P. 175 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17401-17500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	srngmulr 17401	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), ∗ ⟩})    ⇒   ⊢ ( · ∈ 𝑋 → · = (.r‘𝑅))
Theorem	srnginvl 17402	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), ∗ ⟩})    ⇒   ⊢ ( ∗ ∈ 𝑋 → ∗ = (*𝑟‘𝑅))
Theorem	scandx 17403	Index value of the df-sca 17362 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (Scalar‘ndx) = 5
Theorem	scaid 17404	Utility theorem: index-independent form of scalar df-sca 17362. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ Scalar = Slot (Scalar‘ndx)
Theorem	scandxnbasendx 17405	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 17406	The slot for the scalar field is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpsca 20283. (Contributed by AV, 18-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (+g‘ndx)
Theorem	scandxnmulrndx 17407	The slot for the scalar field is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for mgpsca 20283. (Contributed by AV, 29-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (.r‘ndx)
Theorem	vscandx 17408	Index value of the df-vsca 17363 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ ( ·𝑠 ‘ndx) = 6
Theorem	vscaid 17409	Utility theorem: index-independent form of scalar product df-vsca 17363. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
Theorem	vscandxnbasendx 17410	The slot for the scalar product is not the slot for the base set in an extensible structure. Formerly part of proof for rmodislmod 21118. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Base‘ndx)
Theorem	vscandxnplusgndx 17411	The slot for the scalar product is not the slot for the group operation in an extensible structure. Formerly part of proof for rmodislmod 21118. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (+g‘ndx)
Theorem	vscandxnmulrndx 17412	The slot for the scalar product is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for rmodislmod 21118. (Contributed by AV, 29-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (.r‘ndx)
Theorem	vscandxnscandx 17413	The slot for the scalar product is not the slot for the scalar field in an extensible structure. Formerly part of proof for rmodislmod 21118. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx)
Theorem	lmodstr 17414	A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ 𝑊 Struct ⟨1, 6⟩
Theorem	lmodbase 17415	The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (𝐵 ∈ 𝑋 → 𝐵 = (Base‘𝑊))
Theorem	lmodplusg 17416	The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ ( + ∈ 𝑋 → + = (+g‘𝑊))
Theorem	lmodsca 17417	The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (𝐹 ∈ 𝑋 → 𝐹 = (Scalar‘𝑊))
Theorem	lmodvsca 17418	The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ ( · ∈ 𝑋 → · = ( ·𝑠 ‘𝑊))
Theorem	ipndx 17419	Index value of the df-ip 17364 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (·𝑖‘ndx) = 8
Theorem	ipid 17420	Utility theorem: index-independent form of df-ip 17364. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ ·𝑖 = Slot (·𝑖‘ndx)
Theorem	ipndxnbasendx 17421	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 17422	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 17423	The slot for the inner product is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for mgpsca 20283. (Contributed by AV, 29-Oct-2024.)
⊢ (·𝑖‘ndx) ≠ (.r‘ndx)
Theorem	slotsdifipndx 17424	The slot for the scalar is not the index of other slots. Formerly part of proof for srasca 21368 and sravsca 21369. (Contributed by AV, 12-Nov-2024.)
⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx))
Theorem	ipsstr 17425	Lemma to shorten proofs of ipsbase 17426 through ipsvsca 17430. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ 𝐴 Struct ⟨1, 8⟩
Theorem	ipsbase 17426	The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐴))
Theorem	ipsaddg 17427	The additive operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ ( + ∈ 𝑉 → + = (+g‘𝐴))
Theorem	ipsmulr 17428	The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ ( × ∈ 𝑉 → × = (.r‘𝐴))
Theorem	ipssca 17429	The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ (𝑆 ∈ 𝑉 → 𝑆 = (Scalar‘𝐴))
Theorem	ipsvsca 17430	The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ ( · ∈ 𝑉 → · = ( ·𝑠 ‘𝐴))
Theorem	ipsip 17431	The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐼 = (·𝑖‘𝐴))
Theorem	resssca 17432	Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝐹 = (Scalar‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐹 = (Scalar‘𝐻))
Theorem	ressvsca 17433	·𝑠 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ · = ( ·𝑠 ‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘𝐻))
Theorem	ressip 17434	The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ , = (·𝑖‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → , = (·𝑖‘𝐻))
Theorem	phlstr 17435	A constructed pre-Hilbert space is a structure. Starting from lmodstr 17414 (which has 4 members), we chain strleun 17253 once more, adding an ordered pair to the function, to get all 5 members. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})    ⇒   ⊢ 𝐻 Struct ⟨1, 8⟩
Theorem	phlbase 17436	The base set of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})    ⇒   ⊢ (𝐵 ∈ 𝑋 → 𝐵 = (Base‘𝐻))
Theorem	phlplusg 17437	The additive operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})    ⇒   ⊢ ( + ∈ 𝑋 → + = (+g‘𝐻))
Theorem	phlsca 17438	The ring of scalars of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})    ⇒   ⊢ (𝑇 ∈ 𝑋 → 𝑇 = (Scalar‘𝐻))
Theorem	phlvsca 17439	The scalar product operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})    ⇒   ⊢ ( · ∈ 𝑋 → · = ( ·𝑠 ‘𝐻))
Theorem	phlip 17440	The inner product (Hermitian form) operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})    ⇒   ⊢ ( , ∈ 𝑋 → , = (·𝑖‘𝐻))
Theorem	tsetndx 17441	Index value of the df-tset 17365 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (TopSet‘ndx) = 9
Theorem	tsetid 17442	Utility theorem: index-independent form of df-tset 17365. (Contributed by NM, 20-Oct-2012.)
