Metamath Proof Explorer

Theorem prdshom

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)

Ref Expression
Hypotheses prdsbas.p ⊢ 𝑃 = ( 𝑆 Xs 𝑅 )
prdsbas.s ⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
prdsbas.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
prdsbas.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
prdsbas.i ⊢ ( 𝜑 → dom 𝑅 = 𝐼 )
prdshom.h ⊢ 𝐻 = ( Hom ‘ 𝑃 )
Assertion prdshom ( 𝜑 → 𝐻 = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) )

Proof

Step Hyp Ref Expression
1 prdsbas.p ⊢ 𝑃 = ( 𝑆 Xs 𝑅 )
2 prdsbas.s ⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
3 prdsbas.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
4 prdsbas.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
5 prdsbas.i ⊢ ( 𝜑 → dom 𝑅 = 𝐼 )
6 prdshom.h ⊢ 𝐻 = ( Hom ‘ 𝑃 )
7 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
8 1 2 3 4 5 prdsbas ⊢ ( 𝜑 → 𝐵 = X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) )
9 eqid ⊢ ( +g ‘ 𝑃 ) = ( +g ‘ 𝑃 )
10 1 2 3 4 5 9 prdsplusg ⊢ ( 𝜑 → ( +g ‘ 𝑃 ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) )
11 eqid ⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 )
12 1 2 3 4 5 11 prdsmulr ⊢ ( 𝜑 → ( .r ‘ 𝑃 ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) )
13 eqid ⊢ ( ·𝑠 ‘ 𝑃 ) = ( ·𝑠 ‘ 𝑃 )
14 1 2 3 4 5 7 13 prdsvsca ⊢ ( 𝜑 → ( ·𝑠 ‘ 𝑃 ) = ( 𝑓 ∈ ( Base ‘ 𝑆 ) , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( 𝑓 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) )
15 eqidd ⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) )
16 eqid ⊢ ( TopSet ‘ 𝑃 ) = ( TopSet ‘ 𝑃 )
17 1 2 3 4 5 16 prdstset ⊢ ( 𝜑 → ( TopSet ‘ 𝑃 ) = ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) )
18 eqid ⊢ ( le ‘ 𝑃 ) = ( le ‘ 𝑃 )
19 1 2 3 4 5 18 prdsle ⊢ ( 𝜑 → ( le ‘ 𝑃 ) = { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( { 𝑓 , 𝑔 } ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } )
20 eqid ⊢ ( dist ‘ 𝑃 ) = ( dist ‘ 𝑃 )
21 1 2 3 4 5 20 prdsds ⊢ ( 𝜑 → ( dist ‘ 𝑃 ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) )
22 eqidd ⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) )
23 eqidd ⊢ ( 𝜑 → ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) = ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) )
24 1 7 5 8 10 12 14 15 17 19 21 22 23 2 3 prdsval ⊢ ( 𝜑 → 𝑃 = ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑃 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ 𝑃 ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑃 ) ⟩ , ⟨ ( le ‘ ndx ) , ( le ‘ 𝑃 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) ⟩ } ∪ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) ⟩ } ) ) )
25 homid ⊢ Hom = Slot ( Hom ‘ ndx )
26 ovssunirn ⊢ ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ⊆ ∪ ran ( Hom ‘ ( 𝑅 ‘ 𝑥 ) )
27 25 strfvss ⊢ ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ∪ ran ( 𝑅 ‘ 𝑥 )
28 fvssunirn ⊢ ( 𝑅 ‘ 𝑥 ) ⊆ ∪ ran 𝑅
29 rnss ⊢ ( ( 𝑅 ‘ 𝑥 ) ⊆ ∪ ran 𝑅 → ran ( 𝑅 ‘ 𝑥 ) ⊆ ran ∪ ran 𝑅 )
30 uniss ⊢ ( ran ( 𝑅 ‘ 𝑥 ) ⊆ ran ∪ ran 𝑅 → ∪ ran ( 𝑅 ‘ 𝑥 ) ⊆ ∪ ran ∪ ran 𝑅 )
31 28 29 30 mp2b ⊢ ∪ ran ( 𝑅 ‘ 𝑥 ) ⊆ ∪ ran ∪ ran 𝑅
32 27 31 sstri ⊢ ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran 𝑅
33 rnss ⊢ ( ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran 𝑅 → ran ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ran ∪ ran ∪ ran 𝑅 )
34 uniss ⊢ ( ran ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ran ∪ ran ∪ ran 𝑅 → ∪ ran ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran ∪ ran 𝑅 )
35 32 33 34 mp2b ⊢ ∪ ran ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
36 26 35 sstri ⊢ ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
37 36 rgenw ⊢ ∀ 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
38 ss2ixp ⊢ ( ∀ 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran ∪ ran 𝑅 → X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ⊆ X 𝑥 ∈ 𝐼 ∪ ran ∪ ran ∪ ran 𝑅 )
39 37 38 ax-mp ⊢ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ⊆ X 𝑥 ∈ 𝐼 ∪ ran ∪ ran ∪ ran 𝑅
40 3 dmexd ⊢ ( 𝜑 → dom 𝑅 ∈ V )
41 5 40 eqeltrrd ⊢ ( 𝜑 → 𝐼 ∈ V )
42 rnexg ⊢ ( 𝑅 ∈ 𝑊 → ran 𝑅 ∈ V )
43 uniexg ⊢ ( ran 𝑅 ∈ V → ∪ ran 𝑅 ∈ V )
44 3 42 43 3syl ⊢ ( 𝜑 → ∪ ran 𝑅 ∈ V )
45 rnexg ⊢ ( ∪ ran 𝑅 ∈ V → ran ∪ ran 𝑅 ∈ V )
46 uniexg ⊢ ( ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran 𝑅 ∈ V )
