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Theorem prdshom 17371

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 = (𝑆X_s𝑅)
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**
Ref	Expression
prdshom	⊢ (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))

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.

Step	Hyp	Ref	Expression
1		prdsbas.p	. . 3 ⊢ 𝑃 = (𝑆X_s𝑅)
2		eqid 2731	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		prdsbas.i	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐼)
4		prdsbas.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
5		prdsbas.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
6		prdsbas.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
7	1, 4, 5, 6, 3	prdsbas 17361	. . 3 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
8		eqid 2731	. . . 4 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
9	1, 4, 5, 6, 3, 8	prdsplusg 17362	. . 3 ⊢ (𝜑 → (+_g‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+_g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
10		eqid 2731	. . . 4 ⊢ (._r‘𝑃) = (._r‘𝑃)
11	1, 4, 5, 6, 3, 10	prdsmulr 17363	. . 3 ⊢ (𝜑 → (._r‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
12		eqid 2731	. . . 4 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
13	1, 4, 5, 6, 3, 2, 12	prdsvsca 17364	. . 3 ⊢ (𝜑 → ( ·_𝑠 ‘𝑃) = (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
14		eqidd 2732	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))
15		eqid 2731	. . . 4 ⊢ (TopSet‘𝑃) = (TopSet‘𝑃)
16	1, 4, 5, 6, 3, 15	prdstset 17370	. . 3 ⊢ (𝜑 → (TopSet‘𝑃) = (∏_t‘(TopOpen ∘ 𝑅)))
17		eqid 2731	. . . 4 ⊢ (le‘𝑃) = (le‘𝑃)
18	1, 4, 5, 6, 3, 17	prdsle 17366	. . 3 ⊢ (𝜑 → (le‘𝑃) = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
19		eqid 2731	. . . 4 ⊢ (dist‘𝑃) = (dist‘𝑃)
20	1, 4, 5, 6, 3, 19	prdsds 17368	. . 3 ⊢ (𝜑 → (dist‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < )))
21		eqidd 2732	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))
22		eqidd 2732	. . 3 ⊢ (𝜑 → (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))
23	1, 2, 3, 7, 9, 11, 13, 14, 16, 18, 20, 21, 22, 4, 5	prdsval 17359	. 2 ⊢ (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 Σ_g (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 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
24		prdshom.h	. 2 ⊢ 𝐻 = (Hom ‘𝑃)
25		homid 17316	. 2 ⊢ Hom = Slot (Hom ‘ndx)
26		ovssunirn 7382	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (Hom ‘(𝑅‘𝑥))
27	25	strfvss 17098	. . . . . . . . . . . . 13 ⊢ (Hom ‘(𝑅‘𝑥)) ⊆ ∪ ran (𝑅‘𝑥)
28		fvssunirn 6853	. . . . . . . . . . . . . 14 ⊢ (𝑅‘𝑥) ⊆ ∪ ran 𝑅
29		rnss 5878	. . . . . . . . . . . . . 14 ⊢ ((𝑅‘𝑥) ⊆ ∪ ran 𝑅 → ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅)
30		uniss 4864	. . . . . . . . . . . . . 14 ⊢ (ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅 → ∪ ran (𝑅‘𝑥) ⊆ ∪ ran ∪ ran 𝑅)
31	28, 29, 30	mp2b 10	. . . . . . . . . . . . 13 ⊢ ∪ ran (𝑅‘𝑥) ⊆ ∪ ran ∪ ran 𝑅
32	27, 31	sstri 3939	. . . . . . . . . . . 12 ⊢ (Hom ‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅
33		rnss 5878	. . . . . . . . . . . 12 ⊢ ((Hom ‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅 → ran (Hom ‘(𝑅‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑅)
34		uniss 4864	. . . . . . . . . . . 12 ⊢ (ran (Hom ‘(𝑅‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑅 → ∪ ran (Hom ‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅)
35	32, 33, 34	mp2b 10	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
36	26, 35	sstri 3939	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
37	36	rgenw 3051	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
38		ss2ixp 8834	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅 → X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ∪ ran ∪ ran ∪ ran 𝑅)
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ∪ ran ∪ ran ∪ ran 𝑅
40	5	dmexd 7833	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑅 ∈ V)
41	3, 40	eqeltrrd 2832	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ V)
42		rnexg 7832	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑊 → ran 𝑅 ∈ V)
43		uniexg 7673	. . . . . . . . . . . 12 ⊢ (ran 𝑅 ∈ V → ∪ ran 𝑅 ∈ V)
44	5, 42, 43	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ ran 𝑅 ∈ V)
45		rnexg 7832	. . . . . . . . . . 11 ⊢ (∪ ran 𝑅 ∈ V → ran ∪ ran 𝑅 ∈ V)