⊢ TopSet = Slot (TopSet‘ndx)
Theorem	tsetndxnn 17443	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 17444	The index of the slot for the base set is less than the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
⊢ (Base‘ndx) < (TopSet‘ndx)
Theorem	tsetndxnbasendx 17445	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 17446	The slot for the topology is not the slot for the group operation in an extensible structure. Formerly part of proof for oppgtset 19483. (Contributed by AV, 18-Oct-2024.)
⊢ (TopSet‘ndx) ≠ (+g‘ndx)
Theorem	tsetndxnmulrndx 17447	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 17448	The slot for the topology is not the slot for the involution in an extensible structure. Formerly part of proof for cnfldfunALT 21604. (Contributed by AV, 11-Nov-2024.)
⊢ (TopSet‘ndx) ≠ (*𝑟‘ndx)
Theorem	slotstnscsi 17449	The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. Formerly part of sralem 21364 and proofs using it. (Contributed by AV, 29-Oct-2024.)
⊢ ((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx))
Theorem	topgrpstr 17450	A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   ⊢ 𝑊 Struct ⟨1, 9⟩
Theorem	topgrpbas 17451	The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   ⊢ (𝐵 ∈ 𝑋 → 𝐵 = (Base‘𝑊))
Theorem	topgrpplusg 17452	The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   ⊢ ( + ∈ 𝑋 → + = (+g‘𝑊))
Theorem	topgrptset 17453	The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   ⊢ (𝐽 ∈ 𝑋 → 𝐽 = (TopSet‘𝑊))
Theorem	resstset 17454	TopSet is unaffected by restriction. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝐽 = (TopSet‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐽 = (TopSet‘𝐻))
Theorem	plendx 17455	Index value of the df-ple 17366 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) (New usage is discouraged.)
⊢ (le‘ndx) = ;10
Theorem	pleid 17456	Utility theorem: self-referencing, index-independent form of df-ple 17366. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
⊢ le = Slot (le‘ndx)
Theorem	plendxnn 17457	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 17458	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 17459	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 17460	The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. Formerly part of proof for oppgle 19498. (Contributed by AV, 18-Oct-2024.)
⊢ (le‘ndx) ≠ (+g‘ndx)
Theorem	plendxnmulrndx 17461	The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. Formerly part of proof for opsrmulr 22272. (Contributed by AV, 1-Nov-2024.)
⊢ (le‘ndx) ≠ (.r‘ndx)
Theorem	plendxnscandx 17462	The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. Formerly part of proof for opsrsca 22274. (Contributed by AV, 1-Nov-2024.)
⊢ (le‘ndx) ≠ (Scalar‘ndx)
Theorem	plendxnvscandx 17463	The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. Formerly part of proof for opsrvsca 22273. (Contributed by AV, 1-Nov-2024.)
⊢ (le‘ndx) ≠ ( ·𝑠 ‘ndx)
Theorem	slotsdifplendx 17464	The index of the slot for the distance is not the index of other slots. Formerly part of proof for cnfldfunALT 21604. (Contributed by AV, 11-Nov-2024.)
⊢ ((*𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx))
Theorem	otpsstr 17465	Functionality of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}    ⇒   ⊢ 𝐾 Struct ⟨1, ;10⟩
Theorem	otpsbas 17466	The base set of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐾))
Theorem	otpstset 17467	The open sets of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}    ⇒   ⊢ (𝐽 ∈ 𝑉 → 𝐽 = (TopSet‘𝐾))
Theorem	otpsle 17468	The order of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}    ⇒   ⊢ ( ≤ ∈ 𝑉 → ≤ = (le‘𝐾))
Theorem	ressle 17469	le is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
⊢ 𝑊 = (𝐾 ↾s 𝐴)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ≤ = (le‘𝑊))
Theorem	ocndx 17470	Index value of the df-ocomp 17367 slot. (Contributed by Mario Carneiro, 25-Oct-2015.) (New usage is discouraged.)
⊢ (oc‘ndx) = ;11
Theorem	ocid 17471	Utility theorem: index-independent form of df-ocomp 17367. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ oc = Slot (oc‘ndx)
Theorem	basendxnocndx 17472	The slot for the orthocomplementation is not the slot for the base set in an extensible structure. Formerly part of proof for thlbas 21913. (Contributed by AV, 11-Nov-2024.)
