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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | axc5c711toc7 38901 | Rederivation of ax-c7 38866 from axc5c711 38899. Note that ax-c7 38866 and ax-11 2154 are not used by the rederivation. The use of alimi 1807 (which uses ax-c5 38864) is allowed since we have already proved axc5c711toc5 38900. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) | ||
Theorem | axc5c711to11 38902 | Rederivation of ax-11 2154 from axc5c711 38899. Note that ax-c7 38866 and ax-11 2154 are not used by the rederivation. The use of alimi 1807 (which uses ax-c5 38864) is allowed since we have already proved axc5c711toc5 38900. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | equidqe 38903 | equid 2008 with existential quantifier without using ax-c5 38864 or ax-5 1907. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥 | ||
Theorem | axc5sp1 38904 | A special case of ax-c5 38864 without using ax-c5 38864 or ax-5 1907. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥) | ||
Theorem | equidq 38905 | equid 2008 with universal quantifier without using ax-c5 38864 or ax-5 1907. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∀𝑦 𝑥 = 𝑥 | ||
Theorem | equid1ALT 38906 | Alternate proof of equid 2008 and equid1 38880 from older axioms ax-c7 38866, ax-c10 38867 and ax-c9 38871. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑥 = 𝑥 | ||
Theorem | axc11nfromc11 38907 |
Rederivation of ax-c11n 38869 from original version ax-c11 38868. See Theorem
axc11 2432 for the derivation of ax-c11 38868 from ax-c11n 38869.
This theorem should not be referenced in any proof. Instead, use ax-c11n 38869 above so that uses of ax-c11n 38869 can be more easily identified, or use aecom-o 38882 when this form is needed for studies involving ax-c11 38868 and omitting ax-5 1907. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Theorem | naecoms-o 38908 | A commutation rule for distinct variable specifiers. Version of naecoms 2431 using ax-c11 38868. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑) | ||
Theorem | hbnae-o 38909 | All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 2434 using ax-c11 38868. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | dvelimf-o 38910 | Proof of dvelimh 2452 that uses ax-c11 38868 but not ax-c15 38870, ax-c11n 38869, or ax-12 2174. Version of dvelimh 2452 using ax-c11 38868 instead of axc11 2432. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑧𝜓) & ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | dral2-o 38911 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral2 2440 using ax-c11 38868. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓)) | ||
Theorem | aev-o 38912* | A "distinctor elimination" lemma with no disjoint variable conditions on variables in the consequent, proved without using ax-c16 38873. Version of aev 2054 using ax-c11 38868. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣) | ||
Theorem | ax5eq 38913* | Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 1907 considered as a metatheorem. Do not use it for later proofs - use ax-5 1907 instead, to avoid reference to the redundant axiom ax-c16 38873.) (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) | ||
Theorem | dveeq2-o 38914* | Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2380 using ax-c15 38870. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||
Theorem | axc16g-o 38915* | A generalization of Axiom ax-c16 38873. Version of axc16g 2257 using ax-c11 38868. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | ||
Theorem | dveeq1-o 38916* | Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2382 using ax-c11 . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
Theorem | dveeq1-o16 38917* | Version of dveeq1 2382 using ax-c16 38873 instead of ax-5 1907. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1907. (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
Theorem | ax5el 38918* | Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1907 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦) | ||
Theorem | axc11n-16 38919* | This theorem shows that, given ax-c16 38873, we can derive a version of ax-c11n 38869. However, it is weaker than ax-c11n 38869 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥) | ||
Theorem | dveel2ALT 38920* | Alternate proof of dveel2 2464 using ax-c16 38873 instead of ax-5 1907. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦)) | ||
Theorem | ax12f 38921 | Basis step for constructing a substitution instance of ax-c15 38870 without using ax-c15 38870. We can start with any formula 𝜑 in which 𝑥 is not free. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | ||
Theorem | ax12eq 38922 | Basis step for constructing a substitution instance of ax-c15 38870 without using ax-c15 38870. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦 → 𝑧 = 𝑤)))) | ||
