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Theorem rlocbas 33271
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 2737 . . . 4 (+g𝑅) = (+g𝑅)
7 eqid 2737 . . . 4 (le‘𝑅) = (le‘𝑅)
8 eqid 2737 . . . 4 (Scalar‘𝑅) = (Scalar‘𝑅)
9 eqid 2737 . . . 4 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
10 eqid 2737 . . . 4 ( ·𝑠𝑅) = ( ·𝑠𝑅)
11 rlocbas.w . . . 4 𝑊 = (𝐵 × 𝑆)
12 rlocbas.4 . . . 4 = (𝑅 ~RL 𝑆)
13 eqid 2737 . . . 4 (TopSet‘𝑅) = (TopSet‘𝑅)
14 eqid 2737 . . . 4 (dist‘𝑅) = (dist‘𝑅)
15 eqid 2737 . . . 4 (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩) = (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏))(+g𝑅)((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)
16 eqid 2737 . . . 4 (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩) = (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)
17 eqid 2737 . . . 4 (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩) = (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)
18 eqid 2737 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏))(le‘𝑅)((1st𝑏) · (2nd𝑎)))}
19 eqid 2737 . . . 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 33263 . . 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 2789 . 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 2738 . . . 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 2737 . . . . 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 17496 . . . 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 17250 . . . 4 Base = Slot (Base‘ndx)
28 snsstp1 4816 . . . . 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 4178 . . . . . 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 4178 . . . . . 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 3993 . . . . 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 3993 . . . 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 6920 . . . . . . 7 𝐵 ∈ V
3433a1i 11 . . . . . 6 (𝜑𝐵 ∈ V)
3534, 21ssexd 5324 . . . . . 6 (𝜑𝑆 ∈ V)
3634, 35xpexd 7771 . . . . 5 (𝜑 → (𝐵 × 𝑆) ∈ V)
3711, 36eqeltrid 2845 . . . 4 (𝜑𝑊 ∈ V)
38 eqid 2737 . . . 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 17241 . . 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 2743 . 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 7465 . . 3 ∈ V
4241a1i 11 . 2 (𝜑 ∈ V)
43 tpex 7766 . . . . 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 7766 . . . . 5 {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎𝑊 ↦ ⟨(𝑘( ·𝑠𝑅)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩} ∈ V
4543, 44unex 7764 . . . 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 7766 . . . 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 7764 . . 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 17590 1 (𝜑 → (𝑊 / ) = (Base‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  wss 3951  c0 4333  {csn 4626  {ctp 4630  cop 4632   class class class wbr 5143  {copab 5205   × cxp 5683  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013   / cqs 8744  1c1 11156  2c2 12321  cdc 12733  ndxcnx 17230  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  ·𝑖cip 17302  TopSetcts 17303  lecple 17304  distcds 17306  t crest 17465  0gc0g 17484   /s cqus 17550  -gcsg 18953   ×t ctx 23568   ~RL cerl 33257   RLocal crloc 33258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-ec 8747  df-qs 8751  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-imas 17553  df-qus 17554  df-rloc 33260
This theorem is referenced by:  rloccring  33274  rloc0g  33275  rloc1r  33276  rlocf1  33277  fracbas  33307  fracfld  33310  zringfrac  33582
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