![]() |
Metamath
Proof Explorer Theorem List (p. 368 of 480) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30435) |
![]() (30436-31958) |
![]() (31959-47941) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | wl-nfnae1 36701 | Unlike nfnae 2432, this specialized theorem avoids ax-11 2153. (Contributed by Wolf Lammen, 27-Jun-2019.) |
⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑥 | ||
Theorem | wl-aetr 36702 | 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-axc11r 36703 | Same as axc11r 2364, but using ax12 2421 instead of ax-12 2170 directly. This better reflects axiom usage in theorems dependent on it. (Contributed by NM, 25-Jul-2015.) Avoid direct use of ax-12 2170. (Revised by Wolf Lammen, 30-Mar-2024.) |
⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
Theorem | wl-dral1d 36704 | A version of dral1 2437 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a 𝑥 = 𝑦, ∀𝑥𝑥 = 𝑦 or 𝜑 antecedent. wl-equsal1i 36716 and nf5di 2280 show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))) | ||
Theorem | wl-cbvalnaed 36705 | wl-cbvalnae 36706 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓)) & ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒)) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | wl-cbvalnae 36706 | A more general version of cbval 2396 when nonfree properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2446, nfsb2 2481 or dveeq1 2378. (Contributed by Wolf Lammen, 4-Jun-2019.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑) & ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | wl-exeq 36707 | The semantics of ∃𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.) |
⊢ (∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧)) | ||
Theorem | wl-aleq 36708 | The semantics of ∀𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.) |
⊢ (∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))) | ||
Theorem | wl-nfeqfb 36709 | Extend nfeqf 2379 to an equivalence. (Contributed by Wolf Lammen, 31-Jul-2019.) |
⊢ (Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)) | ||
Theorem | wl-nfs1t 36710 | If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Closed form of nfs1 2486. (Contributed by Wolf Lammen, 27-Jul-2019.) |
⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | wl-equsalvw 36711* |
Version of equsalv 2257 with a disjoint variable condition, and of equsal 2415
with two disjoint variable conditions, which requires fewer axioms. See
also the dual form equsexvw 2007.
This theorem lays the foundation to a transformation of expressions called substitution of set variables in a wff. Only in this particular context we additionally assume 𝜑 and 𝑦 disjoint, stated here as 𝜑(𝑥). Similarly the disjointness of 𝜓 and 𝑥 is expressed by 𝜓(𝑦). Both 𝜑 and 𝜓 may still depend on other set variables, but that is irrelevant here. We want to transform 𝜑(𝑥) into 𝜓(𝑦) such that 𝜓 depends on 𝑦 the same way as 𝜑 depends on 𝑥. This dependency is expressed in our hypothesis (called implicit substitution): (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)). For primitive enough 𝜑 a sort of textual substitution of 𝑥 by 𝑦 is sufficient for such transformation. But note: 𝜑 must not contain wff variables, and the substitution is no proper textual substitution either. We still need grammar information to not accidently replace the x in a token 'x.' denoting multiplication, but only catch set variables 𝑥. Our current stage of development allows only equations and quantifiers make up such primitives. Thanks to equequ1 2027 and cbvalvw 2038 we can then prove in a mechanical way that in fact the implicit substitution holds for each instance. If 𝜑 contains wff variables we cannot use textual transformation any longer, since we don't know how to replace 𝑦 for 𝑥 in placeholders of unknown structure. Our theorem now states, that the generic expression ∀𝑥(𝑥 = 𝑦 → 𝜑) formally behaves as if such a substitution was possible and made. (Contributed by BJ, 31-May-2019.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | wl-equsald 36712 | Deduction version of equsal 2415. (Contributed by Wolf Lammen, 27-Jul-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒)) | ||
Theorem | wl-equsal 36713 | A useful equivalence related to substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) It seems proving wl-equsald 36712 first, and then deriving more specialized versions wl-equsal 36713 and wl-equsal1t 36714 then is more efficient than the other way round, which is possible, too. See also equsal 2415. (Revised by Wolf Lammen, 27-Jul-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | wl-equsal1t 36714 |
The expression 𝑥 = 𝑦 in antecedent position plays an
important role in
predicate logic, namely in implicit substitution. However, occasionally
it is irrelevant, and can safely be dropped. A sufficient condition for
this is when 𝑥 (or 𝑦 or both) is not free in
𝜑.
