Theorem imassca 17440

Description: The scalar field of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)

Hypotheses

Ref	Expression
imasbas.u	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasbas.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasbas.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasbas.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imassca.g	⊢ 𝐺 = (Scalar‘𝑅)

Assertion

Ref	Expression
imassca	⊢ (𝜑 → 𝐺 = (Scalar‘𝑈))

Proof of Theorem imassca

Dummy variables 𝑔 ℎ 𝑖 𝑛 𝑝 𝑞 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassca.g	. . . 4 ⊢ 𝐺 = (Scalar‘𝑅)
2	1	fvexi 6848	. . 3 ⊢ 𝐺 ∈ V
3		eqid 2736	. . . . 5 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
4	3	imasvalstr 17371	. . . 4 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) Struct ⟨1, ;12⟩
5		scaid 17235	. . . 4 ⊢ Scalar = Slot (Scalar‘ndx)
6		snsstp1 4772	. . . . . 6 ⊢ {⟨(Scalar‘ndx), 𝐺⟩} ⊆ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}
7		ssun2 4131	. . . . . 6 ⊢ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩})
8	6, 7	sstri 3943	. . . . 5 ⊢ {⟨(Scalar‘ndx), 𝐺⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩})
9		ssun1 4130	. . . . 5 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
10	8, 9	sstri 3943	. . . 4 ⊢ {⟨(Scalar‘ndx), 𝐺⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
11	4, 5, 10	strfv 17130	. . 3 ⊢ (𝐺 ∈ V → 𝐺 = (Scalar‘(({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})))
12	2, 11	ax-mp 5	. 2 ⊢ 𝐺 = (Scalar‘(({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}))
13		imasbas.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
14		imasbas.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
15		eqid 2736	. . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅)
16		eqid 2736	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
17		eqid 2736	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
18		eqid 2736	. . . 4 ⊢ ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
19		eqid 2736	. . . 4 ⊢ (·𝑖‘𝑅) = (·𝑖‘𝑅)
20		eqid 2736	. . . 4 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
21		eqid 2736	. . . 4 ⊢ (dist‘𝑅) = (dist‘𝑅)
22		eqid 2736	. . . 4 ⊢ (le‘𝑅) = (le‘𝑅)
23		imasbas.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
24		imasbas.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
25		eqid 2736	. . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈)
26	13, 14, 23, 24, 15, 25	imasplusg 17438	. . . 4 ⊢ (𝜑 → (+g‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
27		eqid 2736	. . . . 5 ⊢ (.r‘𝑈) = (.r‘𝑈)
28	13, 14, 23, 24, 16, 27	imasmulr 17439	. . . 4 ⊢ (𝜑 → (.r‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩})
29		eqidd 2737	. . . 4 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞))) = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞))))
30		eqidd 2737	. . . 4 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩})
31		eqidd 2737	. . . 4 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((TopOpen‘𝑅) qTop 𝐹))
32		eqid 2736	. . . . 5 ⊢ (dist‘𝑈) = (dist‘𝑈)
33	13, 14, 23, 24, 21, 32	imasds 17434	. . . 4 ⊢ (𝜑 → (dist‘𝑈) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑔))), ℝ*, < )))
34		eqidd 2737	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) = ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹))
35	13, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 26, 28, 29, 30, 31, 33, 34, 23, 24	imasval 17432	. . 3 ⊢ (𝜑 → 𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}))
36	35	fveq2d 6838	. 2 ⊢ (𝜑 → (Scalar‘𝑈) = (Scalar‘(({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})))
37	12, 36	eqtr4id 2790	1 ⊢ (𝜑 → 𝐺 = (Scalar‘𝑈))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∪ cun 3899  {csn 4580  {ctp 4584  ⟨cop 4586  ∪ ciun 4946  ^◡ccnv 5623   ∘ ccom 5628  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1c1 11027  2c2 12200  ;cdc 12607  ndxcnx 17120  Basecbs 17136  +_gcplusg 17177  ._rcmulr 17178  Scalarcsca 17180   ·_𝑠 cvsca 17181  ·_𝑖cip 17182  TopSetcts 17183  lecple 17184  distcds 17186  TopOpenctopn 17341   qTop cqtop 17424   “_s cimas 17425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-slot 17109  df-ndx 17121  df-base 17137  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-imas 17429

This theorem is referenced by:  quss  17467  xpssca  17497  imaslmod  33434  imaslmhm  33438  algextdeglem8  33881