P. 426 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42501-42600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hdmaplkr 42501	Kernel of the vector to dual map. Line 16 in [Holland95] p. 14. TODO: eliminate 𝐹 hypothesis. (Contributed by NM, 9-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝑌 = (LKer‘𝑈)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑌‘(𝑆‘𝑋)) = (𝑂‘{𝑋}))
Theorem	hdmapellkr 42502	Membership in the kernel (as shown by hdmaplkr 42501) of the vector to dual map. Line 17 in [Holland95] p. 14. (Contributed by NM, 16-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑌) = 0 ↔ 𝑌 ∈ (𝑂‘{𝑋})))
Theorem	hdmapip0 42503	Zero property that will be used for inner product. (Contributed by NM, 9-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝑍 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 ↔ 𝑋 = 0 ))
Theorem	hdmapip1 42504	Construct a proportional vector 𝑌 whose inner product with the original 𝑋 equals one. (Contributed by NM, 13-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (invr‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ 𝑌 = ((𝑁‘((𝑆‘𝑋)‘𝑋)) · 𝑋)    ⇒   ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 1 )
Theorem	hdmapip0com 42505	Commutation property of Baer's sigma map (Holland's A map). Line 20 of [Holland95] p. 14. Also part of Lemma 1 of [Baer] p. 110 line 7. (Contributed by NM, 9-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑌) = 0 ↔ ((𝑆‘𝑌)‘𝑋) = 0 ))
Theorem	hdmapinvlem1 42506	Line 27 in [Baer] p. 110. We use 𝐶 for Baer's u. Our unit vector 𝐸 has the required properties for his w by hdmapevec2 42424. Our ((𝑆‘𝐸)‘𝐶) means the inner product ⟨𝐶, 𝐸⟩ i.e. his f(u,w) (note argument reversal). (Contributed by NM, 11-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    ⇒   ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐶) = 0 )
Theorem	hdmapinvlem2 42507	Line 28 in [Baer] p. 110, 0 = f(w,u). (Contributed by NM, 11-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    ⇒   ⊢ (𝜑 → ((𝑆‘𝐶)‘𝐸) = 0 )
Theorem	hdmapinvlem3 42508	Line 30 in [Baer] p. 110, f(sw + u, tw - v) = 0. (Contributed by NM, 12-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) = ((𝑆‘𝐷)‘𝐶))    ⇒   ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 )
Theorem	hdmapinvlem4 42509	Part 1.1 of Proposition 1 of [Baer] p. 110. We use 𝐶, 𝐷, 𝐼, and 𝐽 for Baer's u, v, s, and t. Our unit vector 𝐸 has the required properties for his w by hdmapevec2 42424. Our ((𝑆‘𝐷)‘𝐶) means his f(u,v) (note argument reversal). (Contributed by NM, 12-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) = ((𝑆‘𝐷)‘𝐶))    ⇒   ⊢ (𝜑 → (𝐽 × (𝐺‘𝐼)) = ((𝑆‘𝐶)‘𝐷))
Theorem	hdmapglem5 42510	Part 1.2 in [Baer] p. 110 line 34, f(u,v) alpha = f(v,u). (Contributed by NM, 12-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷))
Theorem	hgmapvvlem1 42511	Involution property of scalar sigma map. Line 10 in [Baer] p. 111, t sigma squared = t. Our 𝐸, 𝐶, 𝐷, 𝑌, 𝑋 correspond to Baer's w, h, k, s, t. (Contributed by NM, 13-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (invr‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 )    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑌 × (𝐺‘𝑋)) = 1 )    ⇒   ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)
Theorem	hgmapvvlem2 42512	Lemma for hgmapvv 42514. Eliminate 𝑌 (Baer's s). (Contributed by NM, 13-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (invr‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 )    ⇒   ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)
Theorem	hgmapvvlem3 42513	Lemma for hgmapvv 42514. Eliminate ((𝑆‘𝐷)‘𝐶) = 1 (Baer's f(h,k)=1). (Contributed by NM, 13-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (invr‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)
Theorem	hgmapvv 42514	Value of a double involution. Part 1.2 of [Baer] p. 110 line 37. (Contributed by NM, 13-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)
Theorem	hdmapglem7a 42515*	Lemma for hdmapg 42518. (Contributed by NM, 14-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))
Theorem	hdmapglem7b 42516	Lemma for hdmapg 42518. (Contributed by NM, 14-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ✚ = (+g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → 𝑥 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝑦 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝑚 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑛 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑆‘((𝑚 · 𝐸) + 𝑥))‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦)))
Theorem	hdmapglem7 42517	Lemma for hdmapg 42518. Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). In the proof, our 𝐸, (𝑂‘{𝐸}), 𝑋, 𝑌, 𝑘, 𝑢, 𝑙, and 𝑣 correspond respectively to Baer's w, H, x, y, x', x'', y', and y'', and our ((𝑆‘𝑌)‘𝑋) corresponds to Baer's f(x,y). (Contributed by NM, 14-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ✚ = (+g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))
Theorem	hdmapg 42518	Apply the scalar sigma function (involution) 𝐺 to an inner product reverses the arguments. The inner product of 𝑋 and 𝑌 is represented by ((𝑆‘𝑌)‘𝑋). Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). (Contributed by NM, 14-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))
Theorem	hdmapoc 42519*	Express our constructed orthocomplement (polarity) in terms of the Hilbert space definition of orthocomplement. Lines 24 and 25 in [Holland95] p. 14. (Contributed by NM, 17-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 })
Syntax	chlh 42520	Extend class notation with the final constructed Hilbert space.
