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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hlhilnvl 39101 | The involution operation of the star division ring for the final constructed Hilbert space. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ ∗ = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ∗ = (*𝑟‘𝑅)) | ||
Theorem | hlhillvec 39102 | The final constructed Hilbert space is a vector space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑈 ∈ LVec) | ||
Theorem | hlhildrng 39103 | The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝑅 = (Scalar‘𝑈) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
Theorem | hlhilsrnglem 39104 | Lemma for hlhilsrng 39105. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐿) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑆) & ⊢ · = (.r‘𝑆) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) ⇒ ⊢ (𝜑 → 𝑅 ∈ *-Ring) | ||
Theorem | hlhilsrng 39105 | The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 21-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝑅 = (Scalar‘𝑈) ⇒ ⊢ (𝜑 → 𝑅 ∈ *-Ring) | ||
Theorem | hlhil0 39106 | The zero vector for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 0 = (0g‘𝐿) ⇒ ⊢ (𝜑 → 0 = (0g‘𝑈)) | ||
Theorem | hlhillsm 39107 | The vector sum operation for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ ⊕ = (LSSum‘𝐿) ⇒ ⊢ (𝜑 → ⊕ = (LSSum‘𝑈)) | ||
Theorem | hlhilocv 39108 | The orthocomplement for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝑉 = (Base‘𝐿) & ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑂 = (ocv‘𝑈) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (𝑂‘𝑋) = (𝑁‘𝑋)) | ||
Theorem | hlhillcs 39109 | The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 39087 is applied over and over to conclusion rather than applied once to antecedent - would compressed proof be shorter if applied once to antecedent? (Contributed by NM, 23-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐶 = ran 𝐼) | ||
Theorem | hlhilphllem 39110* | Lemma for hlhil 24046. (Contributed by NM, 23-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐹 = (Scalar‘𝑈) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝐿) & ⊢ + = (+g‘𝐿) & ⊢ · = ( ·𝑠 ‘𝐿) & ⊢ 𝑅 = (Scalar‘𝐿) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 0 = (0g‘𝐿) & ⊢ , = (·𝑖‘𝑈) & ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ 𝐸 = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝐽‘𝑦)‘𝑥)) ⇒ ⊢ (𝜑 → 𝑈 ∈ PreHil) | ||
Theorem | hlhilhillem 39111* | Lemma for hlhil 24046. (Contributed by NM, 23-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐹 = (Scalar‘𝑈) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝐿) & ⊢ + = (+g‘𝐿) & ⊢ · = ( ·𝑠 ‘𝐿) & ⊢ 𝑅 = (Scalar‘𝐿) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 0 = (0g‘𝐿) & ⊢ , = (·𝑖‘𝑈) & ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ 𝐸 = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝐽‘𝑦)‘𝑥)) & ⊢ 𝑂 = (ocv‘𝑈) & ⊢ 𝐶 = (ClSubSp‘𝑈) ⇒ ⊢ (𝜑 → 𝑈 ∈ Hil) | ||
Theorem | hlathil 39112 |
Construction of a Hilbert space (df-hil 20848) 𝑈 from a Hilbert
lattice (df-hlat 36502) 𝐾, where 𝑊 is a fixed but arbitrary
hyperplane (co-atom) in 𝐾.