Theorem | ax12el 38923 | Basis step for constructing a substitution instance of ax-c15 38870 without using ax-c15 38870. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝑧 ∈ 𝑤 → ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑤)))) | ||
Theorem | ax12indn 38924 | Induction step for constructing a substitution instance of ax-c15 38870 without using ax-c15 38870. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))) | ||
Theorem | ax12indi 38925 | Induction step for constructing a substitution instance of ax-c15 38870 without using ax-c15 38870. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) & ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜓 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → ((𝜑 → 𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))))) | ||
Theorem | ax12indalem 38926 | Lemma for ax12inda2 38928 and ax12inda 38929. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) ⇒ ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))) | ||
Theorem | ax12inda2ALT 38927* | Alternate proof of ax12inda2 38928, slightly more direct and not requiring ax-c16 38873. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))) | ||
Theorem | ax12inda2 38928* | Induction step for constructing a substitution instance of ax-c15 38870 without using ax-c15 38870. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 38929. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))) | ||
Theorem | ax12inda 38929* | Induction step for constructing a substitution instance of ax-c15 38870 without using ax-c15 38870. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 38928 may be used instead to avoid the dummy variable 𝑤 in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑤 → (𝑥 = 𝑤 → (𝜑 → ∀𝑥(𝑥 = 𝑤 → 𝜑)))) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))) | ||
Theorem | ax12v2-o 38930* | Rederivation of ax-c15 38870 from ax12v 2175 (without using ax-c15 38870 or the full ax-12 2174). Thus, the hypothesis (ax12v 2175) provides an alternate axiom that can be used in place of ax-c15 38870. See also axc15 2424. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | ||
Theorem | ax12a2-o 38931* | Derive ax-c15 38870 from a hypothesis in the form of ax-12 2174, without using ax-12 2174 or ax-c15 38870. The hypothesis is weaker than ax-12 2174, with 𝑧 both distinct from 𝑥 and not occurring in 𝜑. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-12 2174, if we also have ax-c11 38868, which this proof uses. As Theorem ax12 2425 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 38869 instead of ax-c11 38868. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | ||
Theorem | axc11-o 38932 |
Show that ax-c11 38868 can be derived from ax-c11n 38869 and ax-12 2174. An open
problem is whether this theorem can be derived from ax-c11n 38869 and the
others when ax-12 2174 is replaced with ax-c15 38870 or ax12v 2175. See Theorems
axc11nfromc11 38907 for the rederivation of ax-c11n 38869 from axc11 2432.
Normally, axc11 2432 should be used rather than ax-c11 38868 or axc11-o 38932, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
Theorem | fsumshftd 38933* | Index shift of a finite sum with a weaker "implicit substitution" hypothesis than fsumshft 15812. The proof demonstrates how this can be derived starting from from fsumshft 15812. (Contributed by NM, 1-Nov-2019.) |
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑗 = (𝑘 − 𝐾)) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵) | ||
Axiom | ax-riotaBAD 38934 | Define restricted description binder. In case it doesn't exist, we return a set which is not a member of the domain of discourse 𝐴. See also comments for df-iota 6515. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) WARNING: THIS "AXIOM", WHICH IS THE OLD df-riota 7387, CONFLICTS WITH (THE NEW) df-riota 7387 AND MAKES THE SYSTEM IN set.mm INCONSISTENT. IT IS TEMPORARY AND WILL BE DELETED AFTER ALL USES ARE ELIMINATED. |
⊢ (℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴})) | ||
Theorem | riotaclbgBAD 38935* | Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)) | ||
Theorem | riotaclbBAD 38936* | Closure of restricted iota. (Contributed by NM, 15-Sep-2011.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | ||
Theorem | riotasvd 38937* | Deduction version of riotasv 38940. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))) & ⊢ (𝜑 → 𝐷 ∈ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶)) | ||
Theorem | riotasv2d 38938* | Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5408). Special case of riota2f 7411. (Contributed by NM, 2-Mar-2013.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝐹) & ⊢ (𝜑 → Ⅎ𝑦𝜒) & ⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))) & ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → 𝐶 = 𝐹) & ⊢ (𝜑 → 𝐷 ∈ 𝐴) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ (𝜑 → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = 𝐹) | ||
