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Most recent proofs    These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the iset.mm database for the Intuitionistic Logic Explorer. The iset.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from the commit given on the MPE Most Recent Proofs page. The database from that commit is also available here: iset.mm.

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Last updated on 23-Feb-2024 at 6:35 AM ET.
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DateLabelDescription
Theorem

15-Feb-2024dedekindicclemeu 12673 Lemma for dedekindicc 12674. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐶 ∈ (𝐴[,]𝐵))    &   (𝜑 → (∀𝑞𝐿 𝑞 < 𝐶 ∧ ∀𝑟𝑈 𝐶 < 𝑟))    &   (𝜑𝐷 ∈ (𝐴[,]𝐵))    &   (𝜑 → (∀𝑞𝐿 𝑞 < 𝐷 ∧ ∀𝑟𝑈 𝐷 < 𝑟))    &   (𝜑𝐶 < 𝐷)       (𝜑 → ⊥)

15-Feb-2024dedekindicclemlu 12672 Lemma for dedekindicc 12674. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴 < 𝐵)       (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑞𝐿 𝑞 < 𝑥 ∧ ∀𝑟𝑈 𝑥 < 𝑟))

15-Feb-2024dedekindicclemlub 12671 Lemma for dedekindicc 12674. The set L has a least upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴 < 𝐵)       (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑦𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧𝐿 𝑦 < 𝑧)))

15-Feb-2024dedekindicclemloc 12670 Lemma for dedekindicc 12674. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (∃𝑧𝐿 𝑥 < 𝑧 ∨ ∀𝑧𝐿 𝑧 < 𝑦)))

15-Feb-2024dedekindicclemub 12669 Lemma for dedekindicc 12674. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦𝐿 𝑦 < 𝑥)

15-Feb-2024dedekindicclemuub 12668 Lemma for dedekindicc 12674. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐶𝑈)       (𝜑 → ∀𝑧𝐿 𝑧 < 𝐶)

14-Feb-2024suplociccex 12667 An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7801 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 < 𝐶)    &   (𝜑𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ (𝐵[,]𝐶)(𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))       (𝜑 → ∃𝑥 ∈ (𝐵[,]𝐶)(∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐵[,]𝐶)(𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

14-Feb-2024suplociccreex 12666 An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7801 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 < 𝐶)    &   (𝜑𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ (𝐵[,]𝐶)(𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))       (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

2-Feb-2024dedekindeulemuub 12659 Lemma for dedekindeu 12665. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴𝑈)       (𝜑 → ∀𝑧𝐿 𝑧 < 𝐴)

31-Jan-2024dedekindeulemeu 12664 Lemma for dedekindeu 12665. Part of proving uniqueness. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (∀𝑞𝐿 𝑞 < 𝐴 ∧ ∀𝑟𝑈 𝐴 < 𝑟))    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (∀𝑞𝐿 𝑞 < 𝐵 ∧ ∀𝑟𝑈 𝐵 < 𝑟))    &   (𝜑𝐴 < 𝐵)       (𝜑 → ⊥)

31-Jan-2024dedekindeulemlu 12663 Lemma for dedekindeu 12665. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∃𝑥 ∈ ℝ (∀𝑞𝐿 𝑞 < 𝑥 ∧ ∀𝑟𝑈 𝑥 < 𝑟))

31-Jan-2024dedekindeulemlub 12662 Lemma for dedekindeu 12665. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐿 𝑦 < 𝑧)))

31-Jan-2024dedekindeulemloc 12661 Lemma for dedekindeu 12665. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐿 𝑥 < 𝑧 ∨ ∀𝑧𝐿 𝑧 < 𝑦)))

31-Jan-2024dedekindeulemub 12660 Lemma for dedekindeu 12665. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐿 𝑦 < 𝑥)

30-Jan-2024axsuploc 7801 An inhabited, bounded-above, located set of reals has a supremum. Axiom for real and complex numbers, derived from ZF set theory. (This restates ax-pre-suploc 7705 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.)
(((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

24-Jan-2024axpre-suploclemres 7673 Lemma for axpre-suploc 7674. The result. The proof just needs to define 𝐵 as basically the same set as 𝐴 (but expressed as a subset of R rather than a subset of ), and apply suplocsr 7581. (Contributed by Jim Kingdon, 24-Jan-2024.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥)    &   (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))    &   𝐵 = {𝑤R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴}       (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

23-Jan-2024ax-pre-suploc 7705 An inhabited, bounded-above, located set of reals has a supremum.

