rlocbas - Mathbox for Thierry Arnoux

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Theorem rlocbas 33361

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 2737	. . . 4 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
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 ⊢ ( ·_𝑠 ‘𝑅) = ( ·_𝑠 ‘𝑅)
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 ⊢ (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 2737	. . . 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 2737	. . . 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 2737	. . . 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 2737	. . . 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 33353	. . 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 2784	. 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 2738	. . . 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 2737	. . . . 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 17383	. . . 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 17151	. . . 4 ⊢ Base = Slot (Base‘ndx)
28		snsstp1 4774	. . . . 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 4132	. . . . . 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 4132	. . . . . 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 6856	. . . . . . 7 ⊢ 𝐵 ∈ V
34	33	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
35	34, 21	ssexd 5271	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
36	34, 35	xpexd 7706	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝑆) ∈ V)
37	11, 36	eqeltrid 2841	. . . 4 ⊢ (𝜑 → 𝑊 ∈ V)
38		eqid 2737	. . . 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 17143	. . 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 2743	. 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 7402	. . 3 ⊢ ∼ ∈ V
42	41	a1i 11	. 2 ⊢ (𝜑 → ∼ ∈ V)
43		tpex 7701	. . . . 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 7701	. . . . 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 7699	. . . 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 7701	. . . 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 7699	. . 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 17478	1 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3442 ∪ cun 3901 ⊆ wss 3903 ∅c0 4287 {csn 4582 {ctp 4586 ⟨cop 4588 class class class wbr 5100 {copab 5162 × cxp 5630 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 1^st c1st 7941 2^nd c2nd 7942 / cqs 8644 1c1 11039 2c2 12212 cdc 12619 ndxcnx 17132 Basecbs 17148 +_gcplusg 17189 ._rcmulr 17190 Scalarcsca 17192 ·_𝑠 cvsca 17193 ·_𝑖cip 17194 TopSetcts 17195 lecple 17196 distcds 17198 ↾_t crest 17352 0_gc0g 17371 /_s cqus 17438 -_gcsg 18877 ×_t ctx 23516 ~_RL cerl 33347 RLocal crloc 33348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-ec 8647 df-qs 8651 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-imas 17441 df-qus 17442 df-rloc 33350

This theorem is referenced by: rloccring 33364 rloc0g 33365 rloc1r 33366 rlocf1 33367 fracbas 33399 fracfld 33402 zringfrac 33647

W3C validator