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Theorem rlocbas 33239
Description: The base set of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
Hypotheses
Ref Expression
rlocbas.b 𝐵 = (Base‘𝑅)
rlocbas.1 0 = (0g𝑅)
rlocbas.2 · = (.r𝑅)
rlocbas.3 = (-g𝑅)
rlocbas.w 𝑊 = (𝐵 × 𝑆)
rlocbas.l 𝐿 = (𝑅 RLocal 𝑆)
rlocbas.4 = (𝑅 ~RL 𝑆)
rlocbas.r (𝜑𝑅𝑉)
rlocbas.s (𝜑𝑆𝐵)
Assertion
Ref Expression
rlocbas (𝜑 → (𝑊 / ) = (Base‘𝐿))

Proof of Theorem rlocbas
Dummy variables 𝑎 𝑏 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlocbas.l . . 3 𝐿 = (𝑅 RLocal 𝑆)
2 rlocbas.b . . . 4 𝐵 = (Base‘𝑅)
3 rlocbas.1 . . . 4 0 = (0g𝑅)
4 rlocbas.2 . . . 4 · = (.r𝑅)
5 rlocbas.3 . . . 4 = (-g𝑅)
6 eqid 2740 . . . 4 (+g𝑅) = (+g𝑅)
7 eqid 2740 . . . 4 (le‘𝑅) = (le‘𝑅)
8 eqid 2740 . . . 4 (Scalar‘𝑅) = (Scalar‘𝑅)
9 eqid 2740 . . . 4 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
10 eqid 2740 . . . 4 ( ·𝑠𝑅) = ( ·𝑠𝑅)
11 rlocbas.w . . . 4 𝑊 = (𝐵 × 𝑆)
12 rlocbas.4 . . . 4 = (𝑅 ~RL 𝑆)
13 eqid 2740 . . . 4 (TopSet‘𝑅) = (TopSet‘𝑅)
14 eqid 2740 . . . 4 (dist‘𝑅) = (dist‘𝑅)
15 eqid 2740 . . . 4 (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩) = (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)
16 eqid 2740 . . . 4 (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩) = (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)
17 eqid 2740 . . . 4 (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩) = (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)
18 eqid 2740 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}
19 eqid 2740 . . . 4 (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎)))) = (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))
20 rlocbas.r . . . 4 (𝜑𝑅𝑉)
21 rlocbas.s . . . 4 (𝜑𝑆𝐵)
222, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21rlocval 33231 . . 3 (𝜑 → (𝑅 RLocal 𝑆) = ((({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩}) /s ))
231, 22eqtrid 2792 . 2 (𝜑𝐿 = ((({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩}) /s ))
24 eqidd 2741 . . . 4 (𝜑 → (({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩}) = (({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩}))
25 eqid 2740 . . . . 5 (({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩}) = (({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩})
2625imasvalstr 17511 . . . 4 (({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩}) Struct ⟨1, 12⟩
27 baseid 17261 . . . 4 Base = Slot (Base‘ndx)
28 snsstp1 4841 . . . . 5 {⟨(Base‘ndx), 𝑊⟩} ⊆ {⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩}
29 ssun1 4201 . . . . . 6 {⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ⊆ ({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩})
30 ssun1 4201 . . . . . 6 ({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ⊆ (({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩})
3129, 30sstri 4018 . . . . 5 {⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ⊆ (({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩})
3228, 31sstri 4018 . . . 4 {⟨(Base‘ndx), 𝑊⟩} ⊆ (({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩})
332fvexi 6934 . . . . . . 7 𝐵 ∈ V
3433a1i 11 . . . . . 6 (𝜑𝐵 ∈ V)
3534, 21ssexd 5342 . . . . . 6 (𝜑𝑆 ∈ V)
3634, 35xpexd 7786 . . . . 5 (𝜑 → (𝐵 × 𝑆) ∈ V)
3711, 36eqeltrid 2848 . . . 4 (𝜑𝑊 ∈ V)
38 eqid 2740 . . . 4 (Base‘(({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩})) = (Base‘(({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩}))
3924, 26, 27, 32, 37, 38strfv3 17252 . . 3 (𝜑 → (Base‘(({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩})) = 𝑊)
4039eqcomd 2746 . 2 (𝜑𝑊 = (Base‘(({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩})))
4112ovexi 7482 . . 3 ∈ V
4241a1i 11 . 2 (𝜑 ∈ V)
43 tpex 7781 . . . . 5 {⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∈ V
44 tpex 7781 . . . . 5 {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩} ∈ V
4543, 44unex 7779 . . . 4 ({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∈ V
46 tpex 7781 . . . 4 {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩} ∈ V
4745, 46unex 7779 . . 3 (({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩}) ∈ V
4847a1i 11 . 2 (𝜑 → (({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))(dist‘𝑅)((1st𝑏) · (2nd𝑎))))⟩}) ∈ V)
4923, 40, 42, 48qusbas 17605 1 (𝜑 → (𝑊 / ) = (Base‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  Vcvv 3488  cun 3974  wss 3976  c0 4352  {csn 4648  {ctp 4652  cop 4654   class class class wbr 5166  {copab 5228   × cxp 5698  cfv 6573  (class class class)co 7448  cmpo 7450  1st c1st 8028  2nd c2nd 8029   / cqs 8762  1c1 11185  2c2 12348  cdc 12758  ndxcnx 17240  Basecbs 17258  +gcplusg 17311  .rcmulr 17312  Scalarcsca 17314   ·𝑠 cvsca 17315  ·𝑖cip 17316  TopSetcts 17317  lecple 17318  distcds 17320  t crest 17480  0gc0g 17499   /s cqus 17565  -gcsg 18975   ×t ctx 23589   ~RL cerl 33225   RLocal crloc 33226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-ec 8765  df-qs 8769  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-imas 17568  df-qus 17569  df-rloc 33228
This theorem is referenced by:  rloccring  33242  rloc0g  33243  rloc1r  33244  rlocf1  33245  fracbas  33272  fracfld  33275  zringfrac  33547
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