class HLHil
Definition	df-hlhil 42521*	Define our final Hilbert space constructed from a Hilbert lattice. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ HLHil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})))
Theorem	hlhilset 42522*	The final Hilbert space constructed from a Hilbert lattice 𝐾 and an arbitrary hyperplane 𝑊 in 𝐾. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐿 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐿 = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
Theorem	hlhilsca 42523	The scalar of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩)    ⇒   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑈))
Theorem	hlhilbase 42524	The base set of the final constructed Hilbert space. (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))    &   ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑀 = (Base‘𝐿)    ⇒   ⊢ (𝜑 → 𝑀 = (Base‘𝑈))
Theorem	hlhilplus 42525	The vector addition for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝐿)    ⇒   ⊢ (𝜑 → + = (+g‘𝑈))
Theorem	hlhilslem 42526	Lemma for hlhilsbase 42527 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	hlhilsbase 42527	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	hlhilsplus 42528	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	hlhilsmul 42529	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	hlhilsbase2 42530	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 42531	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 42532	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 42533	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 42534	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 42535	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 42536*	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 42537	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 42538	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 42539	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 42540	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 42541	Lemma for hlhilsrng 42542. (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 42542	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 42543	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 42544	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 42545	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 42546	The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 42524 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 42547*	Lemma for hlhil 25485. (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 42548*	Lemma for hlhil 25485. (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 42549	Construction of a Hilbert space (df-hil 21736) 𝑈 from a Hilbert lattice (df-hlat 39939) 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 41697) 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 41697. 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 21736. 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)
21.29  Mathbox for metakunt
21.29.1  Commutative Semiring
Syntax	ccsrg 42550	Extend class notation with the class of all commutative semirings.
class CSRing
Definition	df-csring 42551	Define the class of all commutative semirings. (Contributed by metakunt, 4-Apr-2025.)
⊢ CSRing = {𝑓 ∈ SRing ∣ (mulGrp‘𝑓) ∈ CMnd}
Theorem	iscsrg 42552	A commutative semiring is a semiring whose multiplication is a commutative monoid. (Contributed by metakunt, 4-Apr-2025.)
⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ CSRing ↔ (𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd))
21.29.2  General helpful statements
Theorem	rhmzrhval 42553	Evaluation of integers across a ring homomorphism. (Contributed by metakunt, 4-Jun-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ ℤ)    &   ⊢ 𝑀 = (ℤRHom‘𝑅)    &   ⊢ 𝑁 = (ℤRHom‘𝑆)    ⇒   ⊢ (𝜑 → (𝐹‘(𝑀‘𝑋)) = (𝑁‘𝑋))
Theorem	zndvdchrrhm 42554*	Construction of a ring homomorphism from ℤ/nℤ to 𝑅 when the characteristic of 𝑅 divides 𝑁. (Contributed by metakunt, 4-Jun-2025.)
⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (chr‘𝑅) ∈ ℤ)    &   ⊢ (𝜑 → (chr‘𝑅) ∥ 𝑁)    &   ⊢ 𝑍 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐹 = (𝑥 ∈ (Base‘𝑍) ↦ ∪ ((ℤRHom‘𝑅) “ 𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑍 RingHom 𝑅))
Theorem	relogbcld 42555	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 42556	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 42557	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 42558	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 42559*	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 42560	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	fzne2d 42561	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 42562*	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 42563	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 42564	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 42565	Associative law for addition. (Contributed by metakunt, 25-Apr-2024.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    &   ⊢ 𝐶 ∈ ℕ    ⇒   ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
Theorem	addcomnni 42566	Commutative law for addition. (Contributed by metakunt, 25-Apr-2024.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴)
Theorem	mulassnni 42567	Associative law for multiplication. (Contributed by metakunt, 25-Apr-2024.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    &   ⊢ 𝐶 ∈ ℕ    ⇒   ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
Theorem	mulcomnni 42568	Commutative law for multiplication. (Contributed by metakunt, 25-Apr-2024.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)
Theorem	gcdcomnni 42569	Commutative law for gcd. (Contributed by metakunt, 25-Apr-2024.)