47 44 45 46 3syl ⊢ ( 𝜑 → ∪ ran ∪ ran 𝑅 ∈ V )
48 rnexg ⊢ ( ∪ ran ∪ ran 𝑅 ∈ V → ran ∪ ran ∪ ran 𝑅 ∈ V )
49 uniexg ⊢ ( ran ∪ ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V )
50 47 48 49 3syl ⊢ ( 𝜑 → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V )
51 ixpconstg ⊢ ( ( 𝐼 ∈ V ∧ ∪ ran ∪ ran ∪ ran 𝑅 ∈ V ) → X 𝑥 ∈ 𝐼 ∪ ran ∪ ran ∪ ran 𝑅 = ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
52 41 50 51 syl2anc ⊢ ( 𝜑 → X 𝑥 ∈ 𝐼 ∪ ran ∪ ran ∪ ran 𝑅 = ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
53 39 52 sseqtrid ⊢ ( 𝜑 → X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ⊆ ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
54 ovex ⊢ ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) ∈ V
55 54 elpw2 ⊢ ( X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ∈ 𝒫 ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) ↔ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ⊆ ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
56 53 55 sylibr ⊢ ( 𝜑 → X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ∈ 𝒫 ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
57 56 ralrimivw ⊢ ( 𝜑 → ∀ 𝑔 ∈ 𝐵 X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ∈ 𝒫 ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
58 57 ralrimivw ⊢ ( 𝜑 → ∀ 𝑓 ∈ 𝐵 ∀ 𝑔 ∈ 𝐵 X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ∈ 𝒫 ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
59 eqid ⊢ ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) )
60 59 fmpo ⊢ ( ∀ 𝑓 ∈ 𝐵 ∀ 𝑔 ∈ 𝐵 X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ∈ 𝒫 ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) ↔ ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) : ( 𝐵 × 𝐵 ) ⟶ 𝒫 ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
61 58 60 sylib ⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) : ( 𝐵 × 𝐵 ) ⟶ 𝒫 ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
62 4 fvexi ⊢ 𝐵 ∈ V
63 62 62 xpex ⊢ ( 𝐵 × 𝐵 ) ∈ V
64 63 a1i ⊢ ( 𝜑 → ( 𝐵 × 𝐵 ) ∈ V )
65 54 pwex ⊢ 𝒫 ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) ∈ V
66 65 a1i ⊢ ( 𝜑 → 𝒫 ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) ∈ V )
67 fex2 ⊢ ( ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) : ( 𝐵 × 𝐵 ) ⟶ 𝒫 ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) ∧ ( 𝐵 × 𝐵 ) ∈ V ∧ 𝒫 ( ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) ∈ V ) → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∈ V )
68 61 64 66 67 syl3anc ⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∈ V )
69 snsspr1 ⊢ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ } ⊆ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) ⟩ }
70 ssun2 ⊢ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) ⟩ } ⊆ ( { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑃 ) ⟩ , ⟨ ( le ‘ ndx ) , ( le ‘ 𝑃 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) ⟩ } ∪ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) ⟩ } )
71 69 70 sstri ⊢ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ } ⊆ ( { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑃 ) ⟩ , ⟨ ( le ‘ ndx ) , ( le ‘ 𝑃 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) ⟩ } ∪ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) ⟩ } )
72 ssun2 ⊢ ( { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑃 ) ⟩ , ⟨ ( le ‘ ndx ) , ( le ‘ 𝑃 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) ⟩ } ∪ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) ⟩ } ) ⊆ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑃 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ 𝑃 ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑃 ) ⟩ , ⟨ ( le ‘ ndx ) , ( le ‘ 𝑃 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) ⟩ } ∪ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) ⟩ } ) )
73 71 72 sstri ⊢ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ } ⊆ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑃 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ 𝑃 ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑃 ) ⟩ , ⟨ ( le ‘ ndx ) , ( le ‘ 𝑃 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) ⟩ } ∪ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) ⟩ } ) )
74 24 6 25 68 73 prdsbaslem ⊢ ( 𝜑 → 𝐻 = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) )