Theorem imasvsca 17483

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 2729	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2729	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2729	. . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
6		imasvsca.k	. . . 4 ⊢ 𝐾 = (Base‘𝐺)
7		imassca.g	. . . . 5 ⊢ 𝐺 = (Scalar‘𝑅)
8	7	fveq2i 6861	. . . 4 ⊢ (Base‘𝐺) = (Base‘(Scalar‘𝑅))
9	6, 8	eqtri 2752	. . 3 ⊢ 𝐾 = (Base‘(Scalar‘𝑅))
10		imasvsca.q	. . 3 ⊢ · = ( ·𝑠 ‘𝑅)
11		eqid 2729	. . 3 ⊢ (·𝑖‘𝑅) = (·𝑖‘𝑅)
12		eqid 2729	. . 3 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
13		eqid 2729	. . 3 ⊢ (dist‘𝑅) = (dist‘𝑅)
14		eqid 2729	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
15		imasbas.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
16		imasbas.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
17		eqid 2729	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
18	1, 2, 15, 16, 3, 17	imasplusg 17480	. . 3 ⊢ (𝜑 → (+g‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
19		eqid 2729	. . . 4 ⊢ (.r‘𝑈) = (.r‘𝑈)
20	1, 2, 15, 16, 4, 19	imasmulr 17481	. . 3 ⊢ (𝜑 → (.r‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩})
21		eqidd 2730	. . 3 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))
22		eqidd 2730	. . 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 2730	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((TopOpen‘𝑅) qTop 𝐹))
24		eqid 2729	. . . 4 ⊢ (dist‘𝑈) = (dist‘𝑈)
25	1, 2, 15, 16, 13, 24	imasds 17476	. . 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 2730	. . 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 17474	. 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 2729	. . 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 17414	. 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 17283	. 2 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
31		snsstp2 4781	. . 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 4142	. . . 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 4141	. . . 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 3956	. . 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 3956	. 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 6871	. . . 4 ⊢ (Base‘𝑅) ∈ V
37	2, 36	eqeltrdi 2836	. . 3 ⊢ (𝜑 → 𝑉 ∈ V)
38	6	fvexi 6872	. . . . 5 ⊢ 𝐾 ∈ V
39		snex 5391	. . . . 5 ⊢ {(𝐹‘𝑞)} ∈ V
40	38, 39	mpoex 8058	. . . 4 ⊢ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V
41	40	rgenw 3048	. . 3 ⊢ ∀𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V
42		iunexg 7942	. . 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 17174	1 ⊢ (𝜑 → ∙ = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ∪ cun 3912  {csn 4589  {ctp 4593  ⟨cop 4595  ∪ ciun 4955  ^◡ccnv 5637   ∘ ccom 5642  –onto→wfo 6509  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1c1 11069  2c2 12241  ;cdc 12649  ndxcnx 17163  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  Scalarcsca 17223   ·_𝑠 cvsca 17224  ·_𝑖cip 17225  TopSetcts 17226  lecple 17227  distcds 17229  TopOpenctopn 17384   qTop cqtop 17466   “_s cimas 17467

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-imas 17471

This theorem is referenced by:  imasip  17484  imastset  17485  imasle  17486  imasvscafn  17500  imasvscaval  17501  imasvscaf  17502