Locatedness here means that given 𝑥 < 𝑦, either there is an element of the set greater than 𝑥, or 𝑦 is an upper bound.

Although this and ax-caucvg 7704 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 7704.

(Contributed by Jim Kingdon, 23-Jan-2024.)

(((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

23-Jan-2024axpre-suploc 7674 An inhabited, bounded-above, located set of reals has a supremum.

Locatedness here means that given 𝑥 < 𝑦, either there is an element of the set greater than 𝑥, or 𝑦 is an upper bound.

This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 7705. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.)

(((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

22-Jan-2024suplocsr 7581 An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥R𝑦𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥R𝑦R (𝑥 <R 𝑦 → (∃𝑧𝐴 𝑥 <R 𝑧 ∨ ∀𝑧𝐴 𝑧 <R 𝑦)))       (𝜑 → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))

21-Jan-2024bj-el2oss1o 12792 Shorter proof of el2oss1o 12999 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ 2o𝐴 ⊆ 1o)

21-Jan-2024ltm1sr 7549 Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.)
(𝐴R → (𝐴 +R -1R) <R 𝐴)

19-Jan-2024suplocsrlempr 7579 Lemma for suplocsr 7581. The set 𝐵 has a least upper bound. (Contributed by Jim Kingdon, 19-Jan-2024.)
𝐵 = {𝑤P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   (𝜑𝐴R)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥R𝑦𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥R𝑦R (𝑥 <R 𝑦 → (∃𝑧𝐴 𝑥 <R 𝑧 ∨ ∀𝑧𝐴 𝑧 <R 𝑦)))       (𝜑 → ∃𝑣P (∀𝑤𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢)))

18-Jan-2024suplocsrlemb 7578 Lemma for suplocsr 7581. The set 𝐵 is located. (Contributed by Jim Kingdon, 18-Jan-2024.)
𝐵 = {𝑤P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   (𝜑𝐴R)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥R𝑦𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥R𝑦R (𝑥 <R 𝑦 → (∃𝑧𝐴 𝑥 <R 𝑧 ∨ ∀𝑧𝐴 𝑧 <R 𝑦)))       (𝜑 → ∀𝑢P𝑣P (𝑢<P 𝑣 → (∃𝑞𝐵 𝑢<P 𝑞 ∨ ∀𝑞𝐵 𝑞<P 𝑣)))

16-Jan-2024suplocsrlem 7580 Lemma for suplocsr 7581. The set 𝐴 has a least upper bound. (Contributed by Jim Kingdon, 16-Jan-2024.)
𝐵 = {𝑤P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   (𝜑𝐴R)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥R𝑦𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥R𝑦R (𝑥 <R 𝑦 → (∃𝑧𝐴 𝑥 <R 𝑧 ∨ ∀𝑧𝐴 𝑧 <R 𝑦)))       (𝜑 → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))

14-Jan-2024suplocexprlemlub 7496 Lemma for suplocexpr 7497. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑 → (𝑦<P 𝐵 → ∃𝑧𝐴 𝑦<P 𝑧))

14-Jan-2024suplocexprlemub 7495 Lemma for suplocexpr 7497. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑 → ∀𝑦𝐴 ¬ 𝐵<P 𝑦)

9-Jan-2024suplocexprlemloc 7493 Lemma for suplocexpr 7497. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑 → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵))))

9-Jan-2024suplocexprlemdisj 7492 Lemma for suplocexpr 7497. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑 → ∀𝑞Q ¬ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵)))

9-Jan-2024suplocexprlemru 7491 Lemma for suplocexpr 7497. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑 → ∀𝑟Q (𝑟 ∈ (2nd𝐵) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵))))

