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Theorem prdsmulr 16828
Description: Multiplication in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
Hypotheses
Ref Expression
prdsbas.p 𝑃 = (𝑆Xs𝑅)
prdsbas.s (𝜑𝑆𝑉)
prdsbas.r (𝜑𝑅𝑊)
prdsbas.b 𝐵 = (Base‘𝑃)
prdsbas.i (𝜑 → dom 𝑅 = 𝐼)
prdsmulr.t · = (.r𝑃)
Assertion
Ref Expression
prdsmulr (𝜑· = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(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 prdsmulr
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 2738 . . 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 16826 . . 3 (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
8 eqid 2738 . . . 4 (+g𝑃) = (+g𝑃)
91, 4, 5, 6, 3, 8prdsplusg 16827 . . 3 (𝜑 → (+g𝑃) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
10 eqidd 2739 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
11 eqidd 2739 . . 3 (𝜑 → (𝑓 ∈ (Base‘𝑆), 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓 ∈ (Base‘𝑆), 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
12 eqidd 2739 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
13 eqidd 2739 . . 3 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
14 eqidd 2739 . . 3 (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
15 eqidd 2739 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )) = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
16 eqidd 2739 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
17 eqidd 2739 . . 3 (𝜑 → (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ ((2nd𝑎)(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))𝑐), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))) = (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ ((2nd𝑎)(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))𝑐), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))
181, 2, 3, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 4, 5prdsval 16824 . 2 (𝜑𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑃)⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·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 ∈ ((2nd𝑎)(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))𝑐), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})))
19 prdsmulr.t . 2 · = (.r𝑃)
20 mulrid 16712 . 2 .r = Slot (.r‘ndx)
21 ovssunirn 7200 . . . . . . . . . . 11 ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran (.r‘(𝑅𝑥))
2220strfvss 16602 . . . . . . . . . . . . 13 (.r‘(𝑅𝑥)) ⊆ ran (𝑅𝑥)
23 fvssunirn 6697 . . . . . . . . . . . . . 14 (𝑅𝑥) ⊆ ran 𝑅
24 rnss 5776 . . . . . . . . . . . . . 14 ((𝑅𝑥) ⊆ ran 𝑅 → ran (𝑅𝑥) ⊆ ran ran 𝑅)
25 uniss 4801 . . . . . . . . . . . . . 14 (ran (𝑅𝑥) ⊆ ran ran 𝑅 ran (𝑅𝑥) ⊆ ran ran 𝑅)
2623, 24, 25mp2b 10 . . . . . . . . . . . . 13 ran (𝑅𝑥) ⊆ ran ran 𝑅
2722, 26sstri 3884 . . . . . . . . . . . 12 (.r‘(𝑅𝑥)) ⊆ ran ran 𝑅
28 rnss 5776 . . . . . . . . . . . 12 ((.r‘(𝑅𝑥)) ⊆ ran ran 𝑅 → ran (.r‘(𝑅𝑥)) ⊆ ran ran ran 𝑅)
29 uniss 4801 . . . . . . . . . . . 12 (ran (.r‘(𝑅𝑥)) ⊆ ran ran ran 𝑅 ran (.r‘(𝑅𝑥)) ⊆ ran ran ran 𝑅)
3027, 28, 29mp2b 10 . . . . . . . . . . 11 ran (.r‘(𝑅𝑥)) ⊆ ran ran ran 𝑅
3121, 30sstri 3884 . . . . . . . . . 10 ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑅
32 ovex 7197 . . . . . . . . . . 11 ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)) ∈ V
3332elpw 4489 . . . . . . . . . 10 (((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ran ran ran 𝑅 ↔ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑅)
3431, 33mpbir 234 . . . . . . . . 9 ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ran ran ran 𝑅
3534a1i 11 . . . . . . . 8 ((𝜑𝑥𝐼) → ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ran ran ran 𝑅)
3635fmpttd 6883 . . . . . . 7 (𝜑 → (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))):𝐼⟶𝒫 ran ran ran 𝑅)
37 rnexg 7628 . . . . . . . . . . . 12 (𝑅𝑊 → ran 𝑅 ∈ V)
38 uniexg 7478 . . . . . . . . . . . 12 (ran 𝑅 ∈ V → ran 𝑅 ∈ V)
395, 37, 383syl 18 . . . . . . . . . . 11 (𝜑 ran 𝑅 ∈ V)
40 rnexg 7628 . . . . . . . . . . 11 ( ran 𝑅 ∈ V → ran ran 𝑅 ∈ V)
41 uniexg 7478 . . . . . . . . . . 11 (ran ran 𝑅 ∈ V → ran ran 𝑅 ∈ V)
4239, 40, 413syl 18 . . . . . . . . . 10 (𝜑 ran ran 𝑅 ∈ V)
43 rnexg 7628 . . . . . . . . . 10 ( ran ran 𝑅 ∈ V → ran ran ran 𝑅 ∈ V)
44 uniexg 7478 . . . . . . . . . 10 (ran ran ran 𝑅 ∈ V → ran ran ran 𝑅 ∈ V)
4542, 43, 443syl 18 . . . . . . . . 9 (𝜑 ran ran ran 𝑅 ∈ V)
4645pwexd 5243 . . . . . . . 8 (𝜑 → 𝒫 ran ran ran 𝑅 ∈ V)
