rlocbas - Mathbox for Thierry Arnoux

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Theorem rlocbas 33207

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 33199	. . 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 2776	. 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 17355	. . . 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 17123	. . . 4 ⊢ Base = Slot (Base‘ndx)
28		snsstp1 4767	. . . . 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 4129	. . . . . 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 4129	. . . . . 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 3945	. . . . 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 3945	. . . 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 6836	. . . . . . 7 ⊢ 𝐵 ∈ V
34	33	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
35	34, 21	ssexd 5263	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
36	34, 35	xpexd 7687	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝑆) ∈ V)
37	11, 36	eqeltrid 2832	. . . 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 17115	. . 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 7383	. . 3 ⊢ ∼ ∈ V
42	41	a1i 11	. 2 ⊢ (𝜑 → ∼ ∈ V)
43		tpex 7682	. . . . 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 7682	. . . . 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 7680	. . . 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 7682	. . . 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 7680	. . 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 17449	1 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3436 ∪ cun 3901 ⊆ wss 3903 ∅c0 4284 {csn 4577 {ctp 4581 ⟨cop 4583 class class class wbr 5092 {copab 5154 × cxp 5617 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 1^st c1st 7922 2^nd c2nd 7923 / cqs 8624 1c1 11010 2c2 12183 cdc 12591 ndxcnx 17104 Basecbs 17120 +_gcplusg 17161 ._rcmulr 17162 Scalarcsca 17164 ·_𝑠 cvsca 17165 ·_𝑖cip 17166 TopSetcts 17167 lecple 17168 distcds 17170 ↾_t crest 17324 0_gc0g 17343 /_s cqus 17409 -_gcsg 18814 ×_t ctx 23445 ~_RL cerl 33193 RLocal crloc 33194

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-ec 8627 df-qs 8631 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-inf 9333 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-fz 13411 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-imas 17412 df-qus 17413 df-rloc 33196

This theorem is referenced by: rloccring 33210 rloc0g 33211 rloc1r 33212 rlocf1 33213 fracbas 33244 fracfld 33247 zringfrac 33491

W3C validator