rlocbas - Mathbox for Thierry Arnoux

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Theorem rlocbas 33528

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 = (0_g‘𝑅)
rlocbas.2	⊢ · = (._r‘𝑅)
rlocbas.3	⊢ − = (-_g‘𝑅)
rlocbas.w	⊢ 𝑊 = (𝐵 × 𝑆)
rlocbas.l	⊢ 𝐿 = (𝑅 RLocal 𝑆)
rlocbas.4	⊢ ∼ = (𝑅 ~_RL 𝑆)
rlocbas.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)
rlocbas.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)

**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.

Step	Hyp	Ref	Expression
1		rlocbas.l	. . 3 ⊢ 𝐿 = (𝑅 RLocal 𝑆)
2		rlocbas.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3		rlocbas.1	. . . 4 ⊢ 0 = (0_g‘𝑅)
4		rlocbas.2	. . . 4 ⊢ · = (._r‘𝑅)
5		rlocbas.3	. . . 4 ⊢ − = (-_g‘𝑅)
6		eqid 2769	. . . 4 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
7		eqid 2769	. . . 4 ⊢ (le‘𝑅) = (le‘𝑅)
8		eqid 2769	. . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
9		eqid 2769	. . . 4 ⊢ (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
10		eqid 2769	. . . 4 ⊢ ( ·_𝑠 ‘𝑅) = ( ·_𝑠 ‘𝑅)
11		rlocbas.w	. . . 4 ⊢ 𝑊 = (𝐵 × 𝑆)
12		rlocbas.4	. . . 4 ⊢ ∼ = (𝑅 ~_RL 𝑆)
13		eqid 2769	. . . 4 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅)
14		eqid 2769	. . . 4 ⊢ (dist‘𝑅) = (dist‘𝑅)
15		eqid 2769	. . . 4 ⊢ (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩) = (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)
16		eqid 2769	. . . 4 ⊢ (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩) = (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)
17		eqid 2769	. . . 4 ⊢ (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩) = (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)
18		eqid 2769	. . . 4 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}
19		eqid 2769	. . . 4 ⊢ (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))) = (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))
20		rlocbas.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
21		rlocbas.s	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
22	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21	rlocval 33519	. . 3 ⊢ (𝜑 → (𝑅 RLocal 𝑆) = ((({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩}) /_s ∼ ))
23	1, 22	eqtrid 2816	. 2 ⊢ (𝜑 → 𝐿 = ((({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩}) /_s ∼ ))
24		eqidd 2770	. . . 4 ⊢ (𝜑 → (({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩}) = (({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩}))
25		eqid 2769	. . . . 5 ⊢ (({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩}) = (({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩})
26	25	imasvalstr 17503	. . . 4 ⊢ (({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩}) Struct ⟨1, ;12⟩
27		baseid 17271	. . . 4 ⊢ Base = Slot (Base‘ndx)
28		snsstp1 4786	. . . . 5 ⊢ {⟨(Base‘ndx), 𝑊⟩} ⊆ {⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩}
29		ssun1 4139	. . . . . 6 ⊢ {⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ⊆ ({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩})
30		ssun1 4139	. . . . . 6 ⊢ ({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ⊆ (({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩})
31	29, 30	sstri 3954	. . . . 5 ⊢ {⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ⊆ (({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩})
32	28, 31	sstri 3954	. . . 4 ⊢ {⟨(Base‘ndx), 𝑊⟩} ⊆ (({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩})
33	2	fvexi 6896	. . . . . . 7 ⊢ 𝐵 ∈ V
34	33	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
35	34, 21	ssexd 5295	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
36	34, 35	xpexd 7749	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝑆) ∈ V)
37	11, 36	eqeltrid 2873	. . . 4 ⊢ (𝜑 → 𝑊 ∈ V)
38		eqid 2769	. . . 4 ⊢ (Base‘(({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩})) = (Base‘(({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩}))
39	24, 26, 27, 32, 37, 38	strfv3 17263	. . 3 ⊢ (𝜑 → (Base‘(({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩})) = 𝑊)
40	39	eqcomd 2775	. 2 ⊢ (𝜑 → 𝑊 = (Base‘(({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩})))
41	12	ovexi 7445	. . 3 ⊢ ∼ ∈ V
42	41	a1i 11	. 2 ⊢ (𝜑 → ∼ ∈ V)
43		tpex 7744	. . . . 5 ⊢ {⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∈ V
44		tpex 7744	. . . . 5 ⊢ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩} ∈ V
45	43, 44	unex 7742	. . . 4 ⊢ ({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∈ V
46		tpex 7744	. . . 4 ⊢ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩} ∈ V
47	45, 46	unex 7742	. . 3 ⊢ (({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩}) ∈ V
48	47	a1i 11	. 2 ⊢ (𝜑 → (({⟨(Base‘ndx), 𝑊⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1^st ‘𝑎) · (2^nd ‘𝑏))(+_g‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1^st ‘𝑎) · (1^st ‘𝑏)), ((2^nd ‘𝑎) · (2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ 𝑊 ↦ ⟨(𝑘( ·_𝑠 ‘𝑅)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×_t ((TopSet‘𝑅) ↾_t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1^st ‘𝑎) · (2^nd ‘𝑏))(le‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1^st ‘𝑎) · (2^nd ‘𝑏))(dist‘𝑅)((1^st ‘𝑏) · (2^nd ‘𝑎))))⟩}) ∈ V)
49	23, 40, 42, 48	qusbas 17598	1 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 Vcvv 3463 ∪ cun 3911 ⊆ wss 3913 ∅c0 4294 {csn 4594 {ctp 4598 ⟨cop 4600 class class class wbr 5113 {copab 5177 × cxp 5660 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 1^st c1st 7983 2^nd c2nd 7984 / cqs 8692 1c1 11100 2c2 12294 cdc 12710 ndxcnx 17252 Basecbs 17268 +_gcplusg 17309 ._rcmulr 17310 Scalarcsca 17312 ·_𝑠 cvsca 17313 ·_𝑖cip 17314 TopSetcts 17315 lecple 17316 distcds 17318 ↾_t crest 17472 0_gc0g 17491 /_s cqus 17558 -_gcsg 19001 ×_t ctx 23685 ~_RL cerl 33513 RLocal crloc 33514

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-ec 8695 df-qs 8699 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9401 df-inf 9402 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-fz 13535 df-struct 17206 df-slot 17241 df-ndx 17253 df-base 17269 df-plusg 17322 df-mulr 17323 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-imas 17561 df-qus 17562 df-rloc 33516

This theorem is referenced by: rloccring 33531 rloc0g 33532 rloc1r 33533 rlocf1 33534 rlocinvunit 33535 rlocisunit 33536 fracbas 33568 fracfld 33571 zringfrac 33788

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