P. 126 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12501-12600   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-ocomp 12501	Define the orthocomplementation extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ oc = Slot ;11
Definition	df-ds 12502	Define the distance function component of a metric space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ dist = Slot ;12
Definition	df-unif 12503	Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ UnifSet = Slot ;13
Definition	df-hom 12504	Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ Hom = Slot ;14
Definition	df-cco 12505	Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ comp = Slot ;15
Theorem	strleund 12506	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 12507	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	strle1g 12508	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 12509	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 12510	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 12511	Index value of the df-plusg 12493 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (+g‘ndx) = 2
Theorem	plusgid 12512	Utility theorem: index-independent form of df-plusg 12493. (Contributed by NM, 20-Oct-2012.)
⊢ +g = Slot (+g‘ndx)
Theorem	plusgslid 12513	Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
Theorem	opelstrsl 12514	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 12515	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 12516	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 12517	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	2strstrg 12518	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) → 𝐺 Struct ⟨1, 𝑁⟩)
Theorem	2strbasg 12519	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 12520	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 12521	A constructed two-slot structure. Version of 2strstrg 12518 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 12522	The base set of a constructed two-slot structure. Version of 2strbasg 12519 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 12523	The other slot of a constructed two-slot structure. Version of 2stropg 12520 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 12524	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 12525	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 12526	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 12527	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	mulrndx 12528	Index value of the df-mulr 12494 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (.r‘ndx) = 3
Theorem	mulrid 12529	Utility theorem: index-independent form of df-mulr 12494. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ .r = Slot (.r‘ndx)
Theorem	mulrslid 12530	Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
Theorem	plusgndxnmulrndx 12531	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 12532	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 12533	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 12534	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 12535	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 12536	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 12537	Index value of the df-starv 12495 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (*𝑟‘ndx) = 4
Theorem	starvid 12538	Utility theorem: index-independent form of df-starv 12495. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ *𝑟 = Slot (*𝑟‘ndx)
Theorem	starvslid 12539	Slot property of *𝑟. (Contributed by Jim Kingdon, 4-Feb-2023.)
⊢ (*𝑟 = Slot (*𝑟‘ndx) ∧ (*𝑟‘ndx) ∈ ℕ)
Theorem	srngstrd 12540	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 12541	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 12542	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 12543	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 12544	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 12545	Index value of the df-sca 12496 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (Scalar‘ndx) = 5
Theorem	scaid 12546	Utility theorem: index-independent form of scalar df-sca 12496. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ Scalar = Slot (Scalar‘ndx)
Theorem	scaslid 12547	Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
Theorem	vscandx 12548	Index value of the df-vsca 12497 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ( ·𝑠 ‘ndx) = 6
Theorem	vscaid 12549	Utility theorem: index-independent form of scalar product df-vsca 12497. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
Theorem	vscaslid 12550	Slot property of ·𝑠. (Contributed by Jim Kingdon, 5-Feb-2023.)
⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
Theorem	lmodstrd 12551	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 12552	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 12553	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 12554	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 12555	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 12556	Index value of the df-ip 12498 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (·𝑖‘ndx) = 8
Theorem	ipid 12557	Utility theorem: index-independent form of df-ip 12498. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ ·𝑖 = Slot (·𝑖‘ndx)
Theorem	ipslid 12558	Slot property of ·𝑖. (Contributed by Jim Kingdon, 7-Feb-2023.)
⊢ (·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)
Theorem	ipsstrd 12559	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 12560	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 12561	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 12562	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 12563	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 12564	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 12565	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	tsetndx 12566	Index value of the df-tset 12499 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (TopSet‘ndx) = 9
Theorem	tsetid 12567	Utility theorem: index-independent form of df-tset 12499. (Contributed by NM, 20-Oct-2012.)
⊢ TopSet = Slot (TopSet‘ndx)
Theorem	tsetslid 12568	Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ)
Theorem	topgrpstrd 12569	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 12570	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 12571	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 12572	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 12573	Index value of the df-ple 12500 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
⊢ (le‘ndx) = ;10
Theorem	pleid 12574	Utility theorem: self-referencing, index-independent form of df-ple 12500. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
⊢ le = Slot (le‘ndx)
Theorem	pleslid 12575	Slot property of le. (Contributed by Jim Kingdon, 9-Feb-2023.)
