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Theorem prdshom 17475
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 2726 . . 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 17465 . . 3 (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
8 eqid 2726 . . . 4 (+g𝑃) = (+g𝑃)
91, 4, 5, 6, 3, 8prdsplusg 17466 . . 3 (𝜑 → (+g𝑃) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
10 eqid 2726 . . . 4 (.r𝑃) = (.r𝑃)
111, 4, 5, 6, 3, 10prdsmulr 17467 . . 3 (𝜑 → (.r𝑃) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
12 eqid 2726 . . . 4 ( ·𝑠𝑃) = ( ·𝑠𝑃)
131, 4, 5, 6, 3, 2, 12prdsvsca 17468 . . 3 (𝜑 → ( ·𝑠𝑃) = (𝑓 ∈ (Base‘𝑆), 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
14 eqidd 2727 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
15 eqid 2726 . . . 4 (TopSet‘𝑃) = (TopSet‘𝑃)
161, 4, 5, 6, 3, 15prdstset 17474 . . 3 (𝜑 → (TopSet‘𝑃) = (∏t‘(TopOpen ∘ 𝑅)))
17 eqid 2726 . . . 4 (le‘𝑃) = (le‘𝑃)
181, 4, 5, 6, 3, 17prdsle 17470 . . 3 (𝜑 → (le‘𝑃) = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
19 eqid 2726 . . . 4 (dist‘𝑃) = (dist‘𝑃)
201, 4, 5, 6, 3, 19prdsds 17472 . . 3 (𝜑 → (dist‘𝑃) = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
21 eqidd 2727 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
22 eqidd 2727 . . 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 17463 . 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 17419 . 2 Hom = Slot (Hom ‘ndx)
26 ovssunirn 7450 . . . . . . . . . . 11 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran (Hom ‘(𝑅𝑥))
2725strfvss 17182 . . . . . . . . . . . . 13 (Hom ‘(𝑅𝑥)) ⊆ ran (𝑅𝑥)
28 fvssunirn 6924 . . . . . . . . . . . . . 14 (𝑅𝑥) ⊆ ran 𝑅
29 rnss 5936 . . . . . . . . . . . . . 14 ((𝑅𝑥) ⊆ ran 𝑅 → ran (𝑅𝑥) ⊆ ran ran 𝑅)
30 uniss 4914 . . . . . . . . . . . . . 14 (ran (𝑅𝑥) ⊆ ran ran 𝑅 ran (𝑅𝑥) ⊆ ran ran 𝑅)
3128, 29, 30mp2b 10 . . . . . . . . . . . . 13 ran (𝑅𝑥) ⊆ ran ran 𝑅
3227, 31sstri 3989 . . . . . . . . . . . 12 (Hom ‘(𝑅𝑥)) ⊆ ran ran 𝑅
33 rnss 5936 . . . . . . . . . . . 12 ((Hom ‘(𝑅𝑥)) ⊆ ran ran 𝑅 → ran (Hom ‘(𝑅𝑥)) ⊆ ran ran ran 𝑅)
34 uniss 4914 . . . . . . . . . . . 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 3989 . . . . . . . . . 10 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑅
3736rgenw 3055 . . . . . . . . 9 𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑅
38 ss2ixp 8929 . . . . . . . . 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 7906 . . . . . . . . . 10 (𝜑 → dom 𝑅 ∈ V)
413, 40eqeltrrd 2827 . . . . . . . . 9 (𝜑𝐼 ∈ V)
42 rnexg 7905 . . . . . . . . . . . 12 (𝑅𝑊 → ran 𝑅 ∈ V)
43 uniexg 7741 . . . . . . . . . . . 12 (ran 𝑅 ∈ V → ran 𝑅 ∈ V)
445, 42, 433syl 18 . . . . . . . . . . 11 (𝜑 ran 𝑅 ∈ V)
45 rnexg 7905 . . . . . . . . . . 11 ( ran 𝑅 ∈ V → ran ran 𝑅 ∈ V)
46 uniexg 7741 . . . . . . . . . . 11 (ran ran 𝑅 ∈ V → ran ran 𝑅 ∈ V)
4744, 45, 463syl 18 . . . . . . . . . 10 (𝜑 ran ran 𝑅 ∈ V)
48 rnexg 7905 . . . . . . . . . 10 ( ran ran 𝑅 ∈ V → ran ran ran 𝑅 ∈ V)
49 uniexg 7741 . . . . . . . . . 10 (ran ran ran 𝑅 ∈ V → ran ran ran 𝑅 ∈ V)
5047, 48, 493syl 18 . . . . . . . . 9 (𝜑 ran ran ran 𝑅 ∈ V)
51 ixpconstg 8925 . . . . . . . . 9 ((𝐼 ∈ V ∧ ran ran ran 𝑅 ∈ V) → X𝑥𝐼 ran ran ran 𝑅 = ( ran ran ran 𝑅m 𝐼))
