P. 133 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13201-13300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	srngbased 13201	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 13202	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 13203	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 13204	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 13205	Index value of the df-sca 13147 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (Scalar‘ndx) = 5
Theorem	scaid 13206	Utility theorem: index-independent form of scalar df-sca 13147. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ Scalar = Slot (Scalar‘ndx)
Theorem	scaslid 13207	Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
Theorem	scandxnbasendx 13208	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 13209	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 13210	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 13211	Index value of the df-vsca 13148 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ( ·𝑠 ‘ndx) = 6
Theorem	vscaid 13212	Utility theorem: index-independent form of scalar product df-vsca 13148. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
Theorem	vscandxnbasendx 13213	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 13214	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 13215	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 13216	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 13217	Slot property of ·𝑠. (Contributed by Jim Kingdon, 5-Feb-2023.)
⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
Theorem	lmodstrd 13218	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 13219	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 13220	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 13221	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 13222	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 13223	Index value of the df-ip 13149 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (·𝑖‘ndx) = 8
Theorem	ipid 13224	Utility theorem: index-independent form of df-ip 13149. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ ·𝑖 = Slot (·𝑖‘ndx)
Theorem	ipslid 13225	Slot property of ·𝑖. (Contributed by Jim Kingdon, 7-Feb-2023.)
⊢ (·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)
Theorem	ipndxnbasendx 13226	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 13227	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 13228	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 13229	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 13230	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 13231	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 13232	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 13233	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 13234	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 13235	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 13236	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 13237	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 13238	·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 13239	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 13240	Index value of the df-tset 13150 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (TopSet‘ndx) = 9
Theorem	tsetid 13241	Utility theorem: index-independent form of df-tset 13150. (Contributed by NM, 20-Oct-2012.)
⊢ TopSet = Slot (TopSet‘ndx)
Theorem	tsetslid 13242	Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ)
Theorem	tsetndxnn 13243	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 13244	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 13245	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 13246	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 13247	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 13248	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 13249	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 13250	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 13251	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 13252	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 13253	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 13254	Index value of the df-ple 13151 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
⊢ (le‘ndx) = ;10
Theorem	pleid 13255	Utility theorem: self-referencing, index-independent form of df-ple 13151. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
⊢ le = Slot (le‘ndx)
Theorem	pleslid 13256	Slot property of le. (Contributed by Jim Kingdon, 9-Feb-2023.)
⊢ (le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ)
Theorem	plendxnn 13257	The index value of the order slot is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 30-Oct-2024.)
⊢ (le‘ndx) ∈ ℕ
Theorem	basendxltplendx 13258	The index value of the Base slot is less than the index value of the le slot. (Contributed by AV, 30-Oct-2024.)
⊢ (Base‘ndx) < (le‘ndx)
Theorem	plendxnbasendx 13259	The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.)
⊢ (le‘ndx) ≠ (Base‘ndx)
Theorem	plendxnplusgndx 13260	The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ (le‘ndx) ≠ (+g‘ndx)
Theorem	plendxnmulrndx 13261	The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 1-Nov-2024.)
⊢ (le‘ndx) ≠ (.r‘ndx)
Theorem	plendxnscandx 13262	The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. (Contributed by AV, 1-Nov-2024.)
⊢ (le‘ndx) ≠ (Scalar‘ndx)
Theorem	plendxnvscandx 13263	The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. (Contributed by AV, 1-Nov-2024.)
⊢ (le‘ndx) ≠ ( ·𝑠 ‘ndx)
Theorem	slotsdifplendx 13264	The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
⊢ ((*𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx))
Theorem	ocndx 13265	Index value of the df-ocomp 13152 slot. (Contributed by Mario Carneiro, 25-Oct-2015.) (New usage is discouraged.)
⊢ (oc‘ndx) = ;11
Theorem	ocid 13266	Utility theorem: index-independent form of df-ocomp 13152. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ oc = Slot (oc‘ndx)
Theorem	basendxnocndx 13267	The slot for the orthocomplementation is not the slot for the base set in an extensible structure. (Contributed by AV, 11-Nov-2024.)
⊢ (Base‘ndx) ≠ (oc‘ndx)
Theorem	plendxnocndx 13268	The slot for the orthocomplementation is not the slot for the order in an extensible structure. (Contributed by AV, 11-Nov-2024.)
⊢ (le‘ndx) ≠ (oc‘ndx)
Theorem	dsndx 13269	Index value of the df-ds 13153 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (dist‘ndx) = ;12
Theorem	dsid 13270	Utility theorem: index-independent form of df-ds 13153. (Contributed by Mario Carneiro, 23-Dec-2013.)
