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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | wl-sbcom2d-lem2 35901* | Lemma used to prove wl-sbcom2d 35902. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.) |
⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝜑))) | ||
Theorem | wl-sbcom2d 35902 | Version of sbcom2 2162 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.) |
⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑤) & ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧) & ⊢ (𝜑 → ¬ ∀𝑧 𝑧 = 𝑦) ⇒ ⊢ (𝜑 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓)) | ||
Theorem | wl-sbalnae 35903 | A theorem used in elimination of disjoint variable restrictions by replacing them with distinctors. (Contributed by Wolf Lammen, 25-Jul-2019.) |
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | ||
Theorem | wl-sbal1 35904* | A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) Proof is based on wl-sbalnae 35903 now. See also sbal1 2533. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | ||
Theorem | wl-sbal2 35905* | Move quantifier in and out of substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 2-Jan-2002.) Proof is based on wl-sbalnae 35903 now. See also sbal2 2534. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)) | ||
Theorem | wl-2spsbbi 35906 | spsbbi 2077 applied twice. (Contributed by Wolf Lammen, 5-Aug-2023.) |
⊢ (∀𝑎∀𝑏(𝜑 ↔ 𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓)) | ||
Theorem | wl-lem-exsb 35907* | This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | wl-lem-nexmo 35908 | This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.) |
⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑧)) | ||
Theorem | wl-lem-moexsb 35909* |
The antecedent ∀𝑥(𝜑 → 𝑥 = 𝑧) relates to ∃*𝑥𝜑, but is
better suited for usage in proofs. Note that no distinct variable
restriction is placed on 𝜑.
This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.) |
⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑)) | ||
Theorem | wl-alanbii 35910 | This theorem extends alanimi 1819 to a biconditional. Recurrent usage stacks up more quantifiers. (Contributed by Wolf Lammen, 4-Oct-2019.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (∀𝑥𝜑 ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒)) | ||
Theorem | wl-mo2df 35911 | Version of mof 2563 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 11-Aug-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦) & ⊢ (𝜑 → Ⅎ𝑦𝜓) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦))) | ||
Theorem | wl-mo2tf 35912 | Closed form of mof 2563 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 20-Sep-2020.) |
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | wl-eudf 35913 | Version of eu6 2574 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦) & ⊢ (𝜑 → Ⅎ𝑦𝜓) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦))) | ||
Theorem | wl-eutf 35914 | Closed form of eu6 2574 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.) |
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) | ||
Theorem | wl-euequf 35915 | euequ 2597 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦) | ||
Theorem | wl-mo2t 35916* | Closed form of mof 2563. (Contributed by Wolf Lammen, 18-Aug-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | wl-mo3t 35917* | Closed form of mo3 2564. (Contributed by Wolf Lammen, 18-Aug-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) | ||
Theorem | wl-sb8eut 35918 | Substitution of variable in universal quantifier. Closed form of sb8eu 2600. (Contributed by Wolf Lammen, 11-Aug-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)) | ||
Theorem | wl-sb8mot 35919 |
Substitution of variable in universal quantifier. Closed form of
sb8mo 2601.
This theorem relates to wl-mo3t 35917, since replacing 𝜑 with [𝑦 / 𝑥]𝜑 in the latter yields subexpressions like [𝑥 / 𝑦][𝑦 / 𝑥]𝜑, which can be reduced to 𝜑 via sbft 2263 and sbco 2511. So ∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑 is provable from wl-mo3t 35917 in a simple fashion, unfortunately only if 𝑥 and 𝑦 are known to be distinct. To avoid any hassle with distinctors, we prefer to derive this theorem independently, ignoring the close connection between both theorems. From an educational standpoint, one would assume wl-mo3t 35917 to be more fundamental, as it hints how the "at most one" objects on both sides of the biconditional correlate (they are the same), if they exist at all, and then prove this theorem from it. (Contributed by Wolf Lammen, 11-Aug-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)) | ||
Theorem | wl-issetft 35920 | A closed form of issetf 3458. The proof here is a modification of a subproof in vtoclgft 3508, where it could be used to shorten the proof. (Contributed by Wolf Lammen, 25-Jan-2025.) |
⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)) | ||
Theorem | wl-axc11rc11 35921 |
Proving axc11r 2366 from axc11 2430. The hypotheses are two instances of
axc11 2430 used in the proof here. Some systems
introduce axc11 2430 as an
axiom, see for example System S2 in
https://us.metamath.org/downloads/finiteaxiom.pdf 2430.
