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Theorem prdshom 16811
 Description: Structure product hom-sets. (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 (𝜑𝑆𝑉)
prdsbas.r (𝜑𝑅𝑊)
prdsbas.b 𝐵 = (Base‘𝑃)
prdsbas.i (𝜑 → dom 𝑅 = 𝐼)
prdshom.h 𝐻 = (Hom ‘𝑃)
Assertion
Ref Expression
prdshom (𝜑𝐻 = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(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:   𝐻(𝑥,𝑓,𝑔)   𝑉(𝑥,𝑓,𝑔)   𝑊(𝑥,𝑓,𝑔)

Proof of Theorem prdshom
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 2758 . . 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 16801 . . 3 (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
8 eqid 2758 . . . 4 (+g𝑃) = (+g𝑃)
91, 4, 5, 6, 3, 8prdsplusg 16802 . . 3 (𝜑 → (+g𝑃) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
10 eqid 2758 . . . 4 (.r𝑃) = (.r𝑃)
111, 4, 5, 6, 3, 10prdsmulr 16803 . . 3 (𝜑 → (.r𝑃) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
12 eqid 2758 . . . 4 ( ·𝑠𝑃) = ( ·𝑠𝑃)
131, 4, 5, 6, 3, 2, 12prdsvsca 16804 . . 3 (𝜑 → ( ·𝑠𝑃) = (𝑓 ∈ (Base‘𝑆), 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
14 eqidd 2759 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
15 eqid 2758 . . . 4 (TopSet‘𝑃) = (TopSet‘𝑃)
161, 4, 5, 6, 3, 15prdstset 16810 . . 3 (𝜑 → (TopSet‘𝑃) = (∏t‘(TopOpen ∘ 𝑅)))
17 eqid 2758 . . . 4 (le‘𝑃) = (le‘𝑃)
181, 4, 5, 6, 3, 17prdsle 16806 . . 3 (𝜑 → (le‘𝑃) = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
19 eqid 2758 . . . 4 (dist‘𝑃) = (dist‘𝑃)
201, 4, 5, 6, 3, 19prdsds 16808 . . 3 (𝜑 → (dist‘𝑃) = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
21 eqidd 2759 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
22 eqidd 2759 . . 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))))))
231, 2, 3, 7, 9, 11, 13, 14, 16, 18, 20, 21, 22, 4, 5prdsval 16799 . 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), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(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)))))⟩})))
24 prdshom.h . 2 𝐻 = (Hom ‘𝑃)
25 homid 16759 . 2 Hom = Slot (Hom ‘ndx)
26 ovssunirn 7192 . . . . . . . . . . 11 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran (Hom ‘(𝑅𝑥))
2725strfvss 16577 . . . . . . . . . . . . 13 (Hom ‘(𝑅𝑥)) ⊆ ran (𝑅𝑥)
28 fvssunirn 6692 . . . . . . . . . . . . . 14 (𝑅𝑥) ⊆ ran 𝑅
29 rnss 5785 . . . . . . . . . . . . . 14 ((𝑅𝑥) ⊆ ran 𝑅 → ran (𝑅𝑥) ⊆ ran ran 𝑅)
30 uniss 4809 . . . . . . . . . . . . . 14 (ran (𝑅𝑥) ⊆ ran ran 𝑅 ran (𝑅𝑥) ⊆ ran ran 𝑅)
3128, 29, 30mp2b 10 . . . . . . . . . . . . 13 ran (𝑅𝑥) ⊆ ran ran 𝑅
3227, 31sstri 3903 . . . . . . . . . . . 12 (Hom ‘(𝑅𝑥)) ⊆ ran ran 𝑅
33 rnss 5785 . . . . . . . . . . . 12 ((Hom ‘(𝑅𝑥)) ⊆ ran ran 𝑅 → ran (Hom ‘(𝑅𝑥)) ⊆ ran ran ran 𝑅)
34 uniss 4809 . . . . . . . . . . . 12 (ran (Hom ‘(𝑅𝑥)) ⊆ ran ran ran 𝑅 ran (Hom ‘(𝑅𝑥)) ⊆ ran ran ran 𝑅)
3532, 33, 34mp2b 10 . . . . . . . . . . 11 ran (Hom ‘(𝑅𝑥)) ⊆ ran ran ran 𝑅
3626, 35sstri 3903 . . . . . . . . . 10 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑅
3736rgenw 3082 . . . . . . . . 9 𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑅
38 ss2ixp 8505 . . . . . . . . 9 (∀𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑅X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ X𝑥𝐼 ran ran ran 𝑅)
3937, 38ax-mp 5 . . . . . . . 8 X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ X𝑥𝐼 ran ran ran 𝑅
405dmexd 7621 . . . . . . . . . 10 (𝜑 → dom 𝑅 ∈ V)
413, 40eqeltrrd 2853 . . . . . . . . 9 (𝜑𝐼 ∈ V)
42 rnexg 7620 . . . . . . . . . . . 12 (𝑅𝑊 → ran 𝑅 ∈ V)
43 uniexg 7470 . . . . . . . . . . . 12 (ran 𝑅 ∈ V → ran 𝑅 ∈ V)
445, 42, 433syl 18 . . . . . . . . . . 11 (𝜑 ran 𝑅 ∈ V)
45 rnexg 7620 . . . . . . . . . . 11 ( ran 𝑅 ∈ V → ran ran 𝑅 ∈ V)
46 uniexg 7470 . . . . . . . . . . 11 (ran ran 𝑅 ∈ V → ran ran 𝑅 ∈ V)
4744, 45, 463syl 18 . . . . . . . . . 10 (𝜑 ran ran 𝑅 ∈ V)
48 rnexg 7620 . . . . . . . . . 10 ( ran ran 𝑅 ∈ V → ran ran ran 𝑅 ∈ V)
49 uniexg 7470 . . . . . . . . . 10 (ran ran ran 𝑅 ∈ V → ran ran ran 𝑅 ∈ V)
5047, 48, 493syl 18 . . . . . . . . 9 (𝜑 ran ran ran 𝑅 ∈ V)
51 ixpconstg 8501 . . . . . . . . 9 ((𝐼 ∈ V ∧ ran ran ran 𝑅 ∈ V) → X𝑥𝐼 ran ran ran 𝑅 = ( ran ran ran 𝑅m 𝐼))