46		uniexg 7673	. . . . . . . . . . 11 ⊢ (ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran 𝑅 ∈ V)
47	44, 45, 46	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran 𝑅 ∈ V)
48		rnexg 7832	. . . . . . . . . 10 ⊢ (∪ ran ∪ ran 𝑅 ∈ V → ran ∪ ran ∪ ran 𝑅 ∈ V)
49		uniexg 7673	. . . . . . . . . 10 ⊢ (ran ∪ ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
50	47, 48, 49	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
51		ixpconstg 8830	. . . . . . . . 9 ⊢ ((𝐼 ∈ V ∧ ∪ ran ∪ ran ∪ ran 𝑅 ∈ V) → X𝑥 ∈ 𝐼 ∪ ran ∪ ran ∪ ran 𝑅 = (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
52	41, 50, 51	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → X𝑥 ∈ 𝐼 ∪ ran ∪ ran ∪ ran 𝑅 = (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
53	39, 52	sseqtrid 3972	. . . . . . 7 ⊢ (𝜑 → X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
54		ovex 7379	. . . . . . . 8 ⊢ (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ∈ V
55	54	elpw2 5270	. . . . . . 7 ⊢ (X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ↔ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
56	53, 55	sylibr 234	. . . . . 6 ⊢ (𝜑 → X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
57	56	ralrimivw 3128	. . . . 5 ⊢ (𝜑 → ∀𝑔 ∈ 𝐵 X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
58	57	ralrimivw 3128	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
59		eqid 2731	. . . . 5 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))
60	59	fmpo 8000	. . . 4 ⊢ (∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ↔ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))):(𝐵 × 𝐵)⟶𝒫 (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
61	58, 60	sylib 218	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))):(𝐵 × 𝐵)⟶𝒫 (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
62	6	fvexi 6836	. . . . 5 ⊢ 𝐵 ∈ V
63	62, 62	xpex 7686	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
64	63	a1i 11	. . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V)
65	54	pwex 5316	. . . 4 ⊢ 𝒫 (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ∈ V
66	65	a1i 11	. . 3 ⊢ (𝜑 → 𝒫 (∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ∈ V)
67		fex2 7866	. . 3 ⊢ (((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 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 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ V)
68	61, 64, 66, 67	syl3anc 1373	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ V)
69		snsspr1 4763	. . . 4 ⊢ {⟨(Hom ‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩} ⊆ {⟨(Hom ‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}
70		ssun2 4126	. . . 4 ⊢ {⟨(Hom ‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘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 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})
71	69, 70	sstri 3939	. . 3 ⊢ {⟨(Hom ‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 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 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})
72		ssun2 4126	. . 3 ⊢ ({⟨(TopSet‘ndx), (TopSet‘𝑃)⟩, ⟨(le‘ndx), (le‘𝑃)⟩, ⟨(dist‘ndx), (dist‘𝑃)⟩} ∪ {⟨(Hom ‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘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 Σ_g (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 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}))
73	71, 72	sstri 3939	. 2 ⊢ {⟨(Hom ‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 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 Σ_g (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 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}))
74	23, 24, 25, 68, 73	prdsbaslem 17357	1 ⊢ (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 ∀wral 3047 Vcvv 3436 ∪ cun 3895 ⊆ wss 3897 𝒫 cpw 4547 {csn 4573 {cpr 4575 {ctp 4577 ⟨cop 4579 ∪ cuni 4856 ↦ cmpt 5170 × cxp 5612 dom cdm 5614 ran crn 5615 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1^st c1st 7919 2^nd c2nd 7920 ↑_m cmap 8750 Xcixp 8821 ndxcnx 17104 Basecbs 17120 +_gcplusg 17161 ._rcmulr 17162 Scalarcsca 17164 ·_𝑠 cvsca 17165 ·_𝑖cip 17166 TopSetcts 17167 lecple 17168 distcds 17170 Hom chom 17172 compcco 17173 Σ_g cgsu 17344 X_scprds 17349

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-hom 17185 df-cco 17186 df-prds 17351

This theorem is referenced by: prdsco 17372

W3C validator