Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  prdsle Structured version   Visualization version   GIF version

Theorem prdsle 16730
 Description: Structure product weak ordering. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
prdsbas.p 𝑃 = (𝑆Xs𝑅)
prdsbas.s (𝜑𝑆𝑉)
prdsbas.r (𝜑𝑅𝑊)
prdsbas.b 𝐵 = (Base‘𝑃)
prdsbas.i (𝜑 → dom 𝑅 = 𝐼)
prdsle.l = (le‘𝑃)
Assertion
Ref Expression
prdsle (𝜑 = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
Distinct variable groups:   𝑓,𝑔,𝑥,𝐵   𝜑,𝑓,𝑔,𝑥   𝑓,𝐼,𝑔,𝑥   𝑃,𝑓,𝑔,𝑥   𝑅,𝑓,𝑔,𝑥   𝑆,𝑓,𝑔,𝑥
Allowed substitution hints:   (𝑥,𝑓,𝑔)   𝑉(𝑥,𝑓,𝑔)   𝑊(𝑥,𝑓,𝑔)

Proof of Theorem prdsle
Dummy variables 𝑎 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbas.p . . 3 𝑃 = (𝑆Xs𝑅)
2 eqid 2798 . . 3 (Base‘𝑆) = (Base‘𝑆)
3 prdsbas.i . . 3 (𝜑 → dom 𝑅 = 𝐼)
4 prdsbas.s . . . 4 (𝜑𝑆𝑉)
5 prdsbas.r . . . 4 (𝜑𝑅𝑊)
6 prdsbas.b . . . 4 𝐵 = (Base‘𝑃)
71, 4, 5, 6, 3prdsbas 16725 . . 3 (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
8 eqid 2798 . . . 4 (+g𝑃) = (+g𝑃)
91, 4, 5, 6, 3, 8prdsplusg 16726 . . 3 (𝜑 → (+g𝑃) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
10 eqid 2798 . . . 4 (.r𝑃) = (.r𝑃)
111, 4, 5, 6, 3, 10prdsmulr 16727 . . 3 (𝜑 → (.r𝑃) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
12 eqid 2798 . . . 4 ( ·𝑠𝑃) = ( ·𝑠𝑃)
131, 4, 5, 6, 3, 2, 12prdsvsca 16728 . . 3 (𝜑 → ( ·𝑠𝑃) = (𝑓 ∈ (Base‘𝑆), 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
14 eqidd 2799 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
15 eqidd 2799 . . 3 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
16 eqidd 2799 . . 3 (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
17 eqidd 2799 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )) = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
18 eqidd 2799 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
19 eqidd 2799 . . 3 (𝜑 → (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))) = (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))
201, 2, 3, 7, 9, 11, 13, 14, 15, 16, 17, 18, 19, 4, 5prdsval 16723 . 2 (𝜑𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑃)⟩, ⟨(.r‘ndx), (.r𝑃)⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑃)⟩, ⟨(·𝑖‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})))
21 prdsle.l . 2 = (le‘𝑃)
22 pleid 16662 . 2 le = Slot (le‘ndx)
236fvexi 6660 . . . . 5 𝐵 ∈ V
2423, 23xpex 7459 . . . 4 (𝐵 × 𝐵) ∈ V
25 vex 3444 . . . . . . . 8 𝑓 ∈ V
26 vex 3444 . . . . . . . 8 𝑔 ∈ V
2725, 26prss 4713 . . . . . . 7 ((𝑓𝐵𝑔𝐵) ↔ {𝑓, 𝑔} ⊆ 𝐵)
2827anbi1i 626 . . . . . 6 (((𝑓𝐵𝑔𝐵) ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥)) ↔ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥)))
2928opabbii 5098 . . . . 5 {⟨𝑓, 𝑔⟩ ∣ ((𝑓𝐵𝑔𝐵) ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}
30 opabssxp 5608 . . . . 5 {⟨𝑓, 𝑔⟩ ∣ ((𝑓𝐵𝑔𝐵) ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))} ⊆ (𝐵 × 𝐵)
3129, 30eqsstrri 3950 . . . 4 {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))} ⊆ (𝐵 × 𝐵)
3224, 31ssexi 5191 . . 3 {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))} ∈ V
3332a1i 11 . 2 (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))} ∈ V)
34 snsstp2 4710 . . . 4 {⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩} ⊆ {⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩}
35 ssun1 4099 . . . 4 {⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ⊆ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})
3634, 35sstri 3924 . . 3 {⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩} ⊆ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})
37 ssun2 4100 . . 3 ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}) ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑃)⟩, ⟨(.r‘ndx), (.r𝑃)⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑃)⟩, ⟨(·𝑖‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))
3836, 37sstri 3924 . 2 {⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑃)⟩, ⟨(.r‘ndx), (.r𝑃)⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑃)⟩, ⟨(·𝑖‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))
3920, 21, 22, 33, 38prdsvallem 16722 1 (𝜑 = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  {csn 4525  {cpr 4527  {ctp 4529  ⟨cop 4531   class class class wbr 5031  {copab 5093   ↦ cmpt 5111   × cxp 5518  dom cdm 5520  ran crn 5521   ∘ ccom 5524  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  1st c1st 7672  2nd c2nd 7673  Xcixp 8447  supcsup 8891  0cc0 10529  ℝ*cxr 10666   < clt 10667  ndxcnx 16475  Basecbs 16478  +gcplusg 16560  .rcmulr 16561  Scalarcsca 16563   ·𝑠 cvsca 16564  ·𝑖cip 16565  TopSetcts 16566  lecple 16567  distcds 16569  Hom chom 16571  compcco 16572  TopOpenctopn 16690  ∏tcpt 16707   Σg cgsu 16709  Xscprds 16714 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-map 8394  df-ixp 8448  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-sup 8893  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-2 11691  df-3 11692  df-4 11693  df-5 11694  df-6 11695  df-7 11696  df-8 11697  df-9 11698  df-n0 11889  df-z 11973  df-dec 12090  df-uz 12235  df-fz 12889  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-hom 16584  df-cco 16585  df-prds 16716 This theorem is referenced by:  prdsless  16731  prdsds  16732  prdstset  16734  prdshom  16735  prdsco  16736  prdsleval  16745  pwsle  16760
 Copyright terms: Public domain W3C validator