9-Jan-2024suplocexprlemrl 7489 Lemma for suplocexpr 7497. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))       (𝜑 → ∀𝑞Q (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))))

9-Jan-2024suplocexprlem2b 7486 Lemma for suplocexpr 7497. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝐴P → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})

9-Jan-2024suplocexprlemell 7485 Lemma for suplocexpr 7497. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝐵 (1st𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (1st𝑥))

7-Jan-2024suplocexpr 7497 An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))       (𝜑 → ∃𝑥P (∀𝑦𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧)))

7-Jan-2024suplocexprlemex 7494 Lemma for suplocexpr 7497. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑𝐵P)

7-Jan-2024suplocexprlemmu 7490 Lemma for suplocexpr 7497. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑 → ∃𝑠Q 𝑠 ∈ (2nd𝐵))

7-Jan-2024suplocexprlemml 7488 Lemma for suplocexpr 7497. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))       (𝜑 → ∃𝑠Q 𝑠 (1st𝐴))

7-Jan-2024suplocexprlemss 7487 Lemma for suplocexpr 7497. 𝐴 is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))       (𝜑𝐴P)

5-Jan-2024dedekindicc 12674 A Dedekind cut identifies a unique real number. Similar to df-inp 7238 except that the the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 5-Jan-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴 < 𝐵)       (𝜑 → ∃!𝑥 ∈ (𝐴[,]𝐵)(∀𝑞𝐿 𝑞 < 𝑥 ∧ ∀𝑟𝑈 𝑥 < 𝑟))

5-Jan-2024dedekindeu 12665 A Dedekind cut identifies a unique real number. Similar to df-inp 7238 except that the the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 5-Jan-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∃!𝑥 ∈ ℝ (∀𝑞𝐿 𝑞 < 𝑥 ∧ ∀𝑟𝑈 𝑥 < 𝑟))

31-Dec-2023dvmptsubcn 12737 Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
((𝜑𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥 ∈ ℂ) → 𝐵𝑉)    &   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵))    &   ((𝜑𝑥 ∈ ℂ) → 𝐶 ∈ ℂ)    &   ((𝜑𝑥 ∈ ℂ) → 𝐷𝑊)    &   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐶)) = (𝑥 ∈ ℂ ↦ 𝐷))       (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴𝐶))) = (𝑥 ∈ ℂ ↦ (𝐵𝐷)))

31-Dec-2023dvmptnegcn 12736 Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
((𝜑𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥 ∈ ℂ) → 𝐵𝑉)    &   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵))       (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ -𝐴)) = (𝑥 ∈ ℂ ↦ -𝐵))

31-Dec-2023dvmptcmulcn 12735 Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
((𝜑𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥 ∈ ℂ) → 𝐵𝑉)    &   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵))    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝐶 · 𝐴))) = (𝑥 ∈ ℂ ↦ (𝐶 · 𝐵)))

31-Dec-2023brm 3946 If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
(𝐴𝑅𝐵 → ∃𝑥 𝑥𝑅)

30-Dec-2023dvmptccn 12731 Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 0))

30-Dec-2023dvmptidcn 12730 Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
(ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1)

25-Dec-2023ctfoex 6969 A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
(∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝐴 ∈ V)

23-Dec-2023enct 11841 Countability is invariant relative to equinumerosity. (Contributed by Jim Kingdon, 23-Dec-2023.)
(𝐴𝐵 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)))

23-Dec-2023enctlem 11840 Lemma for enct 11841. One direction of the biconditional. (Contributed by Jim Kingdon, 23-Dec-2023.)
(𝐴𝐵 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)))

23-Dec-2023omct 6968 ω is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
𝑓 𝑓:ω–onto→(ω ⊔ 1o)