By contrast, this database sees the variant axc11r 2366, directly derived from ax-12 2172, as foundational. Later axc11 2430 is proven somewhat trickily, requiring ax-10 2138 and ax-13 2372, see its proof. (Contributed by Wolf Lammen, 18-Jul-2023.) |
⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥)) & ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) ⇒ ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
Axiom | ax-wl-11v 35922* | Version of ax-11 2155 with distinct variable conditions. Currently implemented as an axiom to detect unintended references to the foundational axiom ax-11 2155. It will later be converted into a theorem directly based on ax-11 2155. (Contributed by Wolf Lammen, 28-Jun-2019.) |
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | wl-ax11-lem1 35923 | A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 ↔ ∀𝑦 𝑦 = 𝑧)) | ||
Theorem | wl-ax11-lem2 35924* | Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦) | ||
Theorem | wl-ax11-lem3 35925* | Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑢 𝑢 = 𝑦) | ||
Theorem | wl-ax11-lem4 35926* | Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ Ⅎ𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | wl-ax11-lem5 35927 | Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑)) | ||
Theorem | wl-ax11-lem6 35928* | Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑)) | ||
Theorem | wl-ax11-lem7 35929 | Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ (∀𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑)) | ||
Theorem | wl-ax11-lem8 35930* | Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥𝜑)) | ||
Theorem | wl-ax11-lem9 35931 | The easy part when 𝑥 coincides with 𝑦. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)) | ||
Theorem | wl-ax11-lem10 35932* | We now have prepared everything. The unwanted variable 𝑢 is just in one place left. pm2.61 191 can be used in conjunction with wl-ax11-lem9 35931 to eliminate the second antecedent. Missing is something along the lines of ax-6 1972, so we could remove the first antecedent. But the Metamath axioms cannot accomplish this. Such a rule must reside one abstraction level higher than all others: It says that a distinctor implies a distinct variable condition on its contained setvar. This is only needed if such conditions are required, as ax-11v does. The result of this study is for me, that you cannot introduce a setvar capturing this condition, and hope to eliminate it later. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ (∀𝑦 𝑦 = 𝑢 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑))) | ||
Theorem | wl-clabv 35933* |
Variant of df-clab 2716, where the element 𝑥 is required to be
disjoint from the class it is taken from. This restriction meets
similar ones found in other definitions and axioms like ax-ext 2709,
df-clel 2816 and df-cleq 2730. 𝑥 ∈ 𝐴 with 𝐴 depending on 𝑥 can
be the source of side effects, that you rather want to be aware of. So
here we eliminate one possible way of letting this slip in instead.