The Hilbert space 𝑈 is identical to the vector space ((DVecH‘𝐾)‘𝑊) (see dvhlvec 38260) except that it is extended with involution and inner product components. The construction of these two components is provided by Theorem 3.6 in [Holland95] p. 13, whose proof we follow loosely. An example of involution is the complex conjugate when the division ring is the field of complex numbers. The nature of the division ring we constructed is indeterminate, however, until we specialize the initial Hilbert lattice with additional conditions found by Maria Solèr in 1995 and refined by René Mayet in 1998 that result in a division ring isomorphic to ℂ. See additional discussion at https://us.metamath.org/qlegif/mmql.html#what 38260. 𝑊 corresponds to the w in the proof of Theorem 13.4 of [Crawley] p. 111. Such a 𝑊 always exists since HL has lattice rank of at least 4 by df-hil 20848. It can be eliminated if we just want to show the existence of a Hilbert space, as is done in the literature. (Contributed by NM, 23-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑈 ∈ Hil) | ||
Theorem | andiff 39113 | Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.) |
⊢ (𝜑 → (𝜒 → 𝜃)) & ⊢ (𝜓 → (𝜃 → 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) | ||
Theorem | fac2xp3 39114 | Factorial of 2x+3, sublemma for sublemma for AKS. (Contributed by metakunt, 19-Apr-2024.) |
⊢ (𝑥 ∈ ℕ0 → (!‘((2 · 𝑥) + 3)) = ((!‘((2 · 𝑥) + 1)) · (((2 · 𝑥) + 2) · ((2 · 𝑥) + 3)))) | ||
Theorem | facp2 39115 | The factorial of a successor's successor. (Contributed by metakunt, 19-Apr-2024.) |
⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 2)) = ((!‘𝑁) · ((𝑁 + 1) · (𝑁 + 2)))) | ||
Theorem | prodsplit 39116* | Product split into two factors, original by Steven Nguyen. (Contributed by metakunt, 21-Apr-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 𝐾))) → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 𝐾))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · ∏𝑘 ∈ ((𝑁 + 1)...(𝑁 + 𝐾))𝐴)) | ||
Theorem | 2xp3dxp2ge1d 39117 | 2x+3 is greater than or equal to x+2 for x >= -1, a deduction version (Contributed by metakunt, 21-Apr-2024.) |
⊢ (𝜑 → 𝑋 ∈ (-1[,)+∞)) ⇒ ⊢ (𝜑 → 1 ≤ (((2 · 𝑋) + 3) / (𝑋 + 2))) | ||
Theorem | factwoffsmonot 39118 | A factorial with offset is monotonely increasing (Contributed by metakunt, 20-Apr-2024.) |
⊢ (((𝑋 ∈ ℕ0 ∧ 𝑌 ∈ ℕ0 ∧ 𝑋 ≤ 𝑌) ∧ 𝑁 ∈ ℕ0) → (!‘(𝑋 + 𝑁)) ≤ (!‘(𝑌 + 𝑁))) | ||
Theorem | ioin9i8 39119 | Miscellaneous inference creating a biconditional from an implied converse implication. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜒 → ¬ 𝜃) & ⊢ (𝜓 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝜃)) | ||
Theorem | jaodd 39120 | Double deduction form of jaoi 853. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜏) → 𝜃))) | ||
Theorem | nsb 39121 | Generalization rule for negated wff. (Contributed by Steven Nguyen, 3-May-2023.) |
⊢ ¬ 𝜑 ⇒ ⊢ ¬ [𝑥 / 𝑦]𝜑 | ||
Theorem | sbn1 39122 | One direction of sbn 2287, using fewer axioms. Compare 19.2 1981. (Contributed by Steven Nguyen, 18-Aug-2023.) |
⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑) | ||
Theorem | sbor2 39123 | One direction of sbor 2316, using fewer axioms. Compare 19.33 1885. (Contributed by Steven Nguyen, 18-Aug-2023.) |
⊢ (([𝑡 / 𝑥]𝜑 ∨ [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑 ∨ 𝜓)) | ||
Theorem | sn-elabg 39124* | Membership in a class abstraction, using implicit substitution and an intermediate setvar 𝑦 to avoid ax-10 2145, ax-11 2161, ax-12 2177. It also avoids a disjoint variable condition on 𝑥 and 𝐴. Compare sbievw2 2107. (Contributed by SN, 20-Apr-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒)) | ||
Theorem | 3rspcedvd 39125* | Triple application of rspcedvd 3626. (Contributed by Steven Nguyen, 27-Feb-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝜒 ↔ 𝜃)) & ⊢ ((𝜑 ∧ 𝑧 = 𝐶) → (𝜃 ↔ 𝜏)) & ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓) | ||
Theorem | rabeqcda 39126* | When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3678. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴) | ||
Theorem | rabdif 39127* | Move difference in and out of a restricted class abstraction. (Contributed by Steven Nguyen, 6-Jun-2023.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑} | ||
Theorem | sn-axrep5v 39128* | A condensed form of axrep5 5196. (Contributed by SN, 21-Sep-2023.) |
⊢ (∀𝑤 ∈ 𝑥 ∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑)) | ||
Theorem | sn-axprlem3 39129* | axprlem3 5326 using only Tarski's FOL axiom schemes and ax-rep 5190. (Contributed by SN, 22-Sep-2023.) |
⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏)) | ||
Theorem | sn-el 39130* | A version of el 5270 with an inner existential quantifier on 𝑥, which avoids ax-7 2015 and ax-8 2116. (Contributed by SN, 18-Sep-2023.) |
⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦 | ||
Theorem | sn-dtru 39131* | dtru 5271 without ax-8 2116 or ax-12 2177. (Contributed by SN, 21-Sep-2023.) |
⊢ ¬ ∀𝑥 𝑥 = 𝑦 | ||
Theorem | pssexg 39132 | The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) | ||
Theorem | pssn0 39133 | A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅) | ||
Theorem | psspwb 39134 | Classes are proper subclasses if and only if their power classes are proper subclasses. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝐴 ⊊ 𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵) | ||
Theorem | xppss12 39135 | Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐶 ⊊ 𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷)) | ||
Theorem | elpwbi 39136 | Membership in a power set, biconditional. (Contributed by Steven Nguyen, 17-Jul-2022.) (Proof shortened by Steven Nguyen, 16-Sep-2022.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵) | ||
Theorem | opelxpii 39137 | Ordered pair membership in a Cartesian product (implication). (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ 𝐴 ∈ 𝐶 & ⊢ 𝐵 ∈ 𝐷 ⇒ ⊢ 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) | ||
Theorem | iunsn 39138* | Indexed union of a singleton. Compare dfiun2 4958 and rnmpt 5827. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} | ||
Theorem | imaopab 39139* | The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} | ||
Theorem | fnsnbt 39140 | A function's domain is a singleton iff the function is a singleton. (Contributed by Steven Nguyen, 18-Aug-2023.) |
⊢ (𝐴 ∈ V → (𝐹 Fn {𝐴} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) | ||
Theorem | fnimasnd 39141 | The image of a function by a singleton whose element is in the domain of the function. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝑆 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)}) | ||
Theorem | dfqs2 39142* | Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ (𝐴 / 𝑅) = ran (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) | ||
Theorem | dfqs3 39143* | Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} | ||
Theorem | qseq12d 39144 | Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷)) | ||
Theorem | qsalrel 39145* | The quotient set is equal to the singleton of 𝐴 when all elements are related and 𝐴 is nonempty. (Contributed by SN, 8-Jun-2023.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∼ 𝑦) & ⊢ (𝜑 → ∼ Er 𝐴) & ⊢ (𝜑 → 𝑁 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐴 / ∼ ) = {𝐴}) | ||
Theorem | fzosumm1 39146* | Separate out the last term in a finite sum. (Contributed by Steven Nguyen, 22-Aug-2023.) |
⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑘 = (𝑁 − 1) → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = (Σ𝑘 ∈ (𝑀..^(𝑁 − 1))𝐴 + 𝐵)) | ||
Theorem | ccatcan2d 39147 | Cancellation law for concatenation. (Contributed by SN, 6-Sep-2023.) |
⊢ (𝜑 → 𝐴 ∈ Word 𝑉) & ⊢ (𝜑 → 𝐵 ∈ Word 𝑉) & ⊢ (𝜑 → 𝐶 ∈ Word 𝑉) ⇒ ⊢ (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | nelsubginvcld 39148 | The inverse of a non-subgroup-member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.) |
⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆)) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) ∈ (𝐵 ∖ 𝑆)) | ||
Theorem | nelsubgcld 39149 | A non-subgroup-member plus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.) |
⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆)) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐵 ∖ 𝑆)) | ||
Theorem | nelsubgsubcld 39150 | A non-subgroup-member minus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.) |
⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆)) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐵 ∖ 𝑆)) | ||
Theorem | rnasclg 39151 | The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 𝐴 = (algSc‘𝑊) & ⊢ 1 = (1r‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) → ran 𝐴 = (𝑁‘{ 1 })) | ||
Theorem | selvval2lem1 39152 | 𝑇 is an associative algebra. For simplicity, 𝐼 stands for (𝐼 ∖ 𝐽) and we have 𝐽 ∈ 𝑊 instead of 𝐽 ⊆ 𝐼. (Contributed by SN, 15-Dec-2023.) |
⊢ 𝑈 = (𝐼 mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → 𝑇 ∈ AssAlg) | ||
Theorem | selvval2lem2 39153 | 𝐷 is a ring homomorphism. (Contributed by SN, 15-Dec-2023.) |
⊢ 𝑈 = (𝐼 mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ 𝐶 = (algSc‘𝑇) & ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → 𝐷 ∈ (𝑅 RingHom 𝑇)) | ||
Theorem | selvval2lem3 39154 | The third argument passed to evalSub is in the domain. (Contributed by SN, 15-Dec-2023.) |
⊢ 𝑈 = (𝐼 mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ 𝐶 = (algSc‘𝑇) & ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → ran 𝐷 ∈ (SubRing‘𝑇)) | ||
Theorem | selvval2lemn 39155 | A lemma to illustrate the purpose of selvval2lem3 39154 and the value of 𝑄. Will be renamed in the future when this section is moved to main. (Contributed by SN, 5-Nov-2023.) |
⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ 𝐶 = (algSc‘𝑇) & ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) & ⊢ 𝑄 = ((𝐼 evalSub 𝑇)‘ran 𝐷) & ⊢ 𝑊 = (𝐼 mPoly 𝑆) & ⊢ 𝑆 = (𝑇 ↾s ran 𝐷) & ⊢ 𝑋 = (𝑇 ↑s (𝐵 ↑m 𝐼)) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) ⇒ ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑋)) | ||
Theorem | selvval2lem4 39156 | The fourth argument passed to evalSub is in the domain (a polynomial in (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))). (Contributed by SN, 5-Nov-2023.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ 𝐶 = (algSc‘𝑇) & ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) & ⊢ 𝑆 = (𝑇 ↾s ran 𝐷) & ⊢ 𝑊 = (𝐼 mPoly 𝑆) & ⊢ 𝑋 = (Base‘𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷 ∘ 𝐹) ∈ 𝑋) | ||
Theorem | selvval2lem5 39157* | The fifth argument passed to evalSub is in the domain (a function 𝐼⟶𝐸). (Contributed by SN, 22-Feb-2024.) |
⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ 𝐶 = (algSc‘𝑇) & ⊢ 𝐸 = (Base‘𝑇) & ⊢ 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐸 ↑m 𝐼)) | ||
Theorem | selvcl 39158 | Closure of the "variable selection" function. (Contributed by SN, 22-Feb-2024.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ 𝐸 = (Base‘𝑇) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ 𝐸) | ||
Theorem | frlmfielbas 39159 | The vectors of a finite free module are the functions from 𝐼 to 𝑁. (Contributed by SN, 31-Aug-2023.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑁 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ Fin) → (𝑋 ∈ 𝐵 ↔ 𝑋:𝐼⟶𝑁)) | ||
Theorem | frlmfzwrd 39160 | A vector of a module with indices from 0 to 𝑁 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.) |
⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑆 = (Base‘𝐾) ⇒ ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ Word 𝑆) | ||
Theorem | frlmfzowrd 39161 | A vector of a module with indices from 0 to 𝑁 − 1 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.) |
⊢ 𝑊 = (𝐾 freeLMod (0..^𝑁)) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑆 = (Base‘𝐾) ⇒ ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ Word 𝑆) | ||
Theorem | frlmfzolen 39162 | The dimension of a vector of a module with indices from 0 to 𝑁 − 1. (Contributed by SN, 1-Sep-2023.) |
⊢ 𝑊 = (𝐾 freeLMod (0..^𝑁)) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑆 = (Base‘𝐾) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (♯‘𝑋) = 𝑁) | ||
Theorem | frlmfzowrdb 39163 | The vectors of a module with indices 0 to 𝑁 − 1 are the length- 𝑁 words over the scalars of the module. (Contributed by SN, 1-Sep-2023.) |