This theorem is more fundamental than equsal 2415, spimt 2384 or sbft 2260, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)) | ||
Theorem | wl-equsalcom 36715 | This simple equivalence eases substitution of one expression for the other. (Contributed by Wolf Lammen, 1-Sep-2018.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑦 = 𝑥 → 𝜑)) | ||
Theorem | wl-equsal1i 36716 | The antecedent 𝑥 = 𝑦 is irrelevant, if one or both setvar variables are not free in 𝜑. (Contributed by Wolf Lammen, 1-Sep-2018.) |
⊢ (Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑) & ⊢ (𝑥 = 𝑦 → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | wl-sb6rft 36717 | A specialization of wl-equsal1t 36714. Closed form of sb6rf 2466. (Contributed by Wolf Lammen, 27-Jul-2019.) |
⊢ (Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))) | ||
Theorem | wl-cbvalsbi 36718* | Change bounded variables in a special case. The reverse direction seems to involve ax-11 2153. My hope is that I will in some future be able to prove mo3 2557 with reversed quantifiers not using ax-11 2153. See also the remark in mo4 2559, which lead me to this effort. (Contributed by Wolf Lammen, 5-Mar-2024.) |
⊢ (∀𝑥𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | wl-sbrimt 36719 | Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2299. (Contributed by Wolf Lammen, 26-Jul-2019.) |
⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))) | ||
Theorem | wl-sblimt 36720 | Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2299. (Contributed by Wolf Lammen, 26-Jul-2019.) |
⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))) | ||
Theorem | wl-sb8t 36721 | Substitution of variable in universal quantifier. Closed form of sb8 2515. (Contributed by Wolf Lammen, 27-Jul-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)) | ||
Theorem | wl-sb8et 36722 | Substitution of variable in universal quantifier. Closed form of sb8e 2516. (Contributed by Wolf Lammen, 27-Jul-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)) | ||
Theorem | wl-sbhbt 36723 | Closed form of sbhb 2519. Characterizing the expression 𝜑 → ∀𝑥𝜑 using a substitution expression. (Contributed by Wolf Lammen, 28-Jul-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))) | ||
Theorem | wl-sbnf1 36724 | Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2353. Note: This theorem shows that sbnf2 2353 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))) | ||
Theorem | wl-equsb3 36725 | equsb3 2100 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.) |
⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)) | ||
Theorem | wl-equsb4 36726 | Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable condition. (Contributed by Wolf Lammen, 26-Jun-2019.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧)) | ||
Theorem | wl-2sb6d 36727 | Version of 2sb6 2088 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.) |
⊢ (𝜑 → ¬ ∀𝑦 𝑦 = 𝑥) & ⊢ (𝜑 → ¬ ∀𝑦 𝑦 = 𝑤) & ⊢ (𝜑 → ¬ ∀𝑦 𝑦 = 𝑧) & ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧) ⇒ ⊢ (𝜑 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜓 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜓))) | ||
Theorem | wl-sbcom2d-lem1 36728* | Lemma used to prove wl-sbcom2d 36730. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.) |
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))) | ||
Theorem | wl-sbcom2d-lem2 36729* | Lemma used to prove wl-sbcom2d 36730. (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 36730 | Version of sbcom2 2160 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 36731 | 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 36732* | 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 36731 now. See also sbal1 2526. (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 36733* | 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 36731 now. See also sbal2 2527. (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 36734 | spsbbi 2075 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 36735* | 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 36736 | 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 36737* |
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 36738 | This theorem extends alanimi 1817 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 36739 | Version of mof 2556 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 36740 | Closed form of mof 2556 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 36741 | Version of eu6 2567 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 36742 | Closed form of eu6 2567 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 36743 | euequ 2590 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦) | ||
Theorem | wl-mo2t 36744* | Closed form of mof 2556. (Contributed by Wolf Lammen, 18-Aug-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | wl-mo3t 36745* | Closed form of mo3 2557. (Contributed by Wolf Lammen, 18-Aug-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) | ||
Theorem | wl-sb8eut 36746 | Substitution of variable in universal quantifier. Closed form of sb8eu 2593. (Contributed by Wolf Lammen, 11-Aug-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)) | ||
Theorem | wl-sb8mot 36747 |
Substitution of variable in universal quantifier. Closed form of
sb8mo 2594.