⊢ (le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ)
Theorem	dsndx 12576	Index value of the df-ds 12502 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (dist‘ndx) = ;12
Theorem	dsid 12577	Utility theorem: index-independent form of df-ds 12502. (Contributed by Mario Carneiro, 23-Dec-2013.)
⊢ dist = Slot (dist‘ndx)
Theorem	dsslid 12578	Slot property of dist. (Contributed by Jim Kingdon, 6-May-2023.)
⊢ (dist = Slot (dist‘ndx) ∧ (dist‘ndx) ∈ ℕ)
6.1.3  Definition of the structure product
Syntax	crest 12579	Extend class notation with the function returning a subspace topology.
class ↾t
Syntax	ctopn 12580	Extend class notation with the topology extractor function.
class TopOpen
Definition	df-rest 12581*	Function returning the subspace topology induced by the topology 𝑦 and the set 𝑥. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)))
Definition	df-topn 12582	Define the topology extractor function. This differs from df-tset 12499 when a structure has been restricted using df-ress 12424; in this case the TopSet component will still have a topology over the larger set, and this function fixes this by restricting the topology as well. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
Theorem	restfn 12583	The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
⊢ ↾t Fn (V × V)
Theorem	topnfn 12584	The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ TopOpen Fn V
Theorem	restval 12585*	The subspace topology induced by the topology 𝐽 on the set 𝐴. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
Theorem	elrest 12586*	The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵)))
Theorem	elrestr 12587	Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐴 ∈ 𝐽) → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
Theorem	restid2 12588	The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → (𝐽 ↾t 𝐴) = 𝐽)
Theorem	restsspw 12589	The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴
Theorem	restid 12590	The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ 𝑉 → (𝐽 ↾t 𝑋) = 𝐽)
Theorem	topnvalg 12591	Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopSet‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊))
Theorem	topnidg 12592	Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopSet‘𝑊)    ⇒   ⊢ ((𝑊 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐵) → 𝐽 = (TopOpen‘𝑊))
Theorem	topnpropgd 12593	The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.)
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
Syntax	ctg 12594	Extend class notation with a function that converts a basis to its corresponding topology.
class topGen
Syntax	cpt 12595	Extend class notation with a function whose value is a product topology.
class ∏t
Syntax	c0g 12596	Extend class notation with group identity element.
class 0g
Syntax	cgsu 12597	Extend class notation to include finitely supported group sums.
class Σg
Definition	df-0g 12598*	Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-gsum 12599. The related theorems will be provided later. (Contributed by NM, 20-Aug-2011.)
⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))))
Definition	df-gsum 12599*	Define the group sum for the structure 𝐺 of a finite sequence of elements whose values are defined by the expression 𝐵 and whose set of indices is 𝐴. It may be viewed as a product (if 𝐺 is a multiplication), a sum (if 𝐺 is an addition) or any other operation. The variable 𝑘 is normally a free variable in 𝐵 (i.e., 𝐵 can be thought of as 𝐵(𝑘)). The definition is meaningful in different contexts, depending on the size of the index set 𝐴 and each demanding different properties of 𝐺. 1. If 𝐴 = ∅ and 𝐺 has an identity element, then the sum equals this identity. 2. If 𝐴 = (𝑀...𝑁) and 𝐺 is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e., (𝐵(1) + 𝐵(2)) + 𝐵(3), etc. 3. If 𝐴 is a finite set (or is nonzero for finitely many indices) and 𝐺 is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined. 4. If 𝐴 is an infinite set and 𝐺 is a Hausdorff topological group, then there is a meaningful sum, but Σg cannot handle this case. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.)
⊢ Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋{𝑥 ∈ (Base‘𝑤) ∣ ∀𝑦 ∈ (Base‘𝑤)((𝑥(+g‘𝑤)𝑦) = 𝑦 ∧ (𝑦(+g‘𝑤)𝑥) = 𝑦)} / 𝑜⦌if(ran 𝑓 ⊆ 𝑜, (0g‘𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))), (℩𝑥∃𝑔[(◡𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(♯‘𝑦))–1-1-onto→𝑦 ∧ 𝑥 = (seq1((+g‘𝑤), (𝑓 ∘ 𝑔))‘(♯‘𝑦)))))))
Definition	df-topgen 12600*	Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78. (Contributed by NM, 16-Jul-2006.)
⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)})