475dmexd 7629 . . . . . . . . 9 (𝜑 → dom 𝑅 ∈ V)
483, 47eqeltrrd 2834 . . . . . . . 8 (𝜑𝐼 ∈ V)
4946, 48elmapd 8444 . . . . . . 7 (𝜑 → ((𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))) ∈ (𝒫 ran ran ran 𝑅m 𝐼) ↔ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))):𝐼⟶𝒫 ran ran ran 𝑅))
5036, 49mpbird 260 . . . . . 6 (𝜑 → (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))) ∈ (𝒫 ran ran ran 𝑅m 𝐼))
5150ralrimivw 3097 . . . . 5 (𝜑 → ∀𝑔𝐵 (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))) ∈ (𝒫 ran ran ran 𝑅m 𝐼))
5251ralrimivw 3097 . . . 4 (𝜑 → ∀𝑓𝐵𝑔𝐵 (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))) ∈ (𝒫 ran ran ran 𝑅m 𝐼))
53 eqid 2738 . . . . 5 (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))
5453fmpo 7784 . . . 4 (∀𝑓𝐵𝑔𝐵 (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))) ∈ (𝒫 ran ran ran 𝑅m 𝐼) ↔ (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))):(𝐵 × 𝐵)⟶(𝒫 ran ran ran 𝑅m 𝐼))
5552, 54sylib 221 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))):(𝐵 × 𝐵)⟶(𝒫 ran ran ran 𝑅m 𝐼))
566fvexi 6682 . . . . 5 𝐵 ∈ V
5756, 56xpex 7488 . . . 4 (𝐵 × 𝐵) ∈ V
58 ovex 7197 . . . 4 (𝒫 ran ran ran 𝑅m 𝐼) ∈ V
59 fex2 7657 . . . 4 (((𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))):(𝐵 × 𝐵)⟶(𝒫 ran ran ran 𝑅m 𝐼) ∧ (𝐵 × 𝐵) ∈ V ∧ (𝒫 ran ran ran 𝑅m 𝐼) ∈ V) → (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))) ∈ V)
6057, 58, 59mp3an23 1454 . . 3 ((𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))):(𝐵 × 𝐵)⟶(𝒫 ran ran ran 𝑅m 𝐼) → (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))) ∈ V)
6155, 60syl 17 . 2 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))) ∈ V)
62 snsstp3 4703 . . . 4 {⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑃)⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩}
63 ssun1 4060 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑃)⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑃)⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩})
6462, 63sstri 3884 . . 3 {⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑃)⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩})
65 ssun1 4060 . . 3 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑃)⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑃)⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·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 ∈ ((2nd𝑎)(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))𝑐), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))
6664, 65sstri 3884 . 2 {⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑃)⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·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 ∈ ((2nd𝑎)(𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))𝑐), 𝑒 ∈ ((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))
6718, 19, 20, 61, 66prdsvallem 16823 1 (𝜑· = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2113  wral 3053  Vcvv 3397  cun 3839  wss 3841  𝒫 cpw 4485  {csn 4513  {cpr 4515  {ctp 4517  cop 4519   cuni 4793   class class class wbr 5027  {copab 5089  cmpt 5107   × cxp 5517  dom cdm 5519  ran crn 5520  ccom 5523  wf 6329  cfv 6333  (class class class)co 7164  cmpo 7166  1st c1st 7705  2nd c2nd 7706  m cmap 8430  Xcixp 8500  supcsup 8970  0cc0 10608  *cxr 10745   < clt 10746  ndxcnx 16576  Basecbs 16579  +gcplusg 16661  .rcmulr 16662  Scalarcsca 16664   ·𝑠 cvsca 16665  ·𝑖cip 16666  TopSetcts 16667  lecple 16668  distcds 16670  Hom chom 16672  compcco 16673  TopOpenctopn 16791  tcpt 16808   Σg cgsu 16810  Xscprds 16815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-er 8313  df-map 8432  df-ixp 8501  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-sup 8972  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-nn 11710  df-2 11772  df-3 11773  df-4 11774  df-5 11775  df-6 11776  df-7 11777  df-8 11778  df-9 11779  df-n0 11970  df-z 12056  df-dec 12173  df-uz 12318  df-fz 12975  df-struct 16581  df-ndx 16582  df-slot 16583  df-base 16585  df-plusg 16674  df-mulr 16675  df-sca 16677  df-vsca 16678  df-ip 16679  df-tset 16680  df-ple 16681  df-ds 16683  df-hom 16685  df-cco 16686  df-prds 16817
This theorem is referenced by:  prdsvsca  16829  prdsle  16831  prdsds  16833  prdstset  16835  prdshom  16836  prdsco  16837  prdsmulrval  16844  prdsmgp  19475
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