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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hlhilslem 40401 | Lemma for hlhilsbase 40403 etc. (Contributed by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐹 = Slot (𝐹‘ndx) & ⊢ (𝐹‘ndx) ≠ (*𝑟‘ndx) & ⊢ 𝐶 = (𝐹‘𝐸) ⇒ ⊢ (𝜑 → 𝐶 = (𝐹‘𝑅)) | ||
Theorem | hlhilslemOLD 40402 | Obsolete version of hlhilslem 40401 as of 6-Nov-2024. Lemma for hlhilsbase 40403. (Contributed by Mario Carneiro, 28-Jun-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐹 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 < 4 & ⊢ 𝐶 = (𝐹‘𝐸) ⇒ ⊢ (𝜑 → 𝐶 = (𝐹‘𝑅)) | ||
Theorem | hlhilsbase 40403 | The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐶 = (Base‘𝐸) ⇒ ⊢ (𝜑 → 𝐶 = (Base‘𝑅)) | ||
Theorem | hlhilsbaseOLD 40404 | Obsolete version of hlhilsbase 40403 as of 6-Nov-2024. The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐶 = (Base‘𝐸) ⇒ ⊢ (𝜑 → 𝐶 = (Base‘𝑅)) | ||
Theorem | hlhilsplus 40405 | Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ + = (+g‘𝐸) ⇒ ⊢ (𝜑 → + = (+g‘𝑅)) | ||
Theorem | hlhilsplusOLD 40406 | Obsolete version of hlhilsplus 40405 as of 6-Nov-2024. The scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ + = (+g‘𝐸) ⇒ ⊢ (𝜑 → + = (+g‘𝑅)) | ||
Theorem | hlhilsmul 40407 | Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ · = (.r‘𝐸) ⇒ ⊢ (𝜑 → · = (.r‘𝑅)) | ||
Theorem | hlhilsmulOLD 40408 | Obsolete version of hlhilsmul 40407 as of 6-Nov-2024. The scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ · = (.r‘𝐸) ⇒ ⊢ (𝜑 → · = (.r‘𝑅)) | ||
Theorem | hlhilsbase2 40409 | The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐿) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝜑 → 𝐶 = (Base‘𝑅)) | ||
Theorem | hlhilsplus2 40410 | Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐿) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ + = (+g‘𝑆) ⇒ ⊢ (𝜑 → + = (+g‘𝑅)) | ||
Theorem | hlhilsmul2 40411 | Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐿) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ · = (.r‘𝑆) ⇒ ⊢ (𝜑 → · = (.r‘𝑅)) | ||
Theorem | hlhils0 40412 | The scalar ring zero 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) & ⊢ 𝑆 = (Scalar‘𝐿) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ (𝜑 → 0 = (0g‘𝑅)) | ||
Theorem | hlhils1N 40413 | The scalar ring unity for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐿) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 1 = (1r‘𝑆) ⇒ ⊢ (𝜑 → 1 = (1r‘𝑅)) | ||
Theorem | hlhilvsca 40414 | The scalar product for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ · = ( ·𝑠 ‘𝐿) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → · = ( ·𝑠 ‘𝑈)) | ||
Theorem | hlhilip 40415* | Inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝐿) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) ⇒ ⊢ (𝜑 → , = (·𝑖‘𝑈)) | ||
Theorem | hlhilipval 40416 | Value of inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝐿) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ , = (·𝑖‘𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 , 𝑌) = ((𝑆‘𝑌)‘𝑋)) | ||
Theorem | hlhilnvl 40417 | 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 40418 | 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 40419 | 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 40420 | Lemma for hlhilsrng 40421. (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 40421 | 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 40422 | 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 40423 | 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 40424 | 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 40425 | The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 40399 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 40426* | Lemma for hlhil 24807. (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 40427* | Lemma for hlhil 24807. (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 40428 |
Construction of a Hilbert space (df-hil 21110) 𝑈 from a Hilbert