Theorem | riotasv2s 38939* | The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5408) in the form of a substitution instance. Special case of riota2f 7411. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) |
⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶) | ||
Theorem | riotasv 38940* | Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5408). Special case of riota2f 7411. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)) ⇒ ⊢ ((𝐷 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶) | ||
Theorem | riotasv3d 38941* | A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5408) also holds for its description binder 𝐷 (in the form of property 𝜃). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜃) & ⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))) & ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃)) & ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒)) & ⊢ (𝜑 → 𝐷 ∈ 𝐴) & ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝜃) | ||
Theorem | elimhyps 38942 | A version of elimhyp 4595 using explicit substitution. (Contributed by NM, 15-Jun-2019.) |
⊢ [𝐵 / 𝑥]𝜑 ⇒ ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑 | ||
Theorem | dedths 38943 | A version of weak deduction theorem dedth 4588 using explicit substitution. (Contributed by NM, 15-Jun-2019.) |
⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | renegclALT 38944 | Closure law for negative of reals. Demonstrates use of weak deduction theorem with explicit substitution. The proof is much longer than that of renegcl 11569. (Contributed by NM, 15-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | ||
Theorem | elimhyps2 38945 | Generalization of elimhyps 38942 that is not useful unless we can separately prove ⊢ 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.) |
⊢ [𝐵 / 𝑥]𝜑 ⇒ ⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑 | ||
Theorem | dedths2 38946 | Generalization of dedths 38943 that is not useful unless we can separately prove ⊢ 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.) |
⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓 ⇒ ⊢ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) | ||
Theorem | nfcxfrdf 38947 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by NM, 19-Nov-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝐴) | ||
Theorem | nfded 38948 | A deduction theorem that converts a not-free inference directly to deduction form. The first hypothesis is the hypothesis of the deduction form. The second is an equality deduction (e.g., (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴} = ∪ 𝐴)) that starts from abidnf 3710. The last is assigned to the inference form (e.g., Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}) whose hypothesis is satisfied using nfaba1 2910. (Contributed by NM, 19-Nov-2020.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (Ⅎ𝑥𝐴 → 𝐵 = 𝐶) & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝜑 → Ⅎ𝑥𝐶) | ||
Theorem | nfded2 38949 | A deduction theorem that converts a not-free inference directly to deduction form. The first 2 hypotheses are the hypotheses of the deduction form. The third is an equality deduction (e.g., ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 〈{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}〉 = 〈𝐴, 𝐵〉) for nfopd 4894) that starts from abidnf 3710. The last is assigned to the inference form (e.g., Ⅎ𝑥〈{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}〉 for nfop 4893) whose hypotheses are satisfied using nfaba1 2910. (Contributed by NM, 19-Nov-2020.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) & ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 𝐶 = 𝐷) & ⊢ Ⅎ𝑥𝐶 ⇒ ⊢ (𝜑 → Ⅎ𝑥𝐷) | ||
Theorem | nfunidALT2 38950 | Deduction version of nfuni 4918. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) | ||
Theorem | nfunidALT 38951 | Deduction version of nfuni 4918. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) | ||
Theorem | nfopdALT 38952 | Deduction version of bound-variable hypothesis builder nfop 4893. This shows how the deduction version of a not-free theorem such as nfop 4893 can be created from the corresponding not-free inference theorem. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) | ||
Theorem | cnaddcom 38953 | Recover the commutative law of addition for complex numbers from the Abelian group structure. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
Theorem | toycom 38954* | Show the commutative law for an operation 𝑂 on a toy structure class 𝐶 of commutative operations on ℂ. This illustrates how a structure class can be partially specialized. In practice, we would ordinarily define a new constant such as "CAbel" in place of 𝐶. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.) |
