P. 405 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 40401-40500   *Has distinct variable group(s)

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)
21.25  Mathbox for metakunt
21.25.1  General helpful statements
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)
21.25.2  Some gcd and lcm results
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
21.25.3  Least common multiple inequality theorem
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)