lattice (df-hlat 37813) 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 39572) 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 39572. 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 21110. 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 | leexp1ad 40429 | Weak base ordering relationship for exponentiation, a deduction version. (Contributed by metakunt, 22-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ≤ (𝐵↑𝑁)) | ||
Theorem | relogbcld 40430 | Closure of the general logarithm with a positive real base on positive reals, a deduction version. (Contributed by metakunt, 22-May-2024.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐵) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝑋) & ⊢ (𝜑 → 𝐵 ≠ 1) ⇒ ⊢ (𝜑 → (𝐵 logb 𝑋) ∈ ℝ) | ||
Theorem | relogbexpd 40431 | Identity law for general logarithm: the logarithm of a power to the base is the exponent, a deduction version. (Contributed by metakunt, 22-May-2024.) |
⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ≠ 1) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐵 logb (𝐵↑𝑀)) = 𝑀) | ||
Theorem | relogbzexpd 40432 | Power law for the general logarithm for integer powers: The logarithm of a positive real number to the power of an integer is equal to the product of the exponent and the logarithm of the base of the power, a deduction version. (Contributed by metakunt, 22-May-2024.) |
⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ≠ 1) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐵 logb (𝐶↑𝑁)) = (𝑁 · (𝐵 logb 𝐶))) | ||
Theorem | logblebd 40433 | The general logarithm is monotone/increasing, a deduction version. (Contributed by metakunt, 22-May-2024.) |
⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 2 ≤ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝑋) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝑌) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) ⇒ ⊢ (𝜑 → (𝐵 logb 𝑋) ≤ (𝐵 logb 𝑌)) | ||
Theorem | uzindd 40434* | Induction on the upper integers that start at 𝑀. The first four hypotheses give us the substitution instances we need; the following two are the basis and the induction step, a deduction version. (Contributed by metakunt, 8-Jun-2024.) |
⊢ (𝑗 = 𝑀 → (𝜓 ↔ 𝜒)) & ⊢ (𝑗 = 𝑘 → (𝜓 ↔ 𝜃)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜓 ↔ 𝜏)) & ⊢ (𝑗 = 𝑁 → (𝜓 ↔ 𝜂)) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜑 ∧ 𝜃 ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → 𝜏) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝜂) | ||
Theorem | fzadd2d 40435 | Membership of a sum in a finite interval of integers, a deduction version. (Contributed by metakunt, 10-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑂 ∈ ℤ) & ⊢ (𝜑 → 𝑃 ∈ ℤ) & ⊢ (𝜑 → 𝐽 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐾 ∈ (𝑂...𝑃)) & ⊢ (𝜑 → 𝑄 = (𝑀 + 𝑂)) & ⊢ (𝜑 → 𝑅 = (𝑁 + 𝑃)) ⇒ ⊢ (𝜑 → (𝐽 + 𝐾) ∈ (𝑄...𝑅)) | ||
Theorem | zltlem1d 40436 | Integer ordering relation, a deduction version. (Contributed by metakunt, 23-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | ||
Theorem | zltp1led 40437 | Integer ordering relation, a deduction version. (Contributed by metakunt, 23-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) | ||
Theorem | fzne2d 40438 | Elementhood in a finite set of sequential integers, except its upper bound. (Contributed by metakunt, 23-May-2024.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐾 ≠ 𝑁) ⇒ ⊢ (𝜑 → 𝐾 < 𝑁) | ||
Theorem | eqfnfv2d2 40439* | Equality of functions is determined by their values, a deduction version. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | fzsplitnd 40440 | Split a finite interval of integers into two parts. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) ⇒ ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁))) | ||
Theorem | fzsplitnr 40441 | Split a finite interval of integers into two parts. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ≤ 𝐾) & ⊢ (𝜑 → 𝐾 ≤ 𝑁) ⇒ ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁))) | ||