⊢ 𝑀 ∈ ℕ    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)
Theorem	gcdnegnni 42570	Negation invariance for gcd. (Contributed by metakunt, 25-Apr-2024.)
⊢ 𝑀 ∈ ℕ    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝑀 gcd -𝑁) = (𝑀 gcd 𝑁)
Theorem	neggcdnni 42571	Negation invariance for gcd. (Contributed by metakunt, 25-Apr-2024.)
⊢ 𝑀 ∈ ℕ    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (-𝑀 gcd 𝑁) = (𝑀 gcd 𝑁)
Theorem	bccl2d 42572	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 42573	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 42574	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 42575	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 42576	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 42577	Closure of the gcd operator. (Contributed by metakunt, 25-Apr-2024.)
⊢ 𝑀 ∈ ℕ    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝑀 gcd 𝑁) ∈ ℕ
Theorem	muldvds1d 42578	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 42579	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 42580	A positive integer divides a natural number if and only if the quotient is a positive integer, a deduction version of nndivdvds 16278. (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝑀 ∥ 𝑁 ↔ (𝑁 / 𝑀) ∈ ℕ))
Theorem	nnproddivdvdsd 42581	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 42582	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	imadomfi 42583	An image of a function under a finite set is dominated by the set. (Contributed by SN, 10-May-2025.)
⊢ ((𝐴 ∈ Fin ∧ Fun 𝐹) → (𝐹 “ 𝐴) ≼ 𝐴)
21.29.3  Some gcd and lcm results
Theorem	12gcd5e1 42584	The gcd of 12 and 5 is 1. (Contributed by metakunt, 25-Apr-2024.)
⊢ (;12 gcd 5) = 1
Theorem	60gcd6e6 42585	The gcd of 60 and 6 is 6. (Contributed by metakunt, 25-Apr-2024.)
⊢ (;60 gcd 6) = 6
Theorem	60gcd7e1 42586	The gcd of 60 and 7 is 1. (Contributed by metakunt, 25-Apr-2024.)
⊢ (;60 gcd 7) = 1
Theorem	420gcd8e4 42587	The gcd of 420 and 8 is 4. (Contributed by metakunt, 25-Apr-2024.)
⊢ (;;420 gcd 8) = 4
Theorem	lcmeprodgcdi 42588	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 42589	The lcm of 12 and 5 is 60. (Contributed by metakunt, 25-Apr-2024.)
⊢ (;12 lcm 5) = ;60
Theorem	60lcm6e60 42590	The lcm of 60 and 6 is 60. (Contributed by metakunt, 25-Apr-2024.)
⊢ (;60 lcm 6) = ;60
Theorem	60lcm7e420 42591	The lcm of 60 and 7 is 420. (Contributed by metakunt, 25-Apr-2024.)
⊢ (;60 lcm 7) = ;;420
Theorem	420lcm8e840 42592	The lcm of 420 and 8 is 840. (Contributed by metakunt, 25-Apr-2024.)
⊢ (;;420 lcm 8) = ;;840
Theorem	lcmfunnnd 42593	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 42594	Least common multiple of natural numbers up to 1 equals 1. (Contributed by metakunt, 25-Apr-2024.)
⊢ (lcm‘(1...1)) = 1
Theorem	lcm2un 42595	Least common multiple of natural numbers up to 2 equals 2. (Contributed by metakunt, 25-Apr-2024.)
⊢ (lcm‘(1...2)) = 2
Theorem	lcm3un 42596	Least common multiple of natural numbers up to 3 equals 6. (Contributed by metakunt, 25-Apr-2024.)
⊢ (lcm‘(1...3)) = 6
Theorem	lcm4un 42597	Least common multiple of natural numbers up to 4 equals 12. (Contributed by metakunt, 25-Apr-2024.)
⊢ (lcm‘(1...4)) = ;12
Theorem	lcm5un 42598	Least common multiple of natural numbers up to 5 equals 60. (Contributed by metakunt, 25-Apr-2024.)
⊢ (lcm‘(1...5)) = ;60
Theorem	lcm6un 42599	Least common multiple of natural numbers up to 6 equals 60. (Contributed by metakunt, 25-Apr-2024.)
⊢ (lcm‘(1...6)) = ;60
Theorem	lcm7un 42600	Least common multiple of natural numbers up to 7 equals 420. (Contributed by metakunt, 25-Apr-2024.)
⊢ (lcm‘(1...7)) = ;;420