21-Dec-2023dvcoapbr 12723 The chain rule for derivatives at a point. The 𝑢 # 𝐶 → (𝐺𝑢) # (𝐺𝐶) hypothesis constrains what functions work for 𝐺. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 21-Dec-2023.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌𝑋)    &   (𝜑𝑌𝑇)    &   (𝜑 → ∀𝑢𝑌 (𝑢 # 𝐶 → (𝐺𝑢) # (𝐺𝐶)))    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑇 ⊆ ℂ)    &   (𝜑 → (𝐺𝐶)(𝑆 D 𝐹)𝐾)    &   (𝜑𝐶(𝑇 D 𝐺)𝐿)    &   𝐽 = (MetOpen‘(abs ∘ − ))       (𝜑𝐶(𝑇 D (𝐹𝐺))(𝐾 · 𝐿))

19-Dec-2023apsscn 8371 The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.)
{𝑥𝐴𝑥 # 𝐵} ⊆ ℂ

19-Dec-2023aprcl 8370 Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
(𝐴 # 𝐵 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))

18-Dec-2023limccoap 12699 Composition of two limits. This theorem is only usable in the case where 𝑥 # 𝑋 implies R(x) # 𝐶 so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.)
((𝜑𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → 𝑅 ∈ {𝑤𝐵𝑤 # 𝐶})    &   ((𝜑𝑦 ∈ {𝑤𝐵𝑤 # 𝐶}) → 𝑆 ∈ ℂ)    &   (𝜑𝐶 ∈ ((𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑅) lim 𝑋))    &   (𝜑𝐷 ∈ ((𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ↦ 𝑆) lim 𝐶))    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑𝐷 ∈ ((𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑇) lim 𝑋))

16-Dec-2023cnreim 10690 Complex apartness in terms of real and imaginary parts. See also apreim 8328 which is similar but with different notation. (Contributed by Jim Kingdon, 16-Dec-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ ((ℜ‘𝐴) # (ℜ‘𝐵) ∨ (ℑ‘𝐴) # (ℑ‘𝐵))))

14-Dec-2023cnopnap 12658 The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.)
(𝐴 ∈ ℂ → {𝑤 ∈ ℂ ∣ 𝑤 # 𝐴} ∈ (MetOpen‘(abs ∘ − )))

14-Dec-2023cnovex 12260 The class of all continuous functions from a topology to another is a set. (Contributed by Jim Kingdon, 14-Dec-2023.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)

13-Dec-2023reopnap 12602 The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.)
(𝐴 ∈ ℝ → {𝑤 ∈ ℝ ∣ 𝑤 # 𝐴} ∈ (topGen‘ran (,)))

12-Dec-2023cnopncntop 12601 The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
ℂ ∈ (MetOpen‘(abs ∘ − ))

12-Dec-2023unicntopcntop 12600 The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
ℂ = (MetOpen‘(abs ∘ − ))

4-Dec-2023bj-pm2.18st 12769 Clavius law for stable formulas. See pm2.18dc 823. (Contributed by BJ, 4-Dec-2023.)
(STAB 𝜑 → ((¬ 𝜑𝜑) → 𝜑))

4-Dec-2023bj-nnclavius 12761 Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)
((¬ 𝜑𝜑) → ¬ ¬ 𝜑)

2-Dec-2023dvmulxx 12720 The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 12718. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐹))    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐺))       (𝜑 → ((𝑆 D (𝐹𝑓 · 𝐺))‘𝐶) = ((((𝑆 D 𝐹)‘𝐶) · (𝐺𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹𝐶))))

1-Dec-2023dvmulxxbr 12718 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmulxx 12720. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 1-Dec-2023.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (MetOpen‘(abs ∘ − ))       (𝜑𝐶(𝑆 D (𝐹𝑓 · 𝐺))((𝐾 · (𝐺𝐶)) + (𝐿 · (𝐹𝐶))))

29-Nov-2023subctctexmid 13007 If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.)
(𝜑 → ∀𝑥(∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠onto𝑥) → ∃𝑔 𝑔:ω–onto→(𝑥 ⊔ 1o)))    &   (𝜑 → ω ∈ Markov)       (𝜑EXMID)