⊢ 𝐶 = {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} & ⊢ + = (+g‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
Syntax | clsa 38955 | Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space. |
class LSAtoms | ||
Syntax | clsh 38956 | Extend class notation with all subspaces of a left module or left vector space that are hyperplanes. |
class LSHyp | ||
Definition | df-lsatoms 38957* | Define the set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) |
⊢ LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣}))) | ||
Definition | df-lshyp 38958* | Define the set of all hyperplanes of a left module or left vector space. Also called co-atoms, these are subspaces that are one dimension less than the full space. (Contributed by NM, 29-Jun-2014.) |
⊢ LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))}) | ||
Theorem | lshpset 38959* | The set of all hyperplanes of a left module or left vector space. The vector 𝑣 is called a generating vector for the hyperplane. (Contributed by NM, 29-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)}) | ||
Theorem | islshp 38960* | The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))) | ||
Theorem | islshpsm 38961* | Hyperplane properties expressed with subspace sum. (Contributed by NM, 3-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑈 ⊕ (𝑁‘{𝑣})) = 𝑉))) | ||
Theorem | lshplss 38962 | A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) ⇒ ⊢ (𝜑 → 𝑈 ∈ 𝑆) | ||
Theorem | lshpne 38963 | A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) ⇒ ⊢ (𝜑 → 𝑈 ≠ 𝑉) | ||
Theorem | lshpnel 38964 | A hyperplane's generating vector does not belong to the hyperplane. (Contributed by NM, 3-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) ⇒ ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) | ||
Theorem | lshpnelb 38965 | The subspace sum of a hyperplane and the span of an element equals the vector space iff the element is not in the hyperplane. (Contributed by NM, 2-Oct-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (¬ 𝑋 ∈ 𝑈 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) | ||
Theorem | lshpnel2N 38966 | Condition that determines a hyperplane. (Contributed by NM, 3-Oct-2014.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) | ||
Theorem | lshpne0 38967 | The member of the span in the hyperplane definition does not belong to the hyperplane. (Contributed by NM, 14-Jul-2014.) (Proof shortened by AV, 19-Jul-2022.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) ⇒ ⊢ (𝜑 → 𝑋 ≠ 0 ) | ||
Theorem | lshpdisj 38968 | A hyperplane and the span in the hyperplane definition are disjoint. (Contributed by NM, 3-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) ⇒ ⊢ (𝜑 → (𝑈 ∩ (𝑁‘{𝑋})) = { 0 }) | ||
Theorem | lshpcmp 38969 | If two hyperplanes are comparable, they are equal. (Contributed by NM, 9-Oct-2014.) |
⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑇 ∈ 𝐻) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) ⇒ ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈)) | ||
Theorem | lshpinN 38970 | The intersection of two different hyperplanes is not a hyperplane. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.) |
⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑇 ∈ 𝐻) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) ⇒ ⊢ (𝜑 → ((𝑇 ∩ 𝑈) ∈ 𝐻 ↔ 𝑇 = 𝑈)) | ||
Theorem | lsatset 38971* | The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → 𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣}))) | ||
Theorem | islsat 38972* | The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥}))) | ||
Theorem | lsatlspsn2 38973 | The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 38974 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ∈ 𝐴) | ||
Theorem | lsatlspsn 38974 | The span of a nonzero singleton is an atom. (Contributed by NM, 16-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) | ||
Theorem | islsati 38975* | A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑈 ∈ 𝐴) → ∃𝑣 ∈ 𝑉 𝑈 = (𝑁‘{𝑣})) | ||
Theorem | lsateln0 38976* | A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑣 ∈ 𝑈 𝑣 ≠ 0 ) | ||
Theorem | lsatlss 38977 | The set of 1-dim subspaces is a set of subspaces. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → 𝐴 ⊆ 𝑆) | ||
Theorem | lsatlssel 38978 | An atom is a subspace. (Contributed by NM, 25-Aug-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑈 ∈ 𝑆) | ||
Theorem | lsatssv 38979 | An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑄 ⊆ 𝑉) | ||
Theorem | lsatn0 38980 | A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 32373 analog.) (Contributed by NM, 25-Aug-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑈 ≠ { 0 }) | ||