⊢ 𝑊 = (𝐾 freeLMod (0..^𝑁)) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑆 = (Base‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ Word 𝑆 ∧ (♯‘𝑋) = 𝑁))) | ||
Theorem | frlmfzoccat 39164 | The concatenation of two vectors of dimension 𝑁 and 𝑀 forms a vector of dimension 𝑁 + 𝑀. (Contributed by SN, 31-Aug-2023.) |
⊢ 𝑊 = (𝐾 freeLMod (0..^𝐿)) & ⊢ 𝑋 = (𝐾 freeLMod (0..^𝑀)) & ⊢ 𝑌 = (𝐾 freeLMod (0..^𝑁)) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐶 = (Base‘𝑋) & ⊢ 𝐷 = (Base‘𝑌) & ⊢ (𝜑 → 𝐾 ∈ Ring) & ⊢ (𝜑 → (𝑀 + 𝑁) = 𝐿) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑈 ∈ 𝐶) & ⊢ (𝜑 → 𝑉 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵) | ||
Theorem | frlmvscadiccat 39165 | Scalar multiplication distributes over concatenation. (Contributed by SN, 6-Sep-2023.) |
⊢ 𝑊 = (𝐾 freeLMod (0..^𝐿)) & ⊢ 𝑋 = (𝐾 freeLMod (0..^𝑀)) & ⊢ 𝑌 = (𝐾 freeLMod (0..^𝑁)) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐶 = (Base‘𝑋) & ⊢ 𝐷 = (Base‘𝑌) & ⊢ (𝜑 → 𝐾 ∈ Ring) & ⊢ (𝜑 → (𝑀 + 𝑁) = 𝐿) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑈 ∈ 𝐶) & ⊢ (𝜑 → 𝑉 ∈ 𝐷) & ⊢ 𝑂 = ( ·𝑠 ‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝑋) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 𝑆 = (Base‘𝐾) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴𝑂(𝑈 ++ 𝑉)) = ((𝐴 ∙ 𝑈) ++ (𝐴 · 𝑉))) | ||
Theorem | lvecgrp 39166 | A left vector is a group. (Contributed by Steven Nguyen, 28-May-2023.) |
⊢ (𝑊 ∈ LVec → 𝑊 ∈ Grp) | ||
Theorem | lvecring 39167 | The scalar component of a left vector is a ring. (Contributed by Steven Nguyen, 28-May-2023.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → 𝐹 ∈ Ring) | ||
Theorem | lmhmlvec 39168 | The property for modules to be vector spaces is invariant under module isomorphism. (Contributed by Steven Nguyen, 15-Aug-2023.) |
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LVec ↔ 𝑇 ∈ LVec)) | ||
Theorem | frlmsnic 39169* | Given a free module with a singleton as the index set, that is, a free module of one-dimensional vectors, the function that maps each vector to its coordinate is a module isomorphism from that module to its ring of scalars seen as a module. (Contributed by Steven Nguyen, 18-Aug-2023.) |
⊢ 𝑊 = (𝐾 freeLMod {𝐼}) & ⊢ 𝐹 = (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼)) ⇒ ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMIso (ringLMod‘𝐾))) | ||
Theorem | uvccl 39170 | A unit vector is a vector. (Contributed by Steven Nguyen, 16-Jul-2023.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) ∈ 𝐵) | ||
Theorem | uvcn0 39171 | A unit vector is nonzero. (Contributed by Steven Nguyen, 16-Jul-2023.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 0 = (0g‘𝑌) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) ≠ 0 ) | ||
Towards the start of this section are several proofs regarding the different complex number axioms that could be used to prove some results. For example, ax-1rid 10607 is used in mulid1 10639 related theorems, so one could trade off the extra axioms in mulid1 10639 for the axioms needed to prove that something is a real number. Another example is avoiding complex number closure laws by using real number closure laws and then using ax-resscn 10594; in the other direction, real number closure laws can be avoided by using ax-resscn 10594 and then the complex number closure laws. (This only works if the result of (𝐴 + 𝐵) only needs to be a complex number). The natural numbers are especially amenable to axiom reductions, as the set ℕ is the recursive set {1, (1 + 1), ((1 + 1) + 1)}, etc., i.e. the set of numbers formed by only additions of 1. The digits 2 through 9 are defined so that they expand into additions of 1. This makes adding natural numbers conveniently only require the rearrangement of parentheses, as shown with the following: (4 + 3) = 7 ((3 + 1) + (2 + 1)) = (6 + 1) ((((1 + 1) + 1) + 1) + ((1 + 1) + 1)) = ((((((1 + 1) + 1) + 1) + 1) + 1) + 1) This only requires ax-addass 10602, ax-1cn 10595, and ax-addcl 10597. (And in practice, the expression isn't completely expanded into ones.) Multiplication by 1 requires either mulid2i 10646 or (ax-1rid 10607 and 1re 10641) as seen in 1t1e1 11800 and 1t1e1ALT 39175. Multiplying with greater natural numbers uses ax-distr 10604. Still, this takes fewer axioms than adding zero. When zero is involved in the decimal constructor, there's an implicit addition operation which causes such theorems (e.g. (9 + 1) = ;10) to use almost every complex number axiom. | ||