5241, 50, 51syl2anc 582 . . . . . . . 8 (𝜑X𝑥𝐼 ran ran ran 𝑅 = ( ran ran ran 𝑅m 𝐼))
5339, 52sseqtrid 4032 . . . . . . 7 (𝜑X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ( ran ran ran 𝑅m 𝐼))
54 ovex 7447 . . . . . . . 8 ( ran ran ran 𝑅m 𝐼) ∈ V
5554elpw2 5343 . . . . . . 7 (X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑅m 𝐼) ↔ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ( ran ran ran 𝑅m 𝐼))
5653, 55sylibr 233 . . . . . 6 (𝜑X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑅m 𝐼))
5756ralrimivw 3140 . . . . 5 (𝜑 → ∀𝑔𝐵 X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑅m 𝐼))
5857ralrimivw 3140 . . . 4 (𝜑 → ∀𝑓𝐵𝑔𝐵 X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑅m 𝐼))
59 eqid 2726 . . . . 5 (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))
6059fmpo 8072 . . . 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 217 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))):(𝐵 × 𝐵)⟶𝒫 ( ran ran ran 𝑅m 𝐼))
626fvexi 6905 . . . . 5 𝐵 ∈ V
6362, 62xpex 7751 . . . 4 (𝐵 × 𝐵) ∈ V
6463a1i 11 . . 3 (𝜑 → (𝐵 × 𝐵) ∈ V)
6554pwex 5375 . . . 4 𝒫 ( ran ran ran 𝑅m 𝐼) ∈ V
6665a1i 11 . . 3 (𝜑 → 𝒫 ( ran ran ran 𝑅m 𝐼) ∈ V)
67 fex2 7937 . . 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 4814 . . . 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 4172 . . . 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 3989 . . 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 4172 . . 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 3989 . 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, 73prdsbaslem 17461 1 (𝜑𝐻 = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wral 3051  Vcvv 3463  cun 3945  wss 3947  𝒫 cpw 4598  {csn 4624  {cpr 4626  {ctp 4628  cop 4630   cuni 4906  cmpt 5227   × cxp 5671  dom cdm 5673  ran crn 5674  wf 6540  cfv 6544  (class class class)co 7414  cmpo 7416  1st c1st 7991  2nd c2nd 7992  m cmap 8845  Xcixp 8916  ndxcnx 17188  Basecbs 17206  +gcplusg 17259  .rcmulr 17260  Scalarcsca 17262   ·𝑠 cvsca 17263  ·𝑖cip 17264  TopSetcts 17265  lecple 17266  distcds 17268  Hom chom 17270  compcco 17271   Σg cgsu 17448  Xscprds 17453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-cnex 11203  ax-resscn 11204  ax-1cn 11205  ax-icn 11206  ax-addcl 11207  ax-addrcl 11208  ax-mulcl 11209  ax-mulrcl 11210  ax-mulcom 11211  ax-addass 11212  ax-mulass 11213  ax-distr 11214  ax-i2m1 11215  ax-1ne0 11216  ax-1rid 11217  ax-rnegex 11218  ax-rrecex 11219  ax-cnre 11220  ax-pre-lttri 11221  ax-pre-lttrn 11222  ax-pre-ltadd 11223  ax-pre-mulgt0 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-er 8724  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9476  df-pnf 11289  df-mnf 11290  df-xr 11291  df-ltxr 11292  df-le 11293  df-sub 11485  df-neg 11486  df-nn 12257  df-2 12319  df-3 12320  df-4 12321  df-5 12322  df-6 12323  df-7 12324  df-8 12325  df-9 12326  df-n0 12517  df-z 12603  df-dec 12722  df-uz 12867  df-fz 13531  df-struct 17142  df-slot 17177  df-ndx 17189  df-base 17207  df-plusg 17272  df-mulr 17273  df-sca 17275  df-vsca 17276  df-ip 17277  df-tset 17278  df-ple 17279  df-ds 17281  df-hom 17283  df-cco 17284  df-prds 17455
This theorem is referenced by:  prdsco  17476
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