Theorem | lsatspn0 38981 | The span of a vector is an atom iff the vector is nonzero. (Contributed by NM, 4-Feb-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ 𝑋 ≠ 0 )) | ||
Theorem | lsator0sp 38982 | The span of a vector is either an atom or the zero subspace. (Contributed by NM, 15-Mar-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ∨ (𝑁‘{𝑋}) = { 0 })) | ||
Theorem | lsatssn0 38983 | A subspace (or any class) including an atom is nonzero. (Contributed by NM, 3-Feb-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ⊆ 𝑈) ⇒ ⊢ (𝜑 → 𝑈 ≠ { 0 }) | ||
Theorem | lsatcmp 38984 | If two atoms are comparable, they are equal. (atsseq 32375 analog.) TODO: can lspsncmp 21135 shorten this? (Contributed by NM, 25-Aug-2014.) |
⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑇 ∈ 𝐴) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈)) | ||
Theorem | lsatcmp2 38985 | If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 38984. TODO: can lspsncmp 21135 shorten this? (Contributed by NM, 3-Feb-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑇 ∈ 𝐴) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∨ 𝑈 = { 0 })) ⇒ ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈)) | ||
Theorem | lsatel 38986 | A nonzero vector in an atom determines the atom. (Contributed by NM, 25-Aug-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → 𝑈 = (𝑁‘{𝑋})) | ||
Theorem | lsatelbN 38987 | A nonzero vector in an atom determines the atom. (Contributed by NM, 3-Feb-2015.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ 𝑈 = (𝑁‘{𝑋}))) | ||
Theorem | lsat2el 38988 | Two atoms sharing a nonzero vector are equal. (Contributed by NM, 8-Mar-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑋 ≠ 0 ) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑄) ⇒ ⊢ (𝜑 → 𝑃 = 𝑄) | ||
Theorem | lsmsat 38989* | Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 39787 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑇 ≠ { 0 }) & ⊢ (𝜑 → 𝑄 ⊆ (𝑇 ⊕ 𝑈)) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈))) | ||
Theorem | lsatfixedN 38990* | Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 21147. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑋})) & ⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → 𝑄 ⊆ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)})) | ||
Theorem | lsmsatcv 38991 | Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 31680 analog.) Explicit atom version of lsmcv 21160. (Contributed by NM, 29-Oct-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄)) | ||
Theorem | lssatomic 38992* | The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 32386 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ≠ { 0 }) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) | ||
Theorem | lssats 38993* | The lattice of subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. Hypothesis (shatomistici 32389 analog.) (Contributed by NM, 9-Apr-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 = (𝑁‘∪ {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝑈})) | ||
Theorem | lpssat 38994* | Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 32391 analog.) (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑇 ⊊ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇)) | ||
Theorem | lrelat 38995* | Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 32392 analog.) (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑇 ⊊ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) | ||
Theorem | lssatle 38996* | The ordering of two subspaces is determined by the atoms under them. (chrelat3 32399 analog.) (Contributed by NM, 29-Oct-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈))) | ||
Theorem | lssat 38997* | Two subspaces in a proper subset relationship imply the existence of a 1-dim subspace less than or equal to one but not the other. (chpssati 32391 analog.) (Contributed by NM, 9-Apr-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑉 ∈ 𝑆) ∧ 𝑈 ⊊ 𝑉) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ 𝑈)) | ||
Theorem | islshpat 38998* | Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 38961. (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑞 ∈ 𝐴 (𝑈 ⊕ 𝑞) = 𝑉))) | ||
Syntax | clcv 38999 | Extend class notation with the covering relation for a left module or left vector space. |
class ⋖L | ||
Definition | df-lcv 39000* | Define the covering relation for subspaces of a left vector space. Similar to Definition 3.2.18 of [PtakPulmannova] p. 68. Ptak/Pulmannova's notation 𝐴( ⋖L ‘𝑊)𝐵 is read "𝐵 covers 𝐴 " or "𝐴 is covered by 𝐵 " , and it means that 𝐵 is larger than 𝐴 and there is nothing in between. See lcvbr 39002 for binary relation. (df-cv 32307 analog.) (Contributed by NM, 7-Jan-2015.) |
⊢ ⋖L = (𝑤 ∈ V ↦ {〈𝑡, 𝑢〉 ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))}) |
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