rlocbas - Mathbox for Thierry Arnoux

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Theorem rlocbas 33152

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 2729	. . . 4 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
7		eqid 2729	. . . 4 ⊢ (le‘𝑅) = (le‘𝑅)
8		eqid 2729	. . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
9		eqid 2729	. . . 4 ⊢ (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
10		eqid 2729	. . . 4 ⊢ ( ·_𝑠 ‘𝑅) = ( ·_𝑠 ‘𝑅)
11		rlocbas.w	. . . 4 ⊢ 𝑊 = (𝐵 × 𝑆)
12		rlocbas.4	. . . 4 ⊢ ∼ = (𝑅 ~_RL 𝑆)
13		eqid 2729	. . . 4 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅)
14		eqid 2729	. . . 4 ⊢ (dist‘𝑅) = (dist‘𝑅)
15		eqid 2729	. . . 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 2729	. . . 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 2729	. . . 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 2729	. . . 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 2729	. . . 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 33144	. . 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 2781	. 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 2730	. . . 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 2729	. . . . 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 17487	. . . 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 17237	. . . 4 ⊢ Base = Slot (Base‘ndx)
28		snsstp1 4828	. . . . 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 4183	. . . . . 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 4183	. . . . . 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 4001	. . . . 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 4001	. . . 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 6919	. . . . . . 7 ⊢ 𝐵 ∈ V
34	33	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
35	34, 21	ssexd 5332	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
36	34, 35	xpexd 7767	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝑆) ∈ V)
37	11, 36	eqeltrid 2833	. . . 4 ⊢ (𝜑 → 𝑊 ∈ V)
38		eqid 2729	. . . 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 17228	. . 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 2735	. 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 7464	. . 3 ⊢ ∼ ∈ V
42	41	a1i 11	. 2 ⊢ (𝜑 → ∼ ∈ V)
43		tpex 7762	. . . . 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 7762	. . . . 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 7760	. . . 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 7762	. . . 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 7760	. . 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 17581	1 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2100 Vcvv 3472 ∪ cun 3957 ⊆ wss 3959 ∅c0 4335 {csn 4636 {ctp 4640 ⟨cop 4642 class class class wbr 5156 {copab 5218 × cxp 5684 ‘cfv 6558 (class class class)co 7430 ∈ cmpo 7432 1^st c1st 8009 2^nd c2nd 8010 / cqs 8741 1c1 11166 2c2 12329 cdc 12739 ndxcnx 17216 Basecbs 17234 +_gcplusg 17287 ._rcmulr 17288 Scalarcsca 17290 ·_𝑠 cvsca 17291 ·_𝑖cip 17292 TopSetcts 17293 lecple 17294 distcds 17296 ↾_t crest 17456 0_gc0g 17475 /_s cqus 17541 -_gcsg 18951 ×_t ctx 23578 ~_RL cerl 33138 RLocal crloc 33139

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2102 ax-9 2110 ax-10 2133 ax-11 2150 ax-12 2170 ax-ext 2700 ax-rep 5293 ax-sep 5307 ax-nul 5314 ax-pow 5373 ax-pr 5437 ax-un 7751 ax-cnex 11221 ax-resscn 11222 ax-1cn 11223 ax-icn 11224 ax-addcl 11225 ax-addrcl 11226 ax-mulcl 11227 ax-mulrcl 11228 ax-mulcom 11229 ax-addass 11230 ax-mulass 11231 ax-distr 11232 ax-i2m1 11233 ax-1ne0 11234 ax-1rid 11235 ax-rnegex 11236 ax-rrecex 11237 ax-cnre 11238 ax-pre-lttri 11239 ax-pre-lttrn 11240 ax-pre-ltadd 11241 ax-pre-mulgt0 11242

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2062 df-mo 2532 df-eu 2561 df-clab 2707 df-cleq 2721 df-clel 2806 df-nfc 2881 df-ne 2934 df-nel 3040 df-ral 3055 df-rex 3064 df-reu 3374 df-rab 3429 df-v 3474 df-sbc 3789 df-csb 3905 df-dif 3962 df-un 3964 df-in 3966 df-ss 3976 df-pss 3979 df-nul 4336 df-if 4537 df-pw 4612 df-sn 4637 df-pr 4639 df-tp 4641 df-op 4643 df-uni 4919 df-iun 5008 df-br 5157 df-opab 5219 df-mpt 5240 df-tr 5274 df-id 5584 df-eprel 5590 df-po 5598 df-so 5599 df-fr 5641 df-we 5643 df-xp 5692 df-rel 5693 df-cnv 5694 df-co 5695 df-dm 5696 df-rn 5697 df-res 5698 df-ima 5699 df-pred 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7386 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7885 df-1st 8011 df-2nd 8012 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8742 df-ec 8744 df-qs 8748 df-en 8983 df-dom 8984 df-sdom 8985 df-fin 8986 df-sup 9492 df-inf 9493 df-pnf 11307 df-mnf 11308 df-xr 11309 df-ltxr 11310 df-le 11311 df-sub 11503 df-neg 11504 df-nn 12275 df-2 12337 df-3 12338 df-4 12339 df-5 12340 df-6 12341 df-7 12342 df-8 12343 df-9 12344 df-n0 12535 df-z 12621 df-dec 12740 df-uz 12885 df-fz 13549 df-struct 17170 df-slot 17205 df-ndx 17217 df-base 17235 df-plusg 17300 df-mulr 17301 df-sca 17303 df-vsca 17304 df-ip 17305 df-tset 17306 df-ple 17307 df-ds 17309 df-imas 17544 df-qus 17545 df-rloc 33141

This theorem is referenced by: rloccring 33155 rloc0g 33156 rloc1r 33157 rlocf1 33158 fracbas 33185 fracfld 33188 zringfrac 33460

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