Theorem | addassnni 40442 | Associative law for addition. (Contributed by metakunt, 25-Apr-2024.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ & ⊢ 𝐶 ∈ ℕ ⇒ ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)) | ||
Theorem | addcomnni 40443 | Commutative law for addition. (Contributed by metakunt, 25-Apr-2024.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴) | ||
Theorem | mulassnni 40444 | Associative law for multiplication. (Contributed by metakunt, 25-Apr-2024.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ & ⊢ 𝐶 ∈ ℕ ⇒ ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)) | ||
Theorem | mulcomnni 40445 | Commutative law for multiplication. (Contributed by metakunt, 25-Apr-2024.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴) | ||
Theorem | gcdcomnni 40446 | Commutative law for gcd. (Contributed by metakunt, 25-Apr-2024.) |
⊢ 𝑀 ∈ ℕ & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀) | ||
Theorem | gcdnegnni 40447 | Negation invariance for gcd. (Contributed by metakunt, 25-Apr-2024.) |
⊢ 𝑀 ∈ ℕ & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝑀 gcd -𝑁) = (𝑀 gcd 𝑁) | ||
Theorem | neggcdnni 40448 | Negation invariance for gcd. (Contributed by metakunt, 25-Apr-2024.) |
⊢ 𝑀 ∈ ℕ & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (-𝑀 gcd 𝑁) = (𝑀 gcd 𝑁) | ||
Theorem | bccl2d 40449 | Closure of the binomial coefficient, a deduction version. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → 𝐾 ≤ 𝑁) ⇒ ⊢ (𝜑 → (𝑁C𝐾) ∈ ℕ) | ||
Theorem | recbothd 40450 | Take reciprocal on both sides. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ≠ 0) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = (𝐶 / 𝐷) ↔ (𝐵 / 𝐴) = (𝐷 / 𝐶))) | ||
Theorem | gcdmultiplei 40451 | The GCD of a multiple of a positive integer is the positive integer itself. (Contributed by metakunt, 25-Apr-2024.) |
⊢ 𝑀 ∈ ℕ & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝑀 gcd (𝑀 · 𝑁)) = 𝑀 | ||
Theorem | gcdaddmzz2nni 40452 | Adding a multiple of one operand of the gcd operator to the other does not alter the result. (Contributed by metakunt, 25-Apr-2024.) |
⊢ 𝑀 ∈ ℕ & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐾 ∈ ℤ ⇒ ⊢ (𝑀 gcd 𝑁) = (𝑀 gcd (𝑁 + (𝐾 · 𝑀))) | ||
Theorem | gcdaddmzz2nncomi 40453 | Adding a multiple of one operand of the gcd operator to the other does not alter the result. (Contributed by metakunt, 25-Apr-2024.) |
⊢ 𝑀 ∈ ℕ & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐾 ∈ ℤ ⇒ ⊢ (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) | ||
Theorem | gcdnncli 40454 | Closure of the gcd operator. (Contributed by metakunt, 25-Apr-2024.) |
⊢ 𝑀 ∈ ℕ & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝑀 gcd 𝑁) ∈ ℕ | ||
Theorem | muldvds1d 40455 | If a product divides an integer, so does one of its factors, a deduction version. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → (𝐾 · 𝑀) ∥ 𝑁) ⇒ ⊢ (𝜑 → 𝐾 ∥ 𝑁) | ||
Theorem | muldvds2d 40456 | If a product divides an integer, so does one of its factors, a deduction version. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → (𝐾 · 𝑀) ∥ 𝑁) ⇒ ⊢ (𝜑 → 𝑀 ∥ 𝑁) | ||
Theorem | nndivdvdsd 40457 | A positive integer divides a natural number if and only if the quotient is a positive integer, a deduction version of nndivdvds 16145. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝑀 ∥ 𝑁 ↔ (𝑁 / 𝑀) ∈ ℕ)) | ||
Theorem | nnproddivdvdsd 40458 | A product of natural numbers divides a natural number if and only if a factor divides the quotient, a deduction version. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ((𝐾 · 𝑀) ∥ 𝑁 ↔ 𝐾 ∥ (𝑁 / 𝑀))) | ||
Theorem | coprmdvds2d 40459 | If an integer is divisible by two coprime integers, then it is divisible by their product, a deduction version. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → (𝐾 gcd 𝑀) = 1) & ⊢ (𝜑 → 𝐾 ∥ 𝑁) & ⊢ (𝜑 → 𝑀 ∥ 𝑁) ⇒ ⊢ (𝜑 → (𝐾 · 𝑀) ∥ 𝑁) | ||
Theorem | 12gcd5e1 40460 | The gcd of 12 and 5 is 1. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (;12 gcd 5) = 1 | ||
