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Theorem prdshom 17285
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 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 17275 . . 3 (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
8 eqid 2738 . . . 4 (+g𝑃) = (+g𝑃)
91, 4, 5, 6, 3, 8prdsplusg 17276 . . 3 (𝜑 → (+g𝑃) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
10 eqid 2738 . . . 4 (.r𝑃) = (.r𝑃)
111, 4, 5, 6, 3, 10prdsmulr 17277 . . 3 (𝜑 → (.r𝑃) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
12 eqid 2738 . . . 4 ( ·𝑠𝑃) = ( ·𝑠𝑃)
131, 4, 5, 6, 3, 2, 12prdsvsca 17278 . . 3 (𝜑 → ( ·𝑠𝑃) = (𝑓 ∈ (Base‘𝑆), 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
14 eqidd 2739 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
15 eqid 2738 . . . 4 (TopSet‘𝑃) = (TopSet‘𝑃)
161, 4, 5, 6, 3, 15prdstset 17284 . . 3 (𝜑 → (TopSet‘𝑃) = (∏t‘(TopOpen ∘ 𝑅)))
17 eqid 2738 . . . 4 (le‘𝑃) = (le‘𝑃)
181, 4, 5, 6, 3, 17prdsle 17280 . . 3 (𝜑 → (le‘𝑃) = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
19 eqid 2738 . . . 4 (dist‘𝑃) = (dist‘𝑃)
201, 4, 5, 6, 3, 19prdsds 17282 . . 3 (𝜑 → (dist‘𝑃) = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
21 eqidd 2739 . . 3 (𝜑 → (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
22 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))))))
231, 2, 3, 7, 9, 11, 13, 14, 16, 18, 20, 21, 22, 4, 5prdsval 17273 . 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 17229 . 2 Hom = Slot (Hom ‘ndx)
26 ovssunirn 7386 . . . . . . . . . . 11 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran (Hom ‘(𝑅𝑥))
2725strfvss 16995 . . . . . . . . . . . . 13 (Hom ‘(𝑅𝑥)) ⊆ ran (𝑅𝑥)
28 fvssunirn 6871 . . . . . . . . . . . . . 14 (𝑅𝑥) ⊆ ran 𝑅
29 rnss 5891 . . . . . . . . . . . . . 14 ((𝑅𝑥) ⊆ ran 𝑅 → ran (𝑅𝑥) ⊆ ran ran 𝑅)
30 uniss 4872 . . . . . . . . . . . . . 14 (ran (𝑅𝑥) ⊆ ran ran 𝑅 ran (𝑅𝑥) ⊆ ran ran 𝑅)
3128, 29, 30mp2b 10 . . . . . . . . . . . . 13 ran (𝑅𝑥) ⊆ ran ran 𝑅
3227, 31sstri 3952 . . . . . . . . . . . 12 (Hom ‘(𝑅𝑥)) ⊆ ran ran 𝑅
33 rnss 5891 . . . . . . . . . . . 12 ((Hom ‘(𝑅𝑥)) ⊆ ran ran 𝑅 → ran (Hom ‘(𝑅𝑥)) ⊆ ran ran ran 𝑅)
34 uniss 4872 . . . . . . . . . . . 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 3952 . . . . . . . . . 10 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑅
3736rgenw 3067 . . . . . . . . 9 𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑅
38 ss2ixp 8782 . . . . . . . . 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 7833 . . . . . . . . . 10 (𝜑 → dom 𝑅 ∈ V)
413, 40eqeltrrd 2840 . . . . . . . . 9 (𝜑𝐼 ∈ V)
42 rnexg 7832 . . . . . . . . . . . 12 (𝑅𝑊 → ran 𝑅 ∈ V)
43 uniexg 7668 . . . . . . . . . . . 12 (ran 𝑅 ∈ V → ran 𝑅 ∈ V)
445, 42, 433syl 18 . . . . . . . . . . 11 (𝜑 ran 𝑅 ∈ V)
45 rnexg 7832 . . . . . . . . . . 11 ( ran 𝑅 ∈ V → ran ran 𝑅 ∈ V)
46 uniexg 7668 . . . . . . . . . . 11 (ran ran 𝑅 ∈ V → ran ran 𝑅 ∈ V)
4744, 45, 463syl 18 . . . . . . . . . 10 (𝜑 ran ran 𝑅 ∈ V)
48 rnexg 7832 . . . . . . . . . 10 ( ran ran 𝑅 ∈ V → ran ran ran 𝑅 ∈ V)
49 uniexg 7668 . . . . . . . . . 10 (ran ran ran 𝑅 ∈ V → ran ran ran 𝑅 ∈ V)
5047, 48, 493syl 18 . . . . . . . . 9 (𝜑 ran ran ran 𝑅 ∈ V)
51 ixpconstg 8778 . . . . . . . . 9 ((𝐼 ∈ V ∧ ran ran ran 𝑅 ∈ V) → X𝑥𝐼 ran ran ran 𝑅 = ( ran ran ran 𝑅m 𝐼))
