Theorem imasvsca 17490

Description: The scalar multiplication operation 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)
imasvsca.k	⊢ 𝐾 = (Base‘𝐺)
imasvsca.q	⊢ · = ( ·𝑠 ‘𝑅)
imasvsca.s	⊢ ∙ = ( ·𝑠 ‘𝑈)

Assertion

Ref	Expression
imasvsca	⊢ (𝜑 → ∙ = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))

Distinct variable groups:   𝑞,𝑝,𝑥,𝐹   𝑅,𝑝,𝑞,𝑥   𝑥,𝑈   𝑥,𝐵   𝜑,𝑝,𝑞,𝑥   𝐾,𝑝,𝑥   𝑉,𝑝,𝑞

Allowed substitution hints:   𝐵(𝑞,𝑝)   ∙ (𝑥,𝑞,𝑝)   · (𝑥,𝑞,𝑝)   𝑈(𝑞,𝑝)   𝐺(𝑥,𝑞,𝑝)   𝐾(𝑞)   𝑉(𝑥)   𝑍(𝑥,𝑞,𝑝)

Proof of Theorem imasvsca

Dummy variables 𝑢 𝑣 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasbas.u	. . 3 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
2		imasbas.v	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		eqid 2730	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2730	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2730	. . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
6		imasvsca.k	. . . 4 ⊢ 𝐾 = (Base‘𝐺)
7		imassca.g	. . . . 5 ⊢ 𝐺 = (Scalar‘𝑅)
8	7	fveq2i 6864	. . . 4 ⊢ (Base‘𝐺) = (Base‘(Scalar‘𝑅))
9	6, 8	eqtri 2753	. . 3 ⊢ 𝐾 = (Base‘(Scalar‘𝑅))
10		imasvsca.q	. . 3 ⊢ · = ( ·𝑠 ‘𝑅)
11		eqid 2730	. . 3 ⊢ (·𝑖‘𝑅) = (·𝑖‘𝑅)
12		eqid 2730	. . 3 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
13		eqid 2730	. . 3 ⊢ (dist‘𝑅) = (dist‘𝑅)
14		eqid 2730	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
15		imasbas.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
16		imasbas.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
17		eqid 2730	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
18	1, 2, 15, 16, 3, 17	imasplusg 17487	. . 3 ⊢ (𝜑 → (+g‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
19		eqid 2730	. . . 4 ⊢ (.r‘𝑈) = (.r‘𝑈)
20	1, 2, 15, 16, 4, 19	imasmulr 17488	. . 3 ⊢ (𝜑 → (.r‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩})
21		eqidd 2731	. . 3 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))
22		eqidd 2731	. . 3 ⊢ (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)⟩})
23		eqidd 2731	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((TopOpen‘𝑅) qTop 𝐹))
24		eqid 2730	. . . 4 ⊢ (dist‘𝑈) = (dist‘𝑈)
25	1, 2, 15, 16, 13, 24	imasds 17483	. . 3 ⊢ (𝜑 → (dist‘𝑈) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑢 ∈ ℕ ran (𝑧 ∈ {𝑤 ∈ ((𝑉 × 𝑉) ↑m (1...𝑢)) ∣ ((𝐹‘(1st ‘(𝑤‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑤‘𝑢))) = 𝑦 ∧ ∀𝑣 ∈ (1...(𝑢 − 1))(𝐹‘(2nd ‘(𝑤‘𝑣))) = (𝐹‘(1st ‘(𝑤‘(𝑣 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑧))), ℝ*, < )))
26		eqidd 2731	. . 3 ⊢ (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) = ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹))
27	1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 14, 18, 20, 21, 22, 23, 25, 26, 15, 16	imasval 17481	. 2 ⊢ (𝜑 → 𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 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)⟩}))
28		eqid 2730	. . 3 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 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), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 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)⟩})
29	28	imasvalstr 17421	. 2 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 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⟩
30		vscaid 17290	. 2 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
31		snsstp2 4784	. . 3 ⊢ {⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩} ⊆ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}
32		ssun2 4145	. . . 4 ⊢ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 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), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩})
33		ssun1 4144	. . . 4 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 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), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 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)⟩})
34	32, 33	sstri 3959	. . 3 ⊢ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 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), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 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)⟩})
35	31, 34	sstri 3959	. 2 ⊢ {⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 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		fvex 6874	. . . 4 ⊢ (Base‘𝑅) ∈ V
37	2, 36	eqeltrdi 2837	. . 3 ⊢ (𝜑 → 𝑉 ∈ V)
38	6	fvexi 6875	. . . . 5 ⊢ 𝐾 ∈ V
39		snex 5394	. . . . 5 ⊢ {(𝐹‘𝑞)} ∈ V
40	38, 39	mpoex 8061	. . . 4 ⊢ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V
41	40	rgenw 3049	. . 3 ⊢ ∀𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V
42		iunexg 7945	. . 3 ⊢ ((𝑉 ∈ V ∧ ∀𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V) → ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V)
43	37, 41, 42	sylancl 586	. 2 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V)
44		imasvsca.s	. 2 ⊢ ∙ = ( ·𝑠 ‘𝑈)
45	27, 29, 30, 35, 43, 44	strfv3 17181	1 ⊢ (𝜑 → ∙ = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ∪ cun 3915  {csn 4592  {ctp 4596  ⟨cop 4598  ∪ ciun 4958  ^◡ccnv 5640   ∘ ccom 5645  –onto→wfo 6512  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  1c1 11076  2c2 12248  ;cdc 12656  ndxcnx 17170  Basecbs 17186  +_gcplusg 17227  ._rcmulr 17228  Scalarcsca 17230   ·_𝑠 cvsca 17231  ·_𝑖cip 17232  TopSetcts 17233  lecple 17234  distcds 17236  TopOpenctopn 17391   qTop cqtop 17473   “_s cimas 17474

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-slot 17159  df-ndx 17171  df-base 17187  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-imas 17478

This theorem is referenced by:  imasip  17491  imastset  17492  imasle  17493  imasvscafn  17507  imasvscaval  17508  imasvscaf  17509