Theorem | c0exALT 39172 | Alternate proof of c0ex 10635 using more set theory axioms but fewer complex number axioms (add ax-10 2145, ax-11 2161, ax-13 2390, ax-nul 5210, and remove ax-1cn 10595, ax-icn 10596, ax-addcl 10597, and ax-mulcl 10599). (Contributed by Steven Nguyen, 4-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 0 ∈ V | ||
Theorem | 0cnALT3 39173 | Alternate proof of 0cn 10633 using ax-resscn 10594, ax-addrcl 10598, ax-rnegex 10608, ax-cnre 10610 instead of ax-icn 10596, ax-addcl 10597, ax-mulcl 10599, ax-i2m1 10605. Version of 0cnALT 10874 using ax-1cn 10595 instead of ax-icn 10596. (Contributed by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 0 ∈ ℂ | ||
Theorem | elre0re 39174 | Specialized version of 0red 10644 without using ax-1cn 10595 and ax-cnre 10610. (Contributed by Steven Nguyen, 28-Jan-2023.) |
⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | ||
Theorem | 1t1e1ALT 39175 | Alternate proof of 1t1e1 11800 using a different set of axioms (add ax-mulrcl 10600, ax-i2m1 10605, ax-1ne0 10606, ax-rrecex 10609 and remove ax-resscn 10594, ax-mulcom 10601, ax-mulass 10603, ax-distr 10604). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (1 · 1) = 1 | ||
Theorem | remulcan2d 39176 | mulcan2d 11274 for real numbers using fewer axioms. (Contributed by Steven Nguyen, 15-Apr-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | readdid1addid2d 39177 | Given some real number 𝐵 where 𝐴 acts like a right additive identity, derive that 𝐴 is a left additive identity. Note that the hypothesis is weaker than proving that 𝐴 is a right additive identity (for all numbers). Although, if there is a right additive identity, then by readdcan 10814, 𝐴 is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝐵 + 𝐴) = 𝐵) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶) | ||
Theorem | sn-1ne2 39178 | A proof of 1ne2 11846 without using ax-mulcom 10601, ax-mulass 10603, ax-pre-mulgt0 10614. Based on mul02lem2 10817. (Contributed by SN, 13-Dec-2023.) |
⊢ 1 ≠ 2 | ||
Theorem | nnn1suc 39179* | A positive integer that is not 1 is a successor of some other positive integer. (Contributed by Steven Nguyen, 19-Aug-2023.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ≠ 1) → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴) | ||
Theorem | nnadd1com 39180 | Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.) |
⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴)) | ||
Theorem | nnaddcom 39181 | Addition is commutative for natural numbers. Uses fewer axioms than addcom 10826. (Contributed by Steven Nguyen, 9-Dec-2022.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
Theorem | nnaddcomli 39182 | Version of addcomli 10832 for natural numbers. (Contributed by Steven Nguyen, 1-Aug-2023.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ & ⊢ (𝐴 + 𝐵) = 𝐶 ⇒ ⊢ (𝐵 + 𝐴) = 𝐶 | ||
Theorem | nnadddir 39183 | Right-distributivity for natural numbers without ax-mulcom 10601. (Contributed by SN, 5-Feb-2024.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))) | ||
Theorem | nnmul1com 39184 | Multiplication with 1 is commutative for natural numbers, without ax-mulcom 10601. Since (𝐴 · 1) is 𝐴 by ax-1rid 10607, this is equivalent to remulid2 39269 for natural numbers, but using fewer axioms (avoiding ax-resscn 10594, ax-addass 10602, ax-mulass 10603, ax-rnegex 10608, ax-pre-lttri 10611, ax-pre-lttrn 10612, ax-pre-ltadd 10613). (Contributed by SN, 5-Feb-2024.) |
⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1)) | ||
Theorem | nnmulcom 39185 | Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | addsubeq4com 39186 | Relation between sums and differences. (Contributed by Steven Nguyen, 5-Jan-2023.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 − 𝐶) = (𝐷 − 𝐵))) | ||