5241, 50, 51syl2anc 585 . . . . . . . 8 (𝜑X𝑥𝐼 ran ran ran 𝑅 = ( ran ran ran 𝑅m 𝐼))
5339, 52sseqtrid 3995 . . . . . . 7 (𝜑X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ⊆ ( ran ran ran 𝑅m 𝐼))
54 ovex 7383 . . . . . . . 8 ( ran ran ran 𝑅m 𝐼) ∈ V
5554elpw2 5301 . . . . . . 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 3146 . . . . 5 (𝜑 → ∀𝑔𝐵 X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑅m 𝐼))
5857ralrimivw 3146 . . . 4 (𝜑 → ∀𝑓𝐵𝑔𝐵 X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑅m 𝐼))
59 eqid 2738 . . . . 5 (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))
6059fmpo 7989 . . . 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 6852 . . . . 5 𝐵 ∈ V
6362, 62xpex 7678 . . . 4 (𝐵 × 𝐵) ∈ V
6463a1i 11 . . 3 (𝜑 → (𝐵 × 𝐵) ∈ V)
6554pwex 5334 . . . 4 𝒫 ( ran ran ran 𝑅m 𝐼) ∈ V
6665a1i 11 . . 3 (𝜑 → 𝒫 ( ran ran ran 𝑅m 𝐼) ∈ V)
67 fex2 7861 . . 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 1372 . 2 (𝜑 → (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) ∈ V)
69 snsspr1 4773 . . . 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 4132 . . . 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 3952 . . 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 4132 . . 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 3952 . 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 17271 1 (𝜑𝐻 = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wral 3063  Vcvv 3444  cun 3907  wss 3909  𝒫 cpw 4559  {csn 4585  {cpr 4587  {ctp 4589  cop 4591   cuni 4864  cmpt 5187   × cxp 5629  dom cdm 5631  ran crn 5632  wf 6488  cfv 6492  (class class class)co 7350  cmpo 7352  1st c1st 7910  2nd c2nd 7911  m cmap 8699  Xcixp 8769  ndxcnx 17001  Basecbs 17019  +gcplusg 17069  .rcmulr 17070  Scalarcsca 17072   ·𝑠 cvsca 17073  ·𝑖cip 17074  TopSetcts 17075  lecple 17076  distcds 17078  Hom chom 17080  compcco 17081   Σg cgsu 17258  Xscprds 17263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663  ax-cnex 11041  ax-resscn 11042  ax-1cn 11043  ax-icn 11044  ax-addcl 11045  ax-addrcl 11046  ax-mulcl 11047  ax-mulrcl 11048  ax-mulcom 11049  ax-addass 11050  ax-mulass 11051  ax-distr 11052  ax-i2m1 11053  ax-1ne0 11054  ax-1rid 11055  ax-rnegex 11056  ax-rrecex 11057  ax-cnre 11058  ax-pre-lttri 11059  ax-pre-lttrn 11060  ax-pre-ltadd 11061  ax-pre-mulgt0 11062
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6250  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7306  df-ov 7353  df-oprab 7354  df-mpo 7355  df-om 7794  df-1st 7912  df-2nd 7913  df-frecs 8180  df-wrecs 8211  df-recs 8285  df-rdg 8324  df-1o 8380  df-er 8582  df-map 8701  df-ixp 8770  df-en 8818  df-dom 8819  df-sdom 8820  df-fin 8821  df-sup 9312  df-pnf 11125  df-mnf 11126  df-xr 11127  df-ltxr 11128  df-le 11129  df-sub 11321  df-neg 11322  df-nn 12088  df-2 12150  df-3 12151  df-4 12152  df-5 12153  df-6 12154  df-7 12155  df-8 12156  df-9 12157  df-n0 12348  df-z 12434  df-dec 12553  df-uz 12698  df-fz 13355  df-struct 16955  df-slot 16990  df-ndx 17002  df-base 17020  df-plusg 17082  df-mulr 17083  df-sca 17085  df-vsca 17086  df-ip 17087  df-tset 17088  df-ple 17089  df-ds 17091  df-hom 17093  df-cco 17094  df-prds 17265
This theorem is referenced by:  prdsco  17286
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