29-Nov-2023ismkvnex 6995 The predicate of being Markov stated in terms of double negation and comparison with 1o. (Contributed by Jim Kingdon, 29-Nov-2023.)
(𝐴𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o𝑚 𝐴)(¬ ¬ ∃𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = 1o)))

28-Nov-2023exmid1stab 13006 If any proposition is stable, excluded middle follows. We are thinking of 𝑥 as a proposition and 𝑥 = {∅} as "x is true". (Contributed by Jim Kingdon, 28-Nov-2023.)
((𝜑𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})       (𝜑EXMID)

28-Nov-2023ccfunen 7043 Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
(𝜑CCHOICE)    &   (𝜑𝐴 ≈ ω)    &   (𝜑 → ∀𝑥𝐴𝑤 𝑤𝑥)       (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))

27-Nov-2023df-cc 7042 The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7026 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
(CCHOICE ↔ ∀𝑥(dom 𝑥 ≈ ω → ∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥)))

26-Nov-2023offeq 5961 Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐵𝑇)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝐶    &   (𝜑𝐻:𝐶𝑈)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐷)    &   ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐸)    &   ((𝜑𝑥𝐶) → (𝐷𝑅𝐸) = (𝐻𝑥))       (𝜑 → (𝐹𝑓 𝑅𝐺) = 𝐻)

25-Nov-2023dvaddxx 12719 The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 12717. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐹))    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐺))       (𝜑 → ((𝑆 D (𝐹𝑓 + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶)))

25-Nov-2023dvaddxxbr 12717 The sum rule for derivatives at a point. That is, if the derivative of 𝐹 at 𝐶 is 𝐾 and the derivative of 𝐺 at 𝐶 is 𝐿, then the derivative of the pointwise sum of those two functions at 𝐶 is 𝐾 + 𝐿. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (MetOpen‘(abs ∘ − ))       (𝜑𝐶(𝑆 D (𝐹𝑓 + 𝐺))(𝐾 + 𝐿))

25-Nov-2023dcnn 816 Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 810. The relation between dcn 810 and dcnn 816 is analogous to that between notnot 601 and notnotnot 606 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 816 means that a proposition is testable if and only if its negation is testable, and dcn 810 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.)
(DECID ¬ 𝜑DECID ¬ ¬ 𝜑)

24-Nov-2023bj-dcst 12778 Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)
(DECID STAB 𝜑STAB 𝜑)

24-Nov-2023bj-nnbidc 12773 If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 12764. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ 𝜑 → (DECID 𝜑𝜑))

24-Nov-2023bj-dcstab 12772 A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(DECID 𝜑STAB 𝜑)

24-Nov-2023bj-fadc 12771 A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
𝜑DECID 𝜑)

24-Nov-2023bj-trdc 12770 A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(𝜑DECID 𝜑)

24-Nov-2023bj-stal 12768 The universal quantification of stable formula is stable. See bj-stim 12765 for implication, stabnot 801 for negation, and bj-stan 12766 for conjunction. (Contributed by BJ, 24-Nov-2023.)
(∀𝑥STAB 𝜑STAB𝑥𝜑)

24-Nov-2023bj-stand 12767 The conjunction of two stable formulas is stable. Deduction form of bj-stan 12766. Its proof is shorter, so one could prove it first and then bj-stan 12766 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(𝜑STAB 𝜓)    &   (𝜑STAB 𝜒)       (𝜑STAB (𝜓𝜒))

24-Nov-2023bj-stan 12766 The conjunction of two stable formulas is stable. See bj-stim 12765 for implication, stabnot 801 for negation, and bj-stal 12768 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
((STAB 𝜑STAB 𝜓) → STAB (𝜑𝜓))

24-Nov-2023bj-stim 12765 A conjunction with a stable consequent is stable. See stabnot 801 for negation and bj-stan 12766 for conjunction. (Contributed by BJ, 24-Nov-2023.)
(STAB 𝜓STAB (𝜑𝜓))

24-Nov-2023bj-nnbist 12764 If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if 𝜑 is a classical tautology, then ¬ ¬ 𝜑 is an intuitionistic tautology. Therefore, if 𝜑 is a classical tautology, then 𝜑 is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 12773). (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ 𝜑 → (STAB 𝜑𝜑))