Theorem | sqsumi 39187 | A sum squared. (Contributed by Steven Nguyen, 16-Sep-2022.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 · 𝐴) + (𝐵 · 𝐵)) + (2 · (𝐴 · 𝐵))) | ||
Theorem | negn0nposznnd 39188 | Lemma for dffltz 39291. (Contributed by Steven Nguyen, 27-Feb-2023.) |
⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → ¬ 0 < 𝐴) & ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → -𝐴 ∈ ℕ) | ||
Theorem | sqmid3api 39189 | Value of the square of the middle term of a 3-term arithmetic progression. (Contributed by Steven Nguyen, 20-Sep-2022.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝑁 ∈ ℂ & ⊢ (𝐴 + 𝑁) = 𝐵 & ⊢ (𝐵 + 𝑁) = 𝐶 ⇒ ⊢ (𝐵 · 𝐵) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) | ||
Theorem | decaddcom 39190 | Commute ones place in addition. (Contributed by Steven Nguyen, 29-Jan-2023.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 ⇒ ⊢ (;𝐴𝐵 + 𝐶) = (;𝐴𝐶 + 𝐵) | ||
Theorem | sqn5i 39191 | The square of a number ending in 5. This shortcut only works because 5 is half of 10. (Contributed by Steven Nguyen, 16-Sep-2022.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ (;𝐴5 · ;𝐴5) = ;;(𝐴 · (𝐴 + 1))25 | ||
Theorem | sqn5ii 39192 | The square of a number ending in 5. This shortcut only works because 5 is half of 10. (Contributed by Steven Nguyen, 16-Sep-2022.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ (𝐴 + 1) = 𝐵 & ⊢ (𝐴 · 𝐵) = 𝐶 ⇒ ⊢ (;𝐴5 · ;𝐴5) = ;;𝐶25 | ||
Theorem | decpmulnc 39193 | Partial products algorithm for two digit multiplication, no carry. Compare muladdi 11091. (Contributed by Steven Nguyen, 9-Dec-2022.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ (𝐴 · 𝐶) = 𝐸 & ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹 & ⊢ (𝐵 · 𝐷) = 𝐺 ⇒ ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐸𝐹𝐺 | ||
Theorem | decpmul 39194 | Partial products algorithm for two digit multiplication. (Contributed by Steven Nguyen, 10-Dec-2022.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ (𝐴 · 𝐶) = 𝐸 & ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹 & ⊢ (𝐵 · 𝐷) = ;𝐺𝐻 & ⊢ (;𝐸𝐺 + 𝐹) = 𝐼 & ⊢ 𝐺 ∈ ℕ0 & ⊢ 𝐻 ∈ ℕ0 ⇒ ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;𝐼𝐻 | ||
Theorem | sqdeccom12 39195 | The square of a number in terms of its digits switched. (Contributed by Steven Nguyen, 3-Jan-2023.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ ((;𝐴𝐵 · ;𝐴𝐵) − (;𝐵𝐴 · ;𝐵𝐴)) = (;99 · ((𝐴 · 𝐴) − (𝐵 · 𝐵))) | ||
Theorem | sq3deccom12 39196 | Variant of sqdeccom12 39195 with a three digit square. (Contributed by Steven Nguyen, 3-Jan-2023.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ (𝐴 + 𝐶) = 𝐷 ⇒ ⊢ ((;;𝐴𝐵𝐶 · ;;𝐴𝐵𝐶) − (;𝐷𝐵 · ;𝐷𝐵)) = (;99 · ((;𝐴𝐵 · ;𝐴𝐵) − (𝐶 · 𝐶))) | ||
Theorem | 235t711 39197 |
Calculate a product by long multiplication as a base comparison with other
multiplication algorithms.
Conveniently, 711 has two ones which greatly simplifies calculations like 235 · 1. There isn't a higher level mulcomli 10650 saving the lower level uses of mulcomli 10650 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12208 are added then this proof would benefit more than ex-decpmul 39198. For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 11773 or 8t7e56 12219. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.) |
⊢ (;;235 · ;;711) = ;;;;;167085 | ||
Theorem | ex-decpmul 39198 | Example usage of decpmul 39194. This proof is significantly longer than 235t711 39197. There is more unnecessary carrying compared to 235t711 39197. Although saving 5 visual steps, using mulcomli 10650 early on increases the compressed proof length. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (;;235 · ;;711) = ;;;;;167085 | ||
Theorem | oexpreposd 39199 | Lemma for dffltz 39291. (Contributed by Steven Nguyen, 4-Mar-2023.) |
⊢ (𝜑 → 𝑁 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → ¬ (𝑀 / 2) ∈ ℕ) ⇒ ⊢ (𝜑 → (0 < 𝑁 ↔ 0 < (𝑁↑𝑀))) | ||
Theorem | cxpgt0d 39200 | Exponentiation with a positive mantissa is positive. (Contributed by Steven Nguyen, 6-Apr-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝑁 ∈ ℝ) ⇒ ⊢ (𝜑 → 0 < (𝐴↑𝑐𝑁)) |
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