩𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵)) = 𝐶) | ||
Theorem | cover2 35152* | Two ways of expressing the statement "there is a cover of 𝐴 by elements of 𝐵 such that for each set in the cover, 𝜑". Note that 𝜑 and 𝑥 must be distinct. (Contributed by Jeff Madsen, 20-Jun-2010.) |
⊢ 𝐵 ∈ V & ⊢ 𝐴 = ∪ 𝐵 ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)) | ||
Theorem | cover2g 35153* | Two ways of expressing the statement "there is a cover of 𝐴 by elements of 𝐵 such that for each set in the cover, 𝜑". Note that 𝜑 and 𝑥 must be distinct. (Contributed by Jeff Madsen, 21-Jun-2010.) |
⊢ 𝐴 = ∪ 𝐵 ⇒ ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))) | ||
Theorem | brabg2 35154* | Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} & ⊢ (𝜒 → 𝐴 ∈ 𝐶) ⇒ ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 ↔ 𝜒)) | ||
Theorem | opelopab3 35155* | Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝜒 → 𝐴 ∈ 𝐶) ⇒ ⊢ (𝐵 ∈ 𝐷 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | ||
Theorem | cocanfo 35156 | Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.) |
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐺 = 𝐻) | ||
Theorem | brresi2 35157 | Restriction of a binary relation. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴(𝑅 ↾ 𝐶)𝐵 → 𝐴𝑅𝐵) | ||
Theorem | fnopabeqd 35158* | Equality deduction for function abstractions. (Contributed by Jeff Madsen, 19-Jun-2011.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}) | ||
Theorem | fvopabf4g 35159* | Function value of an operator abstraction whose domain is a set of functions with given domain and range. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 12-Sep-2015.) |
⊢ 𝐶 ∈ V & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑m 𝐷) ↦ 𝐵) ⇒ ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌 ∧ 𝐴:𝐷⟶𝑅) → (𝐹‘𝐴) = 𝐶) | ||
Theorem | eqfnun 35160 | Two functions on 𝐴 ∪ 𝐵 are equal if and only if they have equal restrictions to both 𝐴 and 𝐵. (Contributed by Jeff Madsen, 19-Jun-2011.) |
⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)))) | ||
Theorem | fnopabco 35161* | Composition of a function with a function abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.) |
⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶) & ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} & ⊢ 𝐺 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐻‘𝐵))} ⇒ ⊢ (𝐻 Fn 𝐶 → 𝐺 = (𝐻 ∘ 𝐹)) | ||
Theorem | opropabco 35162* | Composition of an operator with a function abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) |
⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑅) & ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑆) & ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 〈𝐵, 𝐶〉)} & ⊢ 𝐺 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶))} ⇒ ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹)) | ||
Theorem | cocnv 35163 | Composition with a function and then with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ↾ ran 𝐺)) | ||
Theorem | f1ocan1fv 35164 | Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.) |
⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘𝑋)) | ||
Theorem | f1ocan2fv 35165 | Cancel a composition by the converse of a bijection by preapplying the bijection. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(𝐺‘𝑋)) = (𝐹‘𝑋)) | ||
Theorem | inixp 35166* | Intersection of Cartesian products over the same base set. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶) = X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) | ||
Theorem | upixp 35167* | Universal property of the indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
⊢ 𝑋 = X𝑏 ∈ 𝐴 (𝐶‘𝑏) & ⊢ 𝑃 = (𝑤 ∈ 𝐴 ↦ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤))) ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → ∃!ℎ(ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) | ||
Theorem | abrexdom 35168* | An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑) ⇒ ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴) | ||
Theorem | abrexdom2 35169* | An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ≼ 𝐴) | ||
Theorem | ac6gf 35170* | Axiom of Choice. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ Ⅎ𝑦𝜓 & ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | ||
Theorem | indexa 35171* | If for every element of an indexing set 𝐴 there exists a corresponding element of another set 𝐵, then there exists a subset of 𝐵 consisting only of those elements which are indexed by 𝐴. Used to avoid the Axiom of Choice in situations where only the range of the choice function is needed. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐵 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑐(𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | indexdom 35172* | If for every element of an indexing set 𝐴 there exists a corresponding element of another set 𝐵, then there exists a subset of 𝐵 consisting only of those elements which are indexed by 𝐴, and which is dominated by the set 𝐴. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐴 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑐((𝑐 ≼ 𝐴 ∧ 𝑐 ⊆ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑))) | ||
