rlocbas - Mathbox for Thierry Arnoux

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Theorem rlocbas 33253

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 2734	. . . 4 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
7		eqid 2734	. . . 4 ⊢ (le‘𝑅) = (le‘𝑅)
8		eqid 2734	. . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
9		eqid 2734	. . . 4 ⊢ (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
10		eqid 2734	. . . 4 ⊢ ( ·_𝑠 ‘𝑅) = ( ·_𝑠 ‘𝑅)
11		rlocbas.w	. . . 4 ⊢ 𝑊 = (𝐵 × 𝑆)
12		rlocbas.4	. . . 4 ⊢ ∼ = (𝑅 ~_RL 𝑆)
13		eqid 2734	. . . 4 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅)
14		eqid 2734	. . . 4 ⊢ (dist‘𝑅) = (dist‘𝑅)
15		eqid 2734	. . . 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 2734	. . . 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 2734	. . . 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 2734	. . . 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 2734	. . . 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 33245	. . 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 2786	. 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 2735	. . . 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 2734	. . . . 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 17497	. . . 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 17247	. . . 4 ⊢ Base = Slot (Base‘ndx)
28		snsstp1 4820	. . . . 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 4187	. . . . . 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 4187	. . . . . 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 4004	. . . . 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 4004	. . . 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 6920	. . . . . . 7 ⊢ 𝐵 ∈ V
34	33	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
35	34, 21	ssexd 5329	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
36	34, 35	xpexd 7769	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝑆) ∈ V)
37	11, 36	eqeltrid 2842	. . . 4 ⊢ (𝜑 → 𝑊 ∈ V)
38		eqid 2734	. . . 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 17238	. . 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 2740	. 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 7764	. . . . 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 7764	. . . . 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 7762	. . . 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 7764	. . . 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 7762	. . 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 17591	1 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∪ cun 3960 ⊆ wss 3962 ∅c0 4338 {csn 4630 {ctp 4634 ⟨cop 4636 class class class wbr 5147 {copab 5209 × cxp 5686 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 1^st c1st 8010 2^nd c2nd 8011 / cqs 8742 1c1 11153 2c2 12318 cdc 12730 ndxcnx 17226 Basecbs 17244 +_gcplusg 17297 ._rcmulr 17298 Scalarcsca 17300 ·_𝑠 cvsca 17301 ·_𝑖cip 17302 TopSetcts 17303 lecple 17304 distcds 17306 ↾_t crest 17466 0_gc0g 17485 /_s cqus 17551 -_gcsg 18965 ×_t ctx 23583 ~_RL cerl 33239 RLocal crloc 33240

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-ec 8745 df-qs 8749 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 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 17554 df-qus 17555 df-rloc 33242

This theorem is referenced by: rloccring 33256 rloc0g 33257 rloc1r 33258 rlocf1 33259 fracbas 33286 fracfld 33289 zringfrac 33561

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