Theorem | 60gcd6e6 40461 | The gcd of 60 and 6 is 6. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (;60 gcd 6) = 6 | ||
Theorem | 60gcd7e1 40462 | The gcd of 60 and 7 is 1. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (;60 gcd 7) = 1 | ||
Theorem | 420gcd8e4 40463 | The gcd of 420 and 8 is 4. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (;;420 gcd 8) = 4 | ||
Theorem | lcmeprodgcdi 40464 | Calculate the least common multiple of two natural numbers. (Contributed by metakunt, 25-Apr-2024.) |
⊢ 𝑀 ∈ ℕ & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐺 ∈ ℕ & ⊢ 𝐻 ∈ ℕ & ⊢ (𝑀 gcd 𝑁) = 𝐺 & ⊢ (𝐺 · 𝐻) = 𝐴 & ⊢ (𝑀 · 𝑁) = 𝐴 ⇒ ⊢ (𝑀 lcm 𝑁) = 𝐻 | ||
Theorem | 12lcm5e60 40465 | The lcm of 12 and 5 is 60. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (;12 lcm 5) = ;60 | ||
Theorem | 60lcm6e60 40466 | The lcm of 60 and 6 is 60. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (;60 lcm 6) = ;60 | ||
Theorem | 60lcm7e420 40467 | The lcm of 60 and 7 is 420. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (;60 lcm 7) = ;;420 | ||
Theorem | 420lcm8e840 40468 | The lcm of 420 and 8 is 840. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (;;420 lcm 8) = ;;840 | ||
Theorem | lcmfunnnd 40469 | Useful equation to calculate the least common multiple of 1 to n. (Contributed by metakunt, 29-Apr-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (lcm‘(1...𝑁)) = ((lcm‘(1...(𝑁 − 1))) lcm 𝑁)) | ||
Theorem | lcm1un 40470 | Least common multiple of natural numbers up to 1 equals 1. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...1)) = 1 | ||
Theorem | lcm2un 40471 | Least common multiple of natural numbers up to 2 equals 2. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...2)) = 2 | ||
Theorem | lcm3un 40472 | Least common multiple of natural numbers up to 3 equals 6. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...3)) = 6 | ||
Theorem | lcm4un 40473 | Least common multiple of natural numbers up to 4 equals 12. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...4)) = ;12 | ||
Theorem | lcm5un 40474 | Least common multiple of natural numbers up to 5 equals 60. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...5)) = ;60 | ||
Theorem | lcm6un 40475 | Least common multiple of natural numbers up to 6 equals 60. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...6)) = ;60 | ||
Theorem | lcm7un 40476 | Least common multiple of natural numbers up to 7 equals 420. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...7)) = ;;420 | ||
Theorem | lcm8un 40477 | Least common multiple of natural numbers up to 8 equals 840. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...8)) = ;;840 | ||
Theorem | 3factsumint1 40478* | Move constants out of integrals or sums and/or commute sum and integral. (Contributed by metakunt, 26-Apr-2024.) |
⊢ 𝐴 = (𝐿[,]𝑈) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ)) ⇒ ⊢ (𝜑 → ∫𝐴Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥) | ||
Theorem | 3factsumint2 40479* | Move constants out of integrals or sums and/or commute sum and integral. (Contributed by metakunt, 26-Apr-2024.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐺 · (𝐹 · 𝐻)) d𝑥) | ||
Theorem | 3factsumint3 40480* | Move constants out of integrals or sums and/or commute sum and integral. (Contributed by metakunt, 26-Apr-2024.) |
⊢ 𝐴 = (𝐿[,]𝑈) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ)) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 ∫𝐴(𝐺 · (𝐹 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 (𝐺 · ∫𝐴(𝐹 · 𝐻) d𝑥)) | ||
Theorem | 3factsumint4 40481* | Move constants out of integrals or sums and/or commute sum and integral. (Contributed by metakunt, 26-Apr-2024.) |
⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ) ⇒ ⊢ (𝜑 → ∫𝐴Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻)) d𝑥 = ∫𝐴(𝐹 · Σ𝑘 ∈ 𝐵 (𝐺 · 𝐻)) d𝑥) | ||
Theorem | 3factsumint 40482* | Helpful equation for lcm inequality proof. (Contributed by metakunt, 26-Apr-2024.) |
⊢ 𝐴 = (𝐿[,]𝑈) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ)) ⇒ ⊢ (𝜑 → ∫𝐴(𝐹 · Σ𝑘 ∈ 𝐵 (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 (𝐺 · ∫𝐴(𝐹 · 𝐻) d𝑥)) | ||
Theorem | resopunitintvd 40483 | Restrict continuous function on open unit interval. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐴) ∈ (ℂ–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (0(,)1) ↦ 𝐴) ∈ ((0(,)1)–cn→ℂ)) | ||
Theorem | resclunitintvd 40484 | Restrict continuous function on closed unit interval. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐴) ∈ (ℂ–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (0[,]1) ↦ 𝐴) ∈ ((0[,]1)–cn→ℂ)) | ||
Theorem | resdvopclptsd 40485* | Restrict derivative on unit interval. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℝ D (𝑥 ∈ (0[,]1) ↦ 𝐴)) = (𝑥 ∈ (0(,)1) ↦ 𝐵)) | ||
Theorem | lcmineqlem1 40486* | Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 29-Apr-2024.) |
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) d𝑥) | ||
Theorem | lcmineqlem2 40487* | Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 29-Apr-2024.) |
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥)) | ||
Theorem | lcmineqlem3 40488* | Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 30-Apr-2024.) |
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (1 / (𝑀 + 𝑘)))) | ||
Theorem | lcmineqlem4 40489 | Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. F is found in lcmineqlem6 40491. (Contributed by metakunt, 10-May-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) & ⊢ (𝜑 → 𝐾 ∈ (0...(𝑁 − 𝑀))) ⇒ ⊢ (𝜑 → ((lcm‘(1...𝑁)) / (𝑀 + 𝐾)) ∈ ℤ) | ||
Theorem | lcmineqlem5 40490 | Technical lemma for reciprocal multiplication in deduction form. (Contributed by metakunt, 10-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 · (𝐵 · (1 / 𝐶))) = (𝐵 · (𝐴 / 𝐶))) | ||
Theorem | lcmineqlem6 40491* | Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 10-May-2024.) |
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℤ) | ||
Theorem | lcmineqlem7 40492 | Derivative of 1-x for chain rule application. (Contributed by metakunt, 12-May-2024.) |
⊢ (ℂ D (𝑥 ∈ ℂ ↦ (1 − 𝑥))) = (𝑥 ∈ ℂ ↦ -1) | ||
Theorem | lcmineqlem8 40493* | Derivative of (1-x)^(N-M). (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 < 𝑁) ⇒ ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((1 − 𝑥)↑(𝑁 − 𝑀)))) = (𝑥 ∈ ℂ ↦ (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))) | ||
Theorem | lcmineqlem9 40494* | (1-x)^(N-M) is continuous. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((1 − 𝑥)↑(𝑁 − 𝑀))) ∈ (ℂ–cn→ℂ)) | ||
Theorem | lcmineqlem10 40495* | Induction step of lcmineqlem13 40498 (deduction form). (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 < 𝑁) ⇒ ⊢ (𝜑 → ∫(0[,]1)((𝑥↑((𝑀 + 1) − 1)) · ((1 − 𝑥)↑(𝑁 − (𝑀 + 1)))) d𝑥 = ((𝑀 / (𝑁 − 𝑀)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥)) | ||
Theorem | lcmineqlem11 40496 | Induction step, continuation for binomial coefficients. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 < 𝑁) ⇒ ⊢ (𝜑 → (1 / ((𝑀 + 1) · (𝑁C(𝑀 + 1)))) = ((𝑀 / (𝑁 − 𝑀)) · (1 / (𝑀 · (𝑁C𝑀))))) | ||
Theorem | lcmineqlem12 40497* | Base case for induction. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ∫(0[,]1)((𝑡↑(1 − 1)) · ((1 − 𝑡)↑(𝑁 − 1))) d𝑡 = (1 / (1 · (𝑁C1)))) | ||
Theorem | lcmineqlem13 40498* | Induction proof for lcm integral. (Contributed by metakunt, 12-May-2024.) |
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝐹 = (1 / (𝑀 · (𝑁C𝑀)))) | ||
Theorem | lcmineqlem14 40499 | Technical lemma for inequality estimate. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℕ) & ⊢ (𝜑 → 𝐷 ∈ ℕ) & ⊢ (𝜑 → 𝐸 ∈ ℕ) & ⊢ (𝜑 → (𝐴 · 𝐶) ∥ 𝐷) & ⊢ (𝜑 → (𝐵 · 𝐶) ∥ 𝐸) & ⊢ (𝜑 → 𝐷 ∥ 𝐸) & ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) ∥ 𝐸) | ||
Theorem | lcmineqlem15 40500* | F times the least common multiple of 1 to n is a natural number. (Contributed by metakunt, 10-May-2024.) |
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℕ) |
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