⊢ dist = Slot (dist‘ndx)
Theorem	dsslid 13271	Slot property of dist. (Contributed by Jim Kingdon, 6-May-2023.)
⊢ (dist = Slot (dist‘ndx) ∧ (dist‘ndx) ∈ ℕ)
Theorem	dsndxnn 13272	The index of the slot for the distance in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
⊢ (dist‘ndx) ∈ ℕ
Theorem	basendxltdsndx 13273	The index of the slot for the base set is less then the index of the slot for the distance in an extensible structure. (Contributed by AV, 28-Oct-2024.)
⊢ (Base‘ndx) < (dist‘ndx)
Theorem	dsndxnbasendx 13274	The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.)
⊢ (dist‘ndx) ≠ (Base‘ndx)
Theorem	dsndxnplusgndx 13275	The slot for the distance function is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ (dist‘ndx) ≠ (+g‘ndx)
Theorem	dsndxnmulrndx 13276	The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
⊢ (dist‘ndx) ≠ (.r‘ndx)
Theorem	slotsdnscsi 13277	The slots Scalar, ·𝑠 and ·𝑖 are different from the slot dist. (Contributed by AV, 29-Oct-2024.)
⊢ ((dist‘ndx) ≠ (Scalar‘ndx) ∧ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (dist‘ndx) ≠ (·𝑖‘ndx))
Theorem	dsndxntsetndx 13278	The slot for the distance function is not the slot for the topology in an extensible structure. (Contributed by AV, 29-Oct-2024.)
⊢ (dist‘ndx) ≠ (TopSet‘ndx)
Theorem	slotsdifdsndx 13279	The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
⊢ ((*𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx))
Theorem	unifndx 13280	Index value of the df-unif 13154 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.) (New usage is discouraged.)
⊢ (UnifSet‘ndx) = ;13
Theorem	unifid 13281	Utility theorem: index-independent form of df-unif 13154. (Contributed by Thierry Arnoux, 17-Dec-2017.)
⊢ UnifSet = Slot (UnifSet‘ndx)
Theorem	unifndxnn 13282	The index of the slot for the uniform set in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
⊢ (UnifSet‘ndx) ∈ ℕ
Theorem	basendxltunifndx 13283	The index of the slot for the base set is less then the index of the slot for the uniform set in an extensible structure. (Contributed by AV, 28-Oct-2024.)
⊢ (Base‘ndx) < (UnifSet‘ndx)
Theorem	unifndxnbasendx 13284	The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
⊢ (UnifSet‘ndx) ≠ (Base‘ndx)
Theorem	unifndxntsetndx 13285	The slot for the uniform set is not the slot for the topology in an extensible structure. (Contributed by AV, 28-Oct-2024.)
⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
Theorem	slotsdifunifndx 13286	The index of the slot for the uniform set is not the index of other slots. (Contributed by AV, 10-Nov-2024.)
⊢ (((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)))
Theorem	homndx 13287	Index value of the df-hom 13155 slot. (Contributed by Mario Carneiro, 7-Jan-2017.) (New usage is discouraged.)
⊢ (Hom ‘ndx) = ;14
Theorem	homid 13288	Utility theorem: index-independent form of df-hom 13155. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ Hom = Slot (Hom ‘ndx)
Theorem	homslid 13289	Slot property of Hom. (Contributed by Jim Kingdon, 20-Mar-2025.)
⊢ (Hom = Slot (Hom ‘ndx) ∧ (Hom ‘ndx) ∈ ℕ)
Theorem	ccondx 13290	Index value of the df-cco 13156 slot. (Contributed by Mario Carneiro, 7-Jan-2017.) (New usage is discouraged.)
⊢ (comp‘ndx) = ;15
Theorem	ccoid 13291	Utility theorem: index-independent form of df-cco 13156. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ comp = Slot (comp‘ndx)
Theorem	ccoslid 13292	Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
⊢ (comp = Slot (comp‘ndx) ∧ (comp‘ndx) ∈ ℕ)
6.1.3  Definition of the structure product
Syntax	crest 13293	Extend class notation with the function returning a subspace topology.
class ↾t
Syntax	ctopn 13294	Extend class notation with the topology extractor function.
class TopOpen
Definition	df-rest 13295*	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 13296	Define the topology extractor function. This differs from df-tset 13150 when a structure has been restricted using df-iress 13061; 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 13297	The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
⊢ ↾t Fn (V × V)
Theorem	topnfn 13298	The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ TopOpen Fn V
Theorem	restval 13299*	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 13300*	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)))