Theorem prdsco 17528

Description: Structure product composition operation. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)

Hypotheses

Ref	Expression
prdsbas.p	⊢ 𝑃 = (𝑆Xs𝑅)
prdsbas.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdsbas.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)
prdsbas.b	⊢ 𝐵 = (Base‘𝑃)
prdsbas.i	⊢ (𝜑 → dom 𝑅 = 𝐼)
prdshom.h	⊢ 𝐻 = (Hom ‘𝑃)
prdsco.o	⊢ ∙ = (comp‘𝑃)

Assertion

Ref	Expression
prdsco	⊢ (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))

Distinct variable groups:   𝑎,𝑐,𝑑,𝑒,𝑥,𝐵   𝐻,𝑎,𝑐,𝑑,𝑒   𝜑,𝑎,𝑐,𝑑,𝑒,𝑥   𝐼,𝑎,𝑐,𝑑,𝑒,𝑥   𝑥,𝑃   𝑅,𝑎,𝑐,𝑑,𝑒,𝑥   𝑆,𝑎,𝑐,𝑑,𝑒,𝑥

Allowed substitution hints:   𝑃(𝑒,𝑎,𝑐,𝑑)   ∙ (𝑥,𝑒,𝑎,𝑐,𝑑)   𝐻(𝑥)   𝑉(𝑥,𝑒,𝑎,𝑐,𝑑)   𝑊(𝑥,𝑒,𝑎,𝑐,𝑑)

Proof of Theorem prdsco

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsbas.p	. . 3 ⊢ 𝑃 = (𝑆Xs𝑅)
2		eqid 2740	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		prdsbas.i	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐼)
4		prdsbas.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
5		prdsbas.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
6		prdsbas.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
7	1, 4, 5, 6, 3	prdsbas 17517	. . 3 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
8		eqid 2740	. . . 4 ⊢ (+g‘𝑃) = (+g‘𝑃)
9	1, 4, 5, 6, 3, 8	prdsplusg 17518	. . 3 ⊢ (𝜑 → (+g‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
10		eqid 2740	. . . 4 ⊢ (.r‘𝑃) = (.r‘𝑃)
11	1, 4, 5, 6, 3, 10	prdsmulr 17519	. . 3 ⊢ (𝜑 → (.r‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
12		eqid 2740	. . . 4 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
13	1, 4, 5, 6, 3, 2, 12	prdsvsca 17520	. . 3 ⊢ (𝜑 → ( ·𝑠 ‘𝑃) = (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
14		eqidd 2741	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))
15		eqid 2740	. . . 4 ⊢ (TopSet‘𝑃) = (TopSet‘𝑃)
16	1, 4, 5, 6, 3, 15	prdstset 17526	. . 3 ⊢ (𝜑 → (TopSet‘𝑃) = (∏t‘(TopOpen ∘ 𝑅)))
17		eqid 2740	. . . 4 ⊢ (le‘𝑃) = (le‘𝑃)
18	1, 4, 5, 6, 3, 17	prdsle 17522	. . 3 ⊢ (𝜑 → (le‘𝑃) = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
19		eqid 2740	. . . 4 ⊢ (dist‘𝑃) = (dist‘𝑃)
20	1, 4, 5, 6, 3, 19	prdsds 17524	. . 3 ⊢ (𝜑 → (dist‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )))
21		prdshom.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝑃)
22	1, 4, 5, 6, 3, 21	prdshom 17527	. . 3 ⊢ (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))
23		eqidd 2741	. . 3 ⊢ (𝜑 → (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))
24	1, 2, 3, 7, 9, 11, 13, 14, 16, 18, 20, 22, 23, 4, 5	prdsval 17515	. 2 ⊢ (𝜑 → 𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑃)⟩, ⟨(.r‘ndx), (.r‘𝑃)⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑃)⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
25		prdsco.o	. 2 ⊢ ∙ = (comp‘𝑃)
26		ccoid 17473	. 2 ⊢ comp = Slot (comp‘ndx)
27	6	fvexi 6934	. . . . 5 ⊢ 𝐵 ∈ V
28	27, 27	xpex 7788	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
29	28, 27	mpoex 8120	. . 3 ⊢ (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) ∈ V
30	29	a1i 11	. 2 ⊢ (𝜑 → (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) ∈ V)
31		snsspr2 4840	. . . 4 ⊢ {⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩} ⊆ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}
32		ssun2 4202	. . . 4 ⊢ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩} ⊆ ({⟨(TopSet‘ndx), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})
33	31, 32	sstri 4018	. . 3 ⊢ {⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩} ⊆ ({⟨(TopSet‘ndx), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})
34		ssun2 4202	. . 3 ⊢ ({⟨(TopSet‘ndx), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}) ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑃)⟩, ⟨(.r‘ndx), (.r‘𝑃)⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑃)⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}))
35	33, 34	sstri 4018	. 2 ⊢ {⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑃)⟩, ⟨(.r‘ndx), (.r‘𝑃)⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑃)⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}))
36	24, 25, 26, 30, 35	prdsbaslem 17513	1 ⊢ (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974  {csn 4648  {cpr 4650  {ctp 4652  ⟨cop 4654   ↦ cmpt 5249   × cxp 5698  dom cdm 5700  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1^st c1st 8028  2^nd c2nd 8029  ndxcnx 17240  Basecbs 17258  +_gcplusg 17311  ._rcmulr 17312  Scalarcsca 17314   ·_𝑠 cvsca 17315  ·_𝑖cip 17316  TopSetcts 17317  lecple 17318  distcds 17320  Hom chom 17322  compcco 17323   Σ_g cgsu 17500  X_scprds 17505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-hom 17335  df-cco 17336  df-prds 17507

This theorem is referenced by: (None)