Theorem | frinfm 35173* | A subset of a well-founded set has an infimum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) | ||
Theorem | welb 35174* | A nonempty subset of a well-ordered set has a lower bound. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → (◡𝑅 Or 𝐵 ∧ ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))) | ||
Theorem | supex2g 35175 | Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 ∈ 𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V) | ||
Theorem | supclt 35176* | Closure of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝑅 Or 𝐴 ∧ ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) | ||
Theorem | supubt 35177* | Upper bound property of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝑅 Or 𝐴 ∧ ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)) | ||
Theorem | filbcmb 35178* | Combine a finite set of lower bounds. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ 𝐵 ⊆ ℝ) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 ≤ 𝑧 → 𝜑) → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 ≤ 𝑧 → ∀𝑥 ∈ 𝐴 𝜑))) | ||
Theorem | fzmul 35179 | Membership of a product in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℕ) → (𝐽 ∈ (𝑀...𝑁) → (𝐾 · 𝐽) ∈ ((𝐾 · 𝑀)...(𝐾 · 𝑁)))) | ||
Theorem | sdclem2 35180* | Lemma for sdc 35182. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓 ↔ 𝜒)) & ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜏)) & ⊢ (𝑛 = 𝑘 → (𝜓 ↔ 𝜃)) & ⊢ ((𝑔 = ℎ ∧ 𝑛 = (𝑘 + 1)) → (𝜓 ↔ 𝜎)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴 ∧ 𝜏)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃) → ∃ℎ(ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑔 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎))) & ⊢ 𝐽 = {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} & ⊢ 𝐹 = (𝑤 ∈ 𝑍, 𝑥 ∈ 𝐽 ↦ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)}) & ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐺:𝑍⟶𝐽) & ⊢ (𝜑 → (𝐺‘𝑀):(𝑀...𝑀)⟶𝐴) & ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝐺‘(𝑤 + 1)) ∈ (𝑤𝐹(𝐺‘𝑤))) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒)) | ||
Theorem | sdclem1 35181* | Lemma for sdc 35182. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓 ↔ 𝜒)) & ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜏)) & ⊢ (𝑛 = 𝑘 → (𝜓 ↔ 𝜃)) & ⊢ ((𝑔 = ℎ ∧ 𝑛 = (𝑘 + 1)) → (𝜓 ↔ 𝜎)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴 ∧ 𝜏)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃) → ∃ℎ(ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑔 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎))) & ⊢ 𝐽 = {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} & ⊢ 𝐹 = (𝑤 ∈ 𝑍, 𝑥 ∈ 𝐽 ↦ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)}) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒)) | ||
Theorem | sdc 35182* | Strong dependent choice. Suppose we may choose an element of 𝐴 such that property 𝜓 holds, and suppose that if we have already chosen the first 𝑘 elements (represented here by a function from 1...𝑘 to 𝐴), we may choose another element so that all 𝑘 + 1 elements taken together have property 𝜓. Then there exists an infinite sequence of elements of 𝐴 such that the first 𝑛 terms of this sequence satisfy 𝜓 for all 𝑛. This theorem allows us to construct infinite seqeunces where each term depends on all the previous terms in the sequence. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 3-Jun-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓 ↔ 𝜒)) & ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜏)) & ⊢ (𝑛 = 𝑘 → (𝜓 ↔ 𝜃)) & ⊢ ((𝑔 = ℎ ∧ 𝑛 = (𝑘 + 1)) → (𝜓 ↔ 𝜎)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴 ∧ 𝜏)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃) → ∃ℎ(ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑔 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎))) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒)) | ||
Theorem | fdc 35183* | Finite version of dependent choice. Construct a function whose value depends on the previous function value, except at a final point at which no new value can be chosen. The final hypothesis ensures that the process will terminate. The proof does not use the Axiom of Choice. (Contributed by Jeff Madsen, 18-Jun-2010.) |