An expression 𝑥 ∈ 𝐴 with 𝑥, 𝐴 not disjoint, is now only introduced either via ax-8 2109, ax-9 2117, or df-clel 2816. Theorem cleljust 2116 shows that a possible choice does not matter. The original df-clab 2716 can be rederived, see wl-dfclab 35934. In an implementation this theorem is the only user of df-clab. (Contributed by NM, 26-May-1993.) Element and class are disjoint. (Revised by Wolf Lammen, 31-May-2023.) |
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) | ||
Theorem | wl-dfclab 35934 | Rederive df-clab 2716 from wl-clabv 35933. (Contributed by Wolf Lammen, 31-May-2023.) (Proof modification is discouraged.) |
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) | ||
Theorem | wl-clabtv 35935* | Using class abstraction in a context, requiring 𝑥 and 𝜑 disjoint, but based on fewer axioms than wl-clabt 35936. (Contributed by Wolf Lammen, 29-May-2023.) |
⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)}) | ||
Theorem | wl-clabt 35936 | Using class abstraction in a context. For a version based on fewer axioms see wl-clabtv 35935. (Contributed by Wolf Lammen, 29-May-2023.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)}) | ||
Theorem | rabiun 35937* | Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017.) |
⊢ {𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∣ 𝜑} = ∪ 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝜑} | ||
Theorem | iundif1 35938* | Indexed union of class difference with the subtrahend held constant. (Contributed by Brendan Leahy, 6-Aug-2018.) |
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶) | ||
Theorem | imadifss 35939 | The difference of images is a subset of the image of the difference. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵)) | ||
Theorem | cureq 35940 | Equality theorem for currying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵) | ||
Theorem | unceq 35941 | Equality theorem for uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵) | ||
Theorem | curf 35942 | Functional property of currying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵)) | ||
Theorem | uncf 35943 | Functional property of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶) | ||
Theorem | curfv 35944 | Value of currying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵)) | ||
Theorem | uncov 35945 | Value of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹‘𝐴)‘𝐵)) | ||
Theorem | curunc 35946 | Currying of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹) | ||
Theorem | unccur 35947 | Uncurrying of currying. (Contributed by Brendan Leahy, 5-Jun-2021.) |
⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = 𝐹) | ||
Theorem | phpreu 35948* | Theorem related to pigeonhole principle. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥 = 𝐶)) | ||
Theorem | finixpnum 35949* | A finite Cartesian product of numerable sets is numerable. (Contributed by Brendan Leahy, 24-Feb-2019.) |
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ dom card) → X𝑥 ∈ 𝐴 𝐵 ∈ dom card) | ||
Theorem | fin2solem 35950* | Lemma for fin2so 35951. (Contributed by Brendan Leahy, 29-Jun-2019.) |
⊢ ((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) → (𝑦𝑅𝑧 → {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} [⊊] {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧})) | ||
Theorem | fin2so 35951 | Any totally ordered Tarski-finite set is finite; in particular, no amorphous set can be ordered. Theorem 2 of [Levy58]] p. 4. (Contributed by Brendan Leahy, 28-Jun-2019.) |
⊢ ((𝐴 ∈ FinII ∧ 𝑅 Or 𝐴) → 𝐴 ∈ Fin) | ||
Theorem | ltflcei 35952 | Theorem to move the floor function across a strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((⌊‘𝐴) < 𝐵 ↔ 𝐴 < -(⌊‘-𝐵))) | ||
Theorem | leceifl 35953 | Theorem to move the floor function across a non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-(⌊‘-𝐴) ≤ 𝐵 ↔ 𝐴 ≤ (⌊‘𝐵))) | ||
Theorem | sin2h 35954 | Half-angle rule for sine. (Contributed by Brendan Leahy, 3-Aug-2018.) |
⊢ (𝐴 ∈ (0[,](2 · π)) → (sin‘(𝐴 / 2)) = (√‘((1 − (cos‘𝐴)) / 2))) | ||
Theorem | cos2h 35955 | Half-angle rule for cosine. (Contributed by Brendan Leahy, 4-Aug-2018.) |
⊢ (𝐴 ∈ (-π[,]π) → (cos‘(𝐴 / 2)) = (√‘((1 + (cos‘𝐴)) / 2))) | ||
Theorem | tan2h 35956 | Half-angle rule for tangent. (Contributed by Brendan Leahy, 4-Aug-2018.) |
⊢ (𝐴 ∈ (0[,)π) → (tan‘(𝐴 / 2)) = (√‘((1 − (cos‘𝐴)) / (1 + (cos‘𝐴))))) | ||
Theorem | lindsadd 35957 | In a vector space, the union of an independent set and a vector not in its span is an independent set. (Contributed by Brendan Leahy, 4-Mar-2023.) |
⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → (𝐹 ∪ {𝑋}) ∈ (LIndS‘𝑊)) | ||
Theorem | lindsdom 35958 | A linearly independent set in a free linear module of finite dimension over a division ring is smaller than the dimension of the module. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝑋 ≼ 𝐼) | ||