This theorem relates to wl-mo3t 36745, 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 2260 and sbco 2505. So ∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑 is provable from wl-mo3t 36745 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 36745 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 36748 | A closed form of issetf 3488. The proof here is a modification of a subproof in vtoclgft 3540, 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 36749 |
Proving axc11r 2364 from axc11 2428. The hypotheses are two instances of
axc11 2428 used in the proof here. Some systems
introduce axc11 2428 as an
axiom, see for example System S2 in
https://us.metamath.org/downloads/finiteaxiom.pdf 2428.
By contrast, this database sees the variant axc11r 2364, directly derived from ax-12 2170, as foundational. Later axc11 2428 is proven somewhat trickily, requiring ax-10 2136 and ax-13 2370, 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 36750* | Version of ax-11 2153 with distinct variable conditions. Currently implemented as an axiom to detect unintended references to the foundational axiom ax-11 2153. It will later be converted into a theorem directly based on ax-11 2153. (Contributed by Wolf Lammen, 28-Jun-2019.) |
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | wl-ax11-lem1 36751 | 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 36752* | Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦) | ||
Theorem | wl-ax11-lem3 36753* | Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑢 𝑢 = 𝑦) | ||
Theorem | wl-ax11-lem4 36754* | Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ Ⅎ𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | wl-ax11-lem5 36755 | Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑)) | ||
Theorem | wl-ax11-lem6 36756* | 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 36757 | Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
⊢ (∀𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑)) | ||
Theorem | wl-ax11-lem8 36758* | 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 36759 | 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 36760* | 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 36759 to eliminate the second antecedent. Missing is something along the lines of ax-6 1970, 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 36761* |
Variant of df-clab 2709, 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 2702,
df-clel 2809 and df-cleq 2723. 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 2107, ax-9 2115, or df-clel 2809. Theorem cleljust 2114 shows that a possible choice does not matter. The original df-clab 2709 can be rederived, see wl-dfclab 36762. 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 36762 | Rederive df-clab 2709 from wl-clabv 36761. (Contributed by Wolf Lammen, 31-May-2023.) (Proof modification is discouraged.) |
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) | ||
Theorem | wl-clabtv 36763* | Using class abstraction in a context, requiring 𝑥 and 𝜑 disjoint, but based on fewer axioms than wl-clabt 36764. (Contributed by Wolf Lammen, 29-May-2023.) |
⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)}) | ||
Theorem | wl-clabt 36764 | Using class abstraction in a context. For a version based on fewer axioms see wl-clabtv 36763. (Contributed by Wolf Lammen, 29-May-2023.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)}) | ||
Theorem | rabiun 36765* | 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 36766* | 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 36767 | 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 36768 | Equality theorem for currying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵) | ||
Theorem | unceq 36769 | Equality theorem for uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵) | ||
Theorem | curf 36770 | 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 36771 | Functional property of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶) | ||
Theorem | curfv 36772 | 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 36773 | Value of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹‘𝐴)‘𝐵)) | ||
Theorem | curunc 36774 | Currying of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹) | ||
Theorem | unccur 36775 | Uncurrying of currying. (Contributed by Brendan Leahy, 5-Jun-2021.) |
⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = 𝐹) | ||
Theorem | phpreu 36776* | 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 36777* | 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 36778* | Lemma for fin2so 36779. (Contributed by Brendan Leahy, 29-Jun-2019.) |
⊢ ((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) → (𝑦𝑅𝑧 → {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} [⊊] {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧})) | ||
Theorem | fin2so 36779 | 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 36780 | 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 36781 | 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 36782 | 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 36783 | Half-angle rule for cosine. (Contributed by Brendan Leahy, 4-Aug-2018.) |
⊢ (𝐴 ∈ (-π[,]π) → (cos‘(𝐴 / 2)) = (√‘((1 + (cos‘𝐴)) / 2))) | ||
Theorem | tan2h 36784 | 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 36785 | 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 36786 | 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 36787 | 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 36788 | One direction of matunitlindf 36790. (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 36789 | One direction of matunitlindf 36790. (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 36790 | 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 36791* | 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 36792* | 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 36793* | Lemma for poimir 36825- 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 36794* | Lemma for poimir 36825- 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 36795* | Lemma for poimir 36825 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 36796* | Lemma for poimir 36825 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 36797* | Lemma for poimir 36825 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 36798* | Lemma for poimir 36825 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 36799* | Lemma for poimir 36825, similar to poimirlem6 36798, 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 36800* | Lemma for poimir 36825, establishing that away from the opposite vertex the walks in poimirlem9 36801 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)}))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |