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Theorem imasvsca 17462

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 2724	. . 3 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
4		eqid 2724	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
5		eqid 2724	. . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
6		imasvsca.k	. . . 4 ⊢ 𝐾 = (Base‘𝐺)
7		imassca.g	. . . . 5 ⊢ 𝐺 = (Scalar‘𝑅)
8	7	fveq2i 6884	. . . 4 ⊢ (Base‘𝐺) = (Base‘(Scalar‘𝑅))
9	6, 8	eqtri 2752	. . 3 ⊢ 𝐾 = (Base‘(Scalar‘𝑅))
10		imasvsca.q	. . 3 ⊢ · = ( ·_𝑠 ‘𝑅)
11		eqid 2724	. . 3 ⊢ (·_𝑖‘𝑅) = (·_𝑖‘𝑅)
12		eqid 2724	. . 3 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
13		eqid 2724	. . 3 ⊢ (dist‘𝑅) = (dist‘𝑅)
14		eqid 2724	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
15		imasbas.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
16		imasbas.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
17		eqid 2724	. . . 4 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
18	1, 2, 15, 16, 3, 17	imasplusg 17459	. . 3 ⊢ (𝜑 → (+_g‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+_g‘𝑅)𝑞))⟩})
19		eqid 2724	. . . 4 ⊢ (._r‘𝑈) = (._r‘𝑈)
20	1, 2, 15, 16, 4, 19	imasmulr 17460	. . 3 ⊢ (𝜑 → (._r‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(._r‘𝑅)𝑞))⟩})
21		eqidd 2725	. . 3 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))
22		eqidd 2725	. . 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 2725	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((TopOpen‘𝑅) qTop 𝐹))
24		eqid 2724	. . . 4 ⊢ (dist‘𝑈) = (dist‘𝑈)
25	1, 2, 15, 16, 13, 24	imasds 17455	. . 3 ⊢ (𝜑 → (dist‘𝑈) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑢 ∈ ℕ ran (𝑧 ∈ {𝑤 ∈ ((𝑉 × 𝑉) ↑_m (1...𝑢)) ∣ ((𝐹‘(1^st ‘(𝑤‘1))) = 𝑥 ∧ (𝐹‘(2^nd ‘(𝑤‘𝑢))) = 𝑦 ∧ ∀𝑣 ∈ (1...(𝑢 − 1))(𝐹‘(2^nd ‘(𝑤‘𝑣))) = (𝐹‘(1^st ‘(𝑤‘(𝑣 + 1)))))} ↦ (ℝ^_𝑠 Σ_g ((dist‘𝑅) ∘ 𝑧))), ℝ^, < )))
26		eqidd 2725	. . 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 17453	. 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 2724	. . 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 17393	. 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 17261	. 2 ⊢ ·_𝑠 = Slot ( ·_𝑠 ‘ndx)
31		snsstp2 4812	. . 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 4165	. . . 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 4164	. . . 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 3983	. . 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 3983	. 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 6894	. . . 4 ⊢ (Base‘𝑅) ∈ V
37	2, 36	eqeltrdi 2833	. . 3 ⊢ (𝜑 → 𝑉 ∈ V)
38	6	fvexi 6895	. . . . 5 ⊢ 𝐾 ∈ V
39		snex 5421	. . . . 5 ⊢ {(𝐹‘𝑞)} ∈ V
40	38, 39	mpoex 8059	. . . 4 ⊢ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V
41	40	rgenw 3057	. . 3 ⊢ ∀𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V
42		iunexg 7943	. . 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 585	. 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 17134	1 ⊢ (𝜑 → ∙ = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 ∪ cun 3938 {csn 4620 {ctp 4624 ⟨cop 4626 ∪ ciun 4987 ^◡ccnv 5665 ∘ ccom 5670 –onto→wfo 6531 ‘cfv 6533 (class class class)co 7401 ∈ cmpo 7403 1c1 11106 2c2 12263 cdc 12673 ndxcnx 17122 Basecbs 17140 +_gcplusg 17193 ._rcmulr 17194 Scalarcsca 17196 ·_𝑠 cvsca 17197 ·_𝑖cip 17198 TopSetcts 17199 lecple 17200 distcds 17202 TopOpenctopn 17363 qTop cqtop 17445 “_s cimas 17446

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-inf 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-imas 17450

This theorem is referenced by: imasip 17463 imastset 17464 imasle 17465 imasvscafn 17479 imasvscaval 17480 imasvscaf 17481

W3C validator