Theorem | lindsenlbs 35959 | A maximal linearly independent set in a free module of finite dimension over a division ring is a basis. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → 𝑋 ∈ (LBasis‘(𝑅 freeLMod 𝐼))) | ||
Theorem | matunitlindflem1 35960 | One direction of matunitlindf 35962. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (¬ curry 𝑀 LIndF (𝑅 freeLMod 𝐼) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅))) | ||
Theorem | matunitlindflem2 35961 | One direction of matunitlindf 35962. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅)) | ||
Theorem | matunitlindf 35962 | A matrix over a field is invertible iff the rows are linearly independent. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ (Unit‘(𝐼 Mat 𝑅)) ↔ curry 𝑀 LIndF (𝑅 freeLMod 𝐼))) | ||
Theorem | ptrest 35963* | Expressing a restriction of a product topology as a product topology. (Contributed by Brendan Leahy, 24-Mar-2019.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶Top) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ∈ 𝑊) ⇒ ⊢ (𝜑 → ((∏t‘𝐹) ↾t X𝑘 ∈ 𝐴 𝑆) = (∏t‘(𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) ↾t 𝑆)))) | ||
Theorem | ptrecube 35964* | Any point in an open set of N-space is surrounded by an open cube within that set. (Contributed by Brendan Leahy, 21-Aug-2020.) (Proof shortened by AV, 28-Sep-2020.) |
⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))})) & ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ ((𝑆 ∈ 𝑅 ∧ 𝑃 ∈ 𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆) | ||
Theorem | poimirlem1 35965* | Lemma for poimir 35997- the vertices on either side of a skipped vertex differ in at least two dimensions. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶ℤ) & ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) ⇒ ⊢ (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)) | ||
Theorem | poimirlem2 35966* | Lemma for poimir 35997- consecutive vertices differ in at most one dimension. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶ℤ) & ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁)) & ⊢ (𝜑 → 𝑉 ∈ (1...(𝑁 − 1))) & ⊢ (𝜑 → 𝑀 ∈ ((0...𝑁) ∖ {𝑉})) ⇒ ⊢ (𝜑 → ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹‘𝑉)‘𝑛)) | ||
Theorem | poimirlem3 35967* | Lemma for poimir 35997 to add an interior point to an admissible face on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 < 𝑁) & ⊢ (𝜑 → 𝑇:(1...𝑀)⟶(0..^𝐾)) & ⊢ (𝜑 → 𝑈:(1...𝑀)–1-1-onto→(1...𝑀)) ⇒ ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))) | ||
Theorem | poimirlem4 35968* | Lemma for poimir 35997 connecting the admissible faces on the back face of the (𝑀 + 1)-cube to admissible simplices in the 𝑀-cube. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 < 𝑁) ⇒ ⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}) | ||
Theorem | poimirlem5 35969* | Lemma for poimir 35997 to establish that, for the simplices defined by a walk along the edges of an 𝑁-cube, if the starting vertex is not opposite a given face, it is the earliest vertex of the face on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 0 < (2nd ‘𝑇)) ⇒ ⊢ (𝜑 → (𝐹‘0) = (1st ‘(1st ‘𝑇))) | ||
Theorem | poimirlem6 35970* | Lemma for poimir 35997 establishing, for a face of a simplex defined by a walk along the edges of an 𝑁-cube, the single dimension in which successive vertices before the opposite vertex differ. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1))) & ⊢ (𝜑 → 𝑀 ∈ (1...((2nd ‘𝑇) − 1))) ⇒ ⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)) = ((2nd ‘(1st ‘𝑇))‘𝑀)) | ||
Theorem | poimirlem7 35971* | Lemma for poimir 35997, similar to poimirlem6 35970, but for vertices after the opposite vertex. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1))) & ⊢ (𝜑 → 𝑀 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) ⇒ ⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)) = ((2nd ‘(1st ‘𝑇))‘𝑀)) | ||
Theorem | poimirlem8 35972* | Lemma for poimir 35997, establishing that away from the opposite vertex the walks in poimirlem9 35973 yield the same vertices. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1))) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) | ||
Theorem | poimirlem9 35973* | Lemma for poimir 35997, establishing the two walks that yield a given face when the opposite vertex is neither first nor last. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1))) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) ≠ (2nd ‘(1st ‘𝑇))) ⇒ ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) = ((2nd ‘(1st ‘𝑇)) ∘ ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))))) | ||
Theorem | poimirlem10 35974* | Lemma for poimir 35997 establishing the cube that yields the simplex that yields a face if the opposite vertex was first on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → (2nd ‘𝑇) = 0) ⇒ ⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st ‘𝑇))) | ||