24-Nov-2023bj-fast 12763 A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)
𝜑STAB 𝜑)

24-Nov-2023bj-trst 12762 A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)
(𝜑STAB 𝜑)

24-Nov-2023bj-nnal 12760 The double negation of a universal quantification implies the universal quantification of the double negation. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)

24-Nov-2023bj-nnan 12759 The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ (𝜑𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))

24-Nov-2023bj-nnim 12758 The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ (𝜑𝜓) → (𝜑 → ¬ ¬ 𝜓))

24-Nov-2023bj-nnsn 12756 As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)
((𝜑 → ¬ 𝜓) ↔ (¬ ¬ 𝜑 → ¬ 𝜓))

22-Nov-2023ofvalg 5957 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)    &   ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)    &   ((𝜑𝑋𝑆) → (𝐶𝑅𝐷) ∈ 𝑈)       ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

21-Nov-2023exmidac 7029 The axiom of choice implies excluded middle. See acexmid 5739 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.)
(CHOICEEXMID)

21-Nov-2023exmidaclem 7028 Lemma for exmidac 7029. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}    &   𝐶 = {𝐴, 𝐵}       (CHOICEEXMID)

21-Nov-2023exmid1dc 4091 A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4085 or ordtriexmid 4405. In this context 𝑥 = {∅} can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
((𝜑𝑥 ⊆ {∅}) → DECID 𝑥 = {∅})       (𝜑EXMID)

20-Nov-2023acfun 7027 A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)
(𝜑CHOICE)    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑥𝐴𝑤 𝑤𝑥)       (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))

18-Nov-2023condc 821 Contraposition of a decidable proposition.

This theorem swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky." This theorem (without the decidability condition, of course) is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103) and is Axiom A3 of [Margaris] p. 49. We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning.

(Contributed by Jim Kingdon, 13-Mar-2018.) (Proof shortened by BJ, 18-Nov-2023.)

(DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))

18-Nov-2023const 820 Contraposition of a stable proposition. See comment of condc 821. (Contributed by BJ, 18-Nov-2023.)
(STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))

18-Nov-2023stdcn 815 A formula is stable if and only if the decidability of its negation implies its decidability. Note that the right-hand side of this biconditional is the converse of dcn 810. (Contributed by BJ, 18-Nov-2023.)
(STAB 𝜑 ↔ (DECID ¬ 𝜑DECID 𝜑))

17-Nov-2023cnplimclemr 12690 Lemma for cnplimccntop 12691. The reverse direction. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = (𝐾t 𝐴)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵𝐴)    &   (𝜑 → (𝐹𝐵) ∈ (𝐹 lim 𝐵))       (𝜑𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))

17-Nov-2023cnplimclemle 12689 Lemma for cnplimccntop 12691. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = (𝐾t 𝐴)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵𝐴)    &   (𝜑 → (𝐹𝐵) ∈ (𝐹 lim 𝐵))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝑍𝐴)    &   ((𝜑𝑍 # 𝐵 ∧ (abs‘(𝑍𝐵)) < 𝐷) → (abs‘((𝐹𝑍) − (𝐹𝐵))) < (𝐸 / 2))    &   (𝜑 → (abs‘(𝑍𝐵)) < 𝐷)       (𝜑 → (abs‘((𝐹𝑍) − (𝐹𝐵))) < 𝐸)

14-Nov-2023limccnp2cntop 12698 The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Nov-2023.)
((𝜑𝑥𝐴) → 𝑅𝑋)    &   ((𝜑𝑥𝐴) → 𝑆𝑌)    &   (𝜑𝑋 ⊆ ℂ)    &   (𝜑𝑌 ⊆ ℂ)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))    &   (𝜑𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵))    &   (𝜑𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵))    &   (𝜑𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))       (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵))

10-Nov-2023rpmaxcl 10935 The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → sup({𝐴, 𝐵}, ℝ, < ) ∈ ℝ+)

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