P. 135 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13401-13500   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-hom 13401	Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ Hom = Slot ;14
Definition	df-cco 13402	Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ comp = Slot ;15
Theorem	strleund 13403	Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
⊢ (𝜑 → 𝐹 Struct ⟨𝐴, 𝐵⟩)    &   ⊢ (𝜑 → 𝐺 Struct ⟨𝐶, 𝐷⟩)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∪ 𝐺) Struct ⟨𝐴, 𝐷⟩)
Theorem	strleun 13404	Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐹 Struct ⟨𝐴, 𝐵⟩    &   ⊢ 𝐺 Struct ⟨𝐶, 𝐷⟩    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ (𝐹 ∪ 𝐺) Struct ⟨𝐴, 𝐷⟩
Theorem	strext 13405	Extending the upper range of a structure. This works because when we say that a structure has components in 𝐴...𝐶 we are not saying that every slot in that range is present, just that all the slots that are present are within that range. (Contributed by Jim Kingdon, 26-Feb-2025.)
⊢ (𝜑 → 𝐹 Struct ⟨𝐴, 𝐵⟩)    &   ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐵))    ⇒   ⊢ (𝜑 → 𝐹 Struct ⟨𝐴, 𝐶⟩)
Theorem	strle1g 13406	Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    ⇒   ⊢ (𝑋 ∈ 𝑉 → {⟨𝐴, 𝑋⟩} Struct ⟨𝐼, 𝐼⟩)
Theorem	strle2g 13407	Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    &   ⊢ 𝐼 < 𝐽    &   ⊢ 𝐽 ∈ ℕ    &   ⊢ 𝐵 = 𝐽    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩} Struct ⟨𝐼, 𝐽⟩)
Theorem	strle3g 13408	Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    &   ⊢ 𝐼 < 𝐽    &   ⊢ 𝐽 ∈ ℕ    &   ⊢ 𝐵 = 𝐽    &   ⊢ 𝐽 < 𝐾    &   ⊢ 𝐾 ∈ ℕ    &   ⊢ 𝐶 = 𝐾    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑃) → {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩, ⟨𝐶, 𝑍⟩} Struct ⟨𝐼, 𝐾⟩)
Theorem	plusgndx 13409	Index value of the df-plusg 13390 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (+g‘ndx) = 2
Theorem	plusgid 13410	Utility theorem: index-independent form of df-plusg 13390. (Contributed by NM, 20-Oct-2012.)
⊢ +g = Slot (+g‘ndx)
Theorem	plusgndxnn 13411	The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.)
⊢ (+g‘ndx) ∈ ℕ
Theorem	plusgslid 13412	Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
Theorem	basendxltplusgndx 13413	The index of the slot for the base set is less then the index of the slot for the group operation in an extensible structure. (Contributed by AV, 17-Oct-2024.)
⊢ (Base‘ndx) < (+g‘ndx)
Theorem	opelstrsl 13414	The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.)
⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   ⊢ (𝜑 → 𝑆 Struct 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → ⟨(𝐸‘ndx), 𝑉⟩ ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝑉 = (𝐸‘𝑆))
Theorem	opelstrbas 13415	The base set of a structure with a base set. (Contributed by AV, 10-Nov-2021.)
⊢ (𝜑 → 𝑆 Struct 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝑉 = (Base‘𝑆))
Theorem	1strstrg 13416	A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Revised by Jim Kingdon, 28-Jan-2023.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐺 Struct ⟨1, 1⟩)
Theorem	1strbas 13417	The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	2strstrndx 13418	A constructed two-slot structure not depending on the hard-coded index value of the base set. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 14-Dec-2025.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩)
Theorem	2strstrg 13419	A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) Use 2strstrndx 13418 instead. (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 1 < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct ⟨1, 𝑁⟩)
Theorem	2strbasg 13420	The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 1 < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐵 = (Base‘𝐺))
Theorem	2stropg 13421	The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 1 < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (𝐸‘𝐺))
Theorem	2strstr1g 13422	A constructed two-slot structure. Version of 2strstrg 13419 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩)
Theorem	2strbas1g 13423	The base set of a constructed two-slot structure. Version of 2strbasg 13420 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐵 = (Base‘𝐺))
Theorem	2strop1g 13424	The other slot of a constructed two-slot structure. Version of 2stropg 13421 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐸 = Slot 𝑁    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (𝐸‘𝐺))
Theorem	basendxnplusgndx 13425	The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.)
⊢ (Base‘ndx) ≠ (+g‘ndx)
Theorem	grpstrg 13426	A constructed group is a structure on 1...2. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct ⟨1, 2⟩)
Theorem	grpbaseg 13427	The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐵 = (Base‘𝐺))
Theorem	grpplusgg 13428	The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (+g‘𝐺))
Theorem	ressplusgd 13429	+g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ (𝜑 → 𝐻 = (𝐺 ↾s 𝐴))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    ⇒   ⊢ (𝜑 → + = (+g‘𝐻))
Theorem	mulrndx 13430	Index value of the df-mulr 13391 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (.r‘ndx) = 3
Theorem	mulridx 13431	Utility theorem: index-independent form of df-mulr 13391. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ .r = Slot (.r‘ndx)
Theorem	mulrslid 13432	Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
Theorem	plusgndxnmulrndx 13433	The slot for the group (addition) operation is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.)
⊢ (+g‘ndx) ≠ (.r‘ndx)
Theorem	basendxnmulrndx 13434	The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.)
⊢ (Base‘ndx) ≠ (.r‘ndx)
Theorem	rngstrg 13435	A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
⊢ 𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝑅 Struct ⟨1, 3⟩)
Theorem	rngbaseg 13436	The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
⊢ 𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝐵 = (Base‘𝑅))
Theorem	rngplusgg 13437	The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → + = (+g‘𝑅))
Theorem	rngmulrg 13438	The multiplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → · = (.r‘𝑅))
Theorem	starvndx 13439	Index value of the df-starv 13392 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (*𝑟‘ndx) = 4
Theorem	starvid 13440	Utility theorem: index-independent form of df-starv 13392. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ *𝑟 = Slot (*𝑟‘ndx)
Theorem	starvslid 13441	Slot property of *𝑟. (Contributed by Jim Kingdon, 4-Feb-2023.)
⊢ (*𝑟 = Slot (*𝑟‘ndx) ∧ (*𝑟‘ndx) ∈ ℕ)
Theorem	starvndxnbasendx 13442	The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (Base‘ndx)
Theorem	starvndxnplusgndx 13443	The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (+g‘ndx)
Theorem	starvndxnmulrndx 13444	The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (.r‘ndx)
Theorem	ressmulrg 13445	.r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → · = (.r‘𝑆))
Theorem	srngstrd 13446	A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝑋)    &   ⊢ (𝜑 → ∗ ∈ 𝑌)    ⇒   ⊢ (𝜑 → 𝑅 Struct ⟨1, 4⟩)
Theorem	srngbased 13447	The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝑋)    &   ⊢ (𝜑 → ∗ ∈ 𝑌)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
Theorem	srngplusgd 13448	The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by Jim Kingdon, 5-Feb-2023.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝑋)    &   ⊢ (𝜑 → ∗ ∈ 𝑌)    ⇒   ⊢ (𝜑 → + = (+g‘𝑅))
Theorem	srngmulrd 13449	The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝑋)    &   ⊢ (𝜑 → ∗ ∈ 𝑌)    ⇒   ⊢ (𝜑 → · = (.r‘𝑅))
Theorem	srnginvld 13450	The involution function of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝑋)    &   ⊢ (𝜑 → ∗ ∈ 𝑌)    ⇒   ⊢ (𝜑 → ∗ = (*𝑟‘𝑅))
Theorem	scandx 13451	Index value of the df-sca 13393 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (Scalar‘ndx) = 5
Theorem	scaid 13452	Utility theorem: index-independent form of scalar df-sca 13393. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ Scalar = Slot (Scalar‘ndx)
Theorem	scaslid 13453	Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
Theorem	scandxnbasendx 13454	The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (Base‘ndx)
Theorem	scandxnplusgndx 13455	The slot for the scalar field is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (+g‘ndx)
Theorem	scandxnmulrndx 13456	The slot for the scalar field is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (.r‘ndx)
Theorem	vscandx 13457	Index value of the df-vsca 13394 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ( ·𝑠 ‘ndx) = 6
Theorem	vscaid 13458	Utility theorem: index-independent form of scalar product df-vsca 13394. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
Theorem	vscandxnbasendx 13459	The slot for the scalar product is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Base‘ndx)
Theorem	vscandxnplusgndx 13460	The slot for the scalar product is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (+g‘ndx)
Theorem	vscandxnmulrndx 13461	The slot for the scalar product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (.r‘ndx)
Theorem	vscandxnscandx 13462	The slot for the scalar product is not the slot for the scalar field in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx)
Theorem	vscaslid 13463	Slot property of ·𝑠. (Contributed by Jim Kingdon, 5-Feb-2023.)
⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
Theorem	lmodstrd 13464	A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑌)    &   ⊢ (𝜑 → · ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝑊 Struct ⟨1, 6⟩)
Theorem	lmodbased 13465	The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑌)    &   ⊢ (𝜑 → · ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝑊))
Theorem	lmodplusgd 13466	The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑌)    &   ⊢ (𝜑 → · ∈ 𝑍)    ⇒   ⊢ (𝜑 → + = (+g‘𝑊))
Theorem	lmodscad 13467	The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑌)    &   ⊢ (𝜑 → · ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))
Theorem	lmodvscad 13468	The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 7-Feb-2023.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑌)    &   ⊢ (𝜑 → · ∈ 𝑍)    ⇒   ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))
Theorem	ipndx 13469	Index value of the df-ip 13395 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (·𝑖‘ndx) = 8
Theorem	ipid 13470	Utility theorem: index-independent form of df-ip 13395. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ ·𝑖 = Slot (·𝑖‘ndx)
Theorem	ipslid 13471	Slot property of ·𝑖. (Contributed by Jim Kingdon, 7-Feb-2023.)
⊢ (·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)
Theorem	ipndxnbasendx 13472	The slot for the inner product is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
⊢ (·𝑖‘ndx) ≠ (Base‘ndx)
Theorem	ipndxnplusgndx 13473	The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
⊢ (·𝑖‘ndx) ≠ (+g‘ndx)
Theorem	ipndxnmulrndx 13474	The slot for the inner product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
⊢ (·𝑖‘ndx) ≠ (.r‘ndx)
Theorem	slotsdifipndx 13475	The slot for the scalar is not the index of other slots. (Contributed by AV, 12-Nov-2024.)
⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx))
Theorem	ipsstrd 13476	A constructed inner product space is a structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → × ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → · ∈ 𝑄)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝐴 Struct ⟨1, 8⟩)
Theorem	ipsbased 13477	The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → × ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → · ∈ 𝑄)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝐴))
Theorem	ipsaddgd 13478	The additive operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → × ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → · ∈ 𝑄)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    ⇒   ⊢ (𝜑 → + = (+g‘𝐴))
Theorem	ipsmulrd 13479	The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → × ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → · ∈ 𝑄)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    ⇒   ⊢ (𝜑 → × = (.r‘𝐴))
Theorem	ipsscad 13480	The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → × ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → · ∈ 𝑄)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝑆 = (Scalar‘𝐴))
Theorem	ipsvscad 13481	The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → × ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → · ∈ 𝑄)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    ⇒   ⊢ (𝜑 → · = ( ·𝑠 ‘𝐴))
Theorem	ipsipd 13482	The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → × ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → · ∈ 𝑄)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝐼 = (·𝑖‘𝐴))
Theorem	ressscag 13483	Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝐹 = (Scalar‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → 𝐹 = (Scalar‘𝐻))
Theorem	ressvscag 13484	·𝑠 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ · = ( ·𝑠 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → · = ( ·𝑠 ‘𝐻))
Theorem	ressipg 13485	The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ , = (·𝑖‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → , = (·𝑖‘𝐻))
Theorem	tsetndx 13486	Index value of the df-tset 13396 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (TopSet‘ndx) = 9
Theorem	tsetid 13487	Utility theorem: index-independent form of df-tset 13396. (Contributed by NM, 20-Oct-2012.)
⊢ TopSet = Slot (TopSet‘ndx)
Theorem	tsetslid 13488	Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ)
Theorem	tsetndxnn 13489	The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.)
⊢ (TopSet‘ndx) ∈ ℕ
Theorem	basendxlttsetndx 13490	The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
⊢ (Base‘ndx) < (TopSet‘ndx)
Theorem	tsetndxnbasendx 13491	The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.)
⊢ (TopSet‘ndx) ≠ (Base‘ndx)
Theorem	tsetndxnplusgndx 13492	The slot for the topology is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ (TopSet‘ndx) ≠ (+g‘ndx)
Theorem	tsetndxnmulrndx 13493	The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
⊢ (TopSet‘ndx) ≠ (.r‘ndx)
Theorem	tsetndxnstarvndx 13494	The slot for the topology is not the slot for the involution in an extensible structure. (Contributed by AV, 11-Nov-2024.)
⊢ (TopSet‘ndx) ≠ (*𝑟‘ndx)
Theorem	slotstnscsi 13495	The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. (Contributed by AV, 29-Oct-2024.)
⊢ ((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx))
Theorem	topgrpstrd 13496	A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝑊 Struct ⟨1, 9⟩)
Theorem	topgrpbasd 13497	The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝑊))
Theorem	topgrpplusgd 13498	The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑋)    ⇒   ⊢ (𝜑 → + = (+g‘𝑊))
Theorem	topgrptsetd 13499	The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝐽 = (TopSet‘𝑊))
Theorem	plendx 13500	Index value of the df-ple 13397 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
⊢ (le‘ndx) = ;10