Theorem | poimirlem11 35975* | Lemma for poimir 35997 connecting walks that could yield from a given cube a given face opposite the first vertex of the walk. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → (2nd ‘𝑇) = 0) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → (2nd ‘𝑈) = 0) & ⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) ⇒ ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ((2nd ‘(1st ‘𝑈)) “ (1...𝑀))) | ||
Theorem | poimirlem12 35976* | Lemma for poimir 35997 connecting walks that could yield from a given cube a given face opposite the final vertex of the walk. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → (2nd ‘𝑈) = 𝑁) & ⊢ (𝜑 → 𝑀 ∈ (0...(𝑁 − 1))) ⇒ ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ((2nd ‘(1st ‘𝑈)) “ (1...𝑀))) | ||
Theorem | poimirlem13 35977* | Lemma for poimir 35997- for at most one simplex associated with a shared face is the opposite vertex first on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) ⇒ ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0) | ||
Theorem | poimirlem14 35978* | Lemma for poimir 35997- for at most one simplex associated with a shared face is the opposite vertex last on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) ⇒ ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁) | ||
Theorem | poimirlem15 35979* | Lemma for poimir 35997, that the face in poimirlem22 35986 is a face. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1))) ⇒ ⊢ (𝜑 → ⟨⟨(1st ‘(1st ‘𝑇)), ((2nd ‘(1st ‘𝑇)) ∘ ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))))⟩, (2nd ‘𝑇)⟩ ∈ 𝑆) | ||
Theorem | poimirlem16 35980* | Lemma for poimir 35997 establishing the vertices of the simplex of poimirlem17 35981. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾) & ⊢ (𝜑 → (2nd ‘𝑇) = 0) ⇒ ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0))) ∘f + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))) | ||
Theorem | poimirlem17 35981* | Lemma for poimir 35997 establishing existence for poimirlem18 35982. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾) & ⊢ (𝜑 → (2nd ‘𝑇) = 0) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇) | ||
Theorem | poimirlem18 35982* | Lemma for poimir 35997 stating that, given a face not on a front face of the main cube and a simplex in which it's opposite the first vertex on the walk, there exists exactly one other simplex containing it. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾) & ⊢ (𝜑 → (2nd ‘𝑇) = 0) ⇒ ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇) | ||
Theorem | poimirlem19 35983* | Lemma for poimir 35997 establishing the vertices of the simplex in poimirlem20 35984. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0) & ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁) ⇒ ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))) | ||
Theorem | poimirlem20 35984* | Lemma for poimir 35997 establishing existence for poimirlem21 35985. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0) & ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇) | ||
Theorem | poimirlem21 35985* | Lemma for poimir 35997 stating that, given a face not on a back face of the cube and a simplex in which it's opposite the final point of the walk, there exists exactly one other simplex containing it. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0) & ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁) ⇒ ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇) | ||
Theorem | poimirlem22 35986* | Lemma for poimir 35997, that a given face belongs to exactly two simplices, provided it's not on the boundary of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} & ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾) ⇒ ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇) | ||
Theorem | poimirlem23 35987* | Lemma for poimir 35997, two ways of expressing the property that a face is not on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾)) & ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁)) & ⊢ (𝜑 → 𝑉 ∈ (0...𝑁)) ⇒ ⊢ (𝜑 → (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁)))) | ||
Theorem | poimirlem24 35988* | Lemma for poimir 35997, two ways of expressing that a simplex has an admissible face on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾)) & ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁)) & ⊢ (𝜑 → 𝑉 ∈ (0...𝑁)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁))))) | ||
Theorem | poimirlem25 35989* | Lemma for poimir 35997 stating that for a given simplex such that no vertex maps to 𝑁, the number of admissible faces is even. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾)) & ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁)) & ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑁 ≠ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶) ⇒ ⊢ (𝜑 → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶})) | ||
Theorem | poimirlem26 35990* | Lemma for poimir 35997 showing an even difference between the number of admissible faces and the number of admissible simplices. Equation (6) of [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) ⇒ ⊢ (𝜑 → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))) | ||