⊢ 𝐴 ∈ V & ⊢ 𝑀 ∈ ℤ & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑁 = (𝑀 + 1) & ⊢ (𝑎 = (𝑓‘(𝑘 − 1)) → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = (𝑓‘𝑘) → (𝜓 ↔ 𝜒)) & ⊢ (𝑎 = (𝑓‘𝑛) → (𝜃 ↔ 𝜏)) & ⊢ (𝜂 → 𝐶 ∈ 𝐴) & ⊢ (𝜂 → 𝑅 Fr 𝐴) & ⊢ ((𝜂 ∧ 𝑎 ∈ 𝐴) → (𝜃 ∨ ∃𝑏 ∈ 𝐴 𝜑)) & ⊢ (((𝜂 ∧ 𝜑) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏𝑅𝑎) ⇒ ⊢ (𝜂 → ∃𝑛 ∈ 𝑍 ∃𝑓(𝑓:(𝑀...𝑛)⟶𝐴 ∧ ((𝑓‘𝑀) = 𝐶 ∧ 𝜏) ∧ ∀𝑘 ∈ (𝑁...𝑛)𝜒)) | ||
Theorem | fdc1 35184* | Variant of fdc 35183 with no specified base value. (Contributed by Jeff Madsen, 18-Jun-2010.) |
⊢ 𝐴 ∈ V & ⊢ 𝑀 ∈ ℤ & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑁 = (𝑀 + 1) & ⊢ (𝑎 = (𝑓‘𝑀) → (𝜁 ↔ 𝜎)) & ⊢ (𝑎 = (𝑓‘(𝑘 − 1)) → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = (𝑓‘𝑘) → (𝜓 ↔ 𝜒)) & ⊢ (𝑎 = (𝑓‘𝑛) → (𝜃 ↔ 𝜏)) & ⊢ (𝜂 → ∃𝑎 ∈ 𝐴 𝜁) & ⊢ (𝜂 → 𝑅 Fr 𝐴) & ⊢ ((𝜂 ∧ 𝑎 ∈ 𝐴) → (𝜃 ∨ ∃𝑏 ∈ 𝐴 𝜑)) & ⊢ (((𝜂 ∧ 𝜑) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏𝑅𝑎) ⇒ ⊢ (𝜂 → ∃𝑛 ∈ 𝑍 ∃𝑓(𝑓:(𝑀...𝑛)⟶𝐴 ∧ (𝜎 ∧ 𝜏) ∧ ∀𝑘 ∈ (𝑁...𝑛)𝜒)) | ||
Theorem | seqpo 35185* | Two ways to say that a sequence respects a partial order. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) → (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ↔ ∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛))) | ||
Theorem | incsequz 35186* | An increasing sequence of positive integers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ (𝐹‘𝑛) ∈ (ℤ≥‘𝐴)) | ||
Theorem | incsequz2 35187* | An increasing sequence of positive integers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)(𝐹‘𝑘) ∈ (ℤ≥‘𝐴)) | ||
Theorem | nnubfi 35188* | A bounded above set of positive integers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.) |
⊢ ((𝐴 ⊆ ℕ ∧ 𝐵 ∈ ℕ) → {𝑥 ∈ 𝐴 ∣ 𝑥 < 𝐵} ∈ Fin) | ||
Theorem | nninfnub 35189* | An infinite set of positive integers is unbounded above. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.) |
⊢ ((𝐴 ⊆ ℕ ∧ ¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ ℕ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑥} ≠ ∅) | ||
Theorem | subspopn 35190 | An open set is open in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ∈ (𝐽 ↾t 𝐴)) | ||
Theorem | neificl 35191 | Neighborhoods are closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Nov-2013.) |
⊢ (((𝐽 ∈ Top ∧ 𝑁 ⊆ ((nei‘𝐽)‘𝑆)) ∧ (𝑁 ∈ Fin ∧ 𝑁 ≠ ∅)) → ∩ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) | ||
Theorem | lpss2 35192 | Limit points of a subset are limit points of the larger set. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((limPt‘𝐽)‘𝐵) ⊆ ((limPt‘𝐽)‘𝐴)) | ||
Theorem | metf1o 35193* | Use a bijection with a metric space to construct a metric on a set. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝑁 = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑌 ↦ ((𝐹‘𝑥)𝑀(𝐹‘𝑦))) ⇒ ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → 𝑁 ∈ (Met‘𝑌)) | ||
Theorem | blssp 35194 | A ball in the subspace metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jan-2014.) |
⊢ 𝑁 = (𝑀 ↾ (𝑆 × 𝑆)) ⇒ ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆)) | ||
Theorem | mettrifi 35195* | Generalized triangle inequality for arbitrary finite sums. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑋) ⇒ ⊢ (𝜑 → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) | ||
Theorem | lmclim2 35196* | A sequence in a metric space converges to a point iff the distance between the point and the elements of the sequence converges to 0. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.) |
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐺 = (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)𝐷𝑌)) & ⊢ (𝜑 → 𝑌 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑌 ↔ 𝐺 ⇝ 0)) | ||
Theorem | geomcau 35197* | If the distance between consecutive points in a sequence is bounded by a geometric sequence, then the sequence is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.) |
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 < 1) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (𝐵↑𝑘))) ⇒ ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) | ||
Theorem | caures 35198 | The restriction of a Cauchy sequence to an upper set of integers is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm ℂ)) ⇒ ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ↾ 𝑍) ∈ (Cau‘𝐷))) | ||
Theorem | caushft 35199* | A shifted Cauchy sequence is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ 𝑊 = (ℤ≥‘(𝑀 + 𝑁)) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝑁))) & ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) & ⊢ (𝜑 → 𝐺:𝑊⟶𝑋) ⇒ ⊢ (𝜑 → 𝐺 ∈ (Cau‘𝐷)) | ||
Theorem | constcncf 35200* | A constant function is a continuous function on ℂ. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved into main set.mm as cncfmptc 23517 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.) |
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝐴) ⇒ ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ)) |
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