5241, 50, 51syl2anc 587 . . . . . . . 8 (𝜑X𝑥𝐼 ran ran ran 𝑅 = ( ran ran ran 𝑅m 𝐼))
5339, 52sseqtrid 3946 . . . . . . 7 (𝜑X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ( ran ran ran 𝑅m 𝐼))
54 ovex 7189 . . . . . . . 8 ( ran ran ran 𝑅m 𝐼) ∈ V
5554elpw2 5219 . . . . . . 7 (X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑅m 𝐼) ↔ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ( ran ran ran 𝑅m 𝐼))
5653, 55sylibr 237 . . . . . 6 (𝜑X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑅m 𝐼))
5756ralrimivw 3114 . . . . 5 (𝜑 → ∀𝑔𝐵 X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑅m 𝐼))
5857ralrimivw 3114 . . . 4 (𝜑 → ∀𝑓𝐵𝑔𝐵 X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑅m 𝐼))
59 eqid 2758 . . . . 5 (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))
6059fmpo 7776 . . . 4 (∀𝑓𝐵𝑔𝐵 X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑅m 𝐼) ↔ (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))):(𝐵 × 𝐵)⟶𝒫 ( ran ran ran 𝑅m 𝐼))
6158, 60sylib 221 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))):(𝐵 × 𝐵)⟶𝒫 ( ran ran ran 𝑅m 𝐼))
626fvexi 6677 . . . . 5 𝐵 ∈ V
6362, 62xpex 7480 . . . 4 (𝐵 × 𝐵) ∈ V
6463a1i 11 . . 3 (𝜑 → (𝐵 × 𝐵) ∈ V)
6554pwex 5253 . . . 4 𝒫 ( ran ran ran 𝑅m 𝐼) ∈ V
6665a1i 11 . . 3 (𝜑 → 𝒫 ( ran ran ran 𝑅m 𝐼) ∈ V)
67 fex2 7649 . . 3 (((𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))):(𝐵 × 𝐵)⟶𝒫 ( ran ran ran 𝑅m 𝐼) ∧ (𝐵 × 𝐵) ∈ V ∧ 𝒫 ( ran ran ran 𝑅m 𝐼) ∈ V) → (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) ∈ V)
6861, 64, 66, 67syl3anc 1368 . 2 (𝜑 → (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) ∈ V)
69 snsspr1 4707 . . . 4 {⟨(Hom ‘ndx), (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩} ⊆ {⟨(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)))))⟩}
70 ssun2 4080 . . . 4 {⟨(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)))))⟩} ⊆ ({⟨(TopSet‘ndx), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(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)))))⟩})
7169, 70sstri 3903 . . 3 {⟨(Hom ‘ndx), (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩} ⊆ ({⟨(TopSet‘ndx), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(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)))))⟩})
72 ssun2 4080 . . 3 ({⟨(TopSet‘ndx), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(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)))))⟩}) ⊆ (({⟨(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), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(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)))))⟩}))
7371, 72sstri 3903 . 2 {⟨(Hom ‘ndx), (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(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), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(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)))))⟩}))
7423, 24, 25, 68, 73prdsvallem 16798 1 (𝜑𝐻 = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ∪ cun 3858   ⊆ wss 3860  𝒫 cpw 4497  {csn 4525  {cpr 4527  {ctp 4529  ⟨cop 4531  ∪ cuni 4801   ↦ cmpt 5116   × cxp 5526  dom cdm 5528  ran crn 5529  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  1st c1st 7697  2nd c2nd 7698   ↑m cmap 8422  Xcixp 8492  ndxcnx 16551  Basecbs 16554  +gcplusg 16636  .rcmulr 16637  Scalarcsca 16639   ·𝑠 cvsca 16640  ·𝑖cip 16641  TopSetcts 16642  lecple 16643  distcds 16645  Hom chom 16647  compcco 16648   Σg cgsu 16785  Xscprds 16790 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 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-er 8305  df-map 8424  df-ixp 8493  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-sup 8952  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-2 11750  df-3 11751  df-4 11752  df-5 11753  df-6 11754  df-7 11755  df-8 11756  df-9 11757  df-n0 11948  df-z 12034  df-dec 12151  df-uz 12296  df-fz 12953  df-struct 16556  df-ndx 16557  df-slot 16558  df-base 16560  df-plusg 16649  df-mulr 16650  df-sca 16652  df-vsca 16653  df-ip 16654  df-tset 16655  df-ple 16656  df-ds 16658  df-hom 16660  df-cco 16661  df-prds 16792 This theorem is referenced by:  prdsco  16812
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