Theorem | poimirlem27 35991* | Lemma for poimir 35997 showing that the difference between admissible faces in the whole cube and admissible faces on the back face is even. Equation (7) of [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) & ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) & ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) ⇒ ⊢ (𝜑 → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}))) | ||
Theorem | poimirlem28 35992* | Lemma for poimir 35997, a variant of Sperner's lemma. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) & ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) & ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) & ⊢ (𝜑 → 𝐾 ∈ ℕ) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) | ||
Theorem | poimirlem29 35993* | Lemma for poimir 35997 connecting cubes of the tessellation to neighborhoods. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁)) & ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))})) & ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅)) & ⊢ 𝑋 = ((𝐹‘(((1st ‘(𝐺‘𝑘)) ∘f + ((((2nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘f / ((1...𝑁) × {𝑘})))‘𝑛) & ⊢ (𝜑 → 𝐺:ℕ⟶((ℕ0 ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘)) & ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋) ⇒ ⊢ (𝜑 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝐶 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))) | ||
Theorem | poimirlem30 35994* | Lemma for poimir 35997 combining poimirlem29 35993 with bwth 22683. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁)) & ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))})) & ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅)) & ⊢ 𝑋 = ((𝐹‘(((1st ‘(𝐺‘𝑘)) ∘f + ((((2nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘f / ((1...𝑁) × {𝑘})))‘𝑛) & ⊢ (𝜑 → 𝐺:ℕ⟶((ℕ0 ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘)) & ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))) | ||
Theorem | poimirlem31 35995* | Lemma for poimir 35997, assigning values to the vertices of the tessellation that meet the hypotheses of both poimirlem30 35994 and poimirlem28 35992. Equation (2) of [Kulpa] p. 547. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁)) & ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))})) & ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅)) & ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0) & ⊢ 𝑃 = ((1st ‘(𝐺‘𝑘)) ∘f + ((((2nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) & ⊢ (𝜑 → 𝐺:ℕ⟶((ℕ0 ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘)) & ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < )) ⇒ ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃 ∘f / ((1...𝑁) × {𝑘})))‘𝑛)) | ||
Theorem | poimirlem32 35996* | Lemma for poimir 35997, combining poimirlem28 35992, poimirlem30 35994, and poimirlem31 35995 to get Equation (1) of [Kulpa] p. 547. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁)) & ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))})) & ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅)) & ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0) & ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛)) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))) | ||
Theorem | poimir 35997* | Poincare-Miranda theorem. Theorem on [Kulpa] p. 547. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁)) & ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))})) & ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅)) & ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0) & ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛)) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 (𝐹‘𝑐) = ((1...𝑁) × {0})) | ||
Theorem | broucube 35998* | Brouwer - or as Kulpa calls it, "Bohl-Brouwer" - fixed point theorem for the unit cube. Theorem on [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁)) & ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))})) & ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn (𝑅 ↾t 𝐼))) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 𝑐 = (𝐹‘𝑐)) | ||
Theorem | heicant 35999 | Heine-Cantor theorem: a continuous mapping between metric spaces whose domain is compact is uniformly continuous. Theorem on [Rosenlicht] p. 80. (Contributed by Brendan Leahy, 13-Aug-2018.) (Proof shortened by AV, 27-Sep-2020.) |
⊢ (𝜑 → 𝐶 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑌)) & ⊢ (𝜑 → (MetOpen‘𝐶) ∈ Comp) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝑌 ≠ ∅) ⇒ ⊢ (𝜑 → ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) = ((MetOpen‘𝐶) Cn (MetOpen‘𝐷))) | ||
Theorem | opnmbllem0 36000* | Lemma for ismblfin 36005; could also be used to shorten proof of opnmbllem 24887. (Contributed by Brendan Leahy, 13-Jul-2018.) |
⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}) = 𝐴) |
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