⊢ (Base‘ndx) ≠ (oc‘ndx)
Theorem	plendxnocndx 17473	The slot for the orthocomplementation is not the slot for the order in an extensible structure. Formerly part of proof for thlle 21914. (Contributed by AV, 11-Nov-2024.)
⊢ (le‘ndx) ≠ (oc‘ndx)
Theorem	dsndx 17474	Index value of the df-ds 17368 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (dist‘ndx) = ;12
Theorem	dsid 17475	Utility theorem: index-independent form of df-ds 17368. (Contributed by Mario Carneiro, 23-Dec-2013.)
⊢ dist = Slot (dist‘ndx)
Theorem	dsndxnn 17476	The index of the slot for the distance in an extensible structure is a positive integer. Formerly part of proof for tmslem 24712. (Contributed by AV, 28-Oct-2024.)
⊢ (dist‘ndx) ∈ ℕ
Theorem	basendxltdsndx 17477	The index of the slot for the base set is less than the index of the slot for the distance in an extensible structure. Formerly part of proof for tmslem 24712. (Contributed by AV, 28-Oct-2024.)
⊢ (Base‘ndx) < (dist‘ndx)
Theorem	dsndxnbasendx 17478	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 17479	The slot for the distance function is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpds 20286. (Contributed by AV, 18-Oct-2024.)
⊢ (dist‘ndx) ≠ (+g‘ndx)
Theorem	dsndxnmulrndx 17480	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 17481	The slots Scalar, ·𝑠 and ·𝑖 are different from the slot dist. Formerly part of sralem 21364 and proofs using it. (Contributed by AV, 29-Oct-2024.)
⊢ ((dist‘ndx) ≠ (Scalar‘ndx) ∧ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (dist‘ndx) ≠ (·𝑖‘ndx))
Theorem	dsndxntsetndx 17482	The slot for the distance function is not the slot for the topology in an extensible structure. Formerly part of proof for tngds 24878. (Contributed by AV, 29-Oct-2024.)
⊢ (dist‘ndx) ≠ (TopSet‘ndx)
Theorem	slotsdifdsndx 17483	The index of the slot for the distance is not the index of other slots. Formerly part of proof for cnfldfunALT 21604. (Contributed by AV, 11-Nov-2024.)
⊢ ((*𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx))
Theorem	unifndx 17484	Index value of the df-unif 17369 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.) (New usage is discouraged.)
⊢ (UnifSet‘ndx) = ;13
Theorem	unifid 17485	Utility theorem: index-independent form of df-unif 17369. (Contributed by Thierry Arnoux, 17-Dec-2017.)
⊢ UnifSet = Slot (UnifSet‘ndx)
Theorem	unifndxnn 17486	The index of the slot for the uniform set in an extensible structure is a positive integer. Formerly part of proof for tuslem 24496. (Contributed by AV, 28-Oct-2024.)
⊢ (UnifSet‘ndx) ∈ ℕ
Theorem	basendxltunifndx 17487	The index of the slot for the base set is less than the index of the slot for the uniform set in an extensible structure. Formerly part of proof for tuslem 24496. (Contributed by AV, 28-Oct-2024.)
⊢ (Base‘ndx) < (UnifSet‘ndx)
Theorem	unifndxnbasendx 17488	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 17489	The slot for the uniform set is not the slot for the topology in an extensible structure. Formerly part of proof for tuslem 24496. (Contributed by AV, 28-Oct-2024.)
⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
Theorem	slotsdifunifndx 17490	The index of the slot for the uniform set is not the index of other slots. Formerly part of proof for cnfldfunALT 21604. (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	ressunif 17491	UnifSet is unaffected by restriction. (Contributed by Thierry Arnoux, 7-Dec-2017.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝑈 = (UnifSet‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝑈 = (UnifSet‘𝐻))
Theorem	odrngstr 17492	Functionality of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.) (Proof shortened by AV, 15-Sep-2021.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ 𝑊 Struct ⟨1, ;12⟩
Theorem	odrngbas 17493	The base set of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝑊))
Theorem	odrngplusg 17494	The addition operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ ( + ∈ 𝑉 → + = (+g‘𝑊))
Theorem	odrngmulr 17495	The multiplication operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ ( · ∈ 𝑉 → · = (.r‘𝑊))
Theorem	odrngtset 17496	The open sets of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ (𝐽 ∈ 𝑉 → 𝐽 = (TopSet‘𝑊))
Theorem	odrngle 17497	The order of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ ( ≤ ∈ 𝑉 → ≤ = (le‘𝑊))
Theorem	odrngds 17498	The metric of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ (𝐷 ∈ 𝑉 → 𝐷 = (dist‘𝑊))
Theorem	ressds 17499	dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝐷 = (dist‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐷 = (dist‘𝐻))
Theorem	homndx 17500	Index value of the df-hom 17370 slot. (Contributed by Mario Carneiro, 7-Jan-2017.) (New usage is discouraged.)
⊢ (Hom ‘ndx) = ;14