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Theorem prdsmulr 17438

Description: Multiplication in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (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)
prdsmulr.t	⊢ · = (._r‘𝑃)

**Assertion**
Ref	Expression
prdsmulr	⊢ (𝜑 → · = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(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: · (𝑥,𝑓,𝑔) 𝑉(𝑥,𝑓,𝑔) 𝑊(𝑥,𝑓,𝑔)

Proof of Theorem prdsmulr 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 2725	. . 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 17436	. . 3 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
8		eqid 2725	. . . 4 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
9	1, 4, 5, 6, 3, 8	prdsplusg 17437	. . 3 ⊢ (𝜑 → (+_g‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+_g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
10		eqidd 2726	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
11		eqidd 2726	. . 3 ⊢ (𝜑 → (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
12		eqidd 2726	. . 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))))))
13		eqidd 2726	. . 3 ⊢ (𝜑 → (∏_t‘(TopOpen ∘ 𝑅)) = (∏_t‘(TopOpen ∘ 𝑅)))
14		eqidd 2726	. . 3 ⊢ (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
15		eqidd 2726	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^, < )) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^, < )))
16		eqidd 2726	. . 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))))
17		eqidd 2726	. . 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))))))
18	1, 2, 3, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 4, 5	prdsval 17434	. 2 ⊢ (𝜑 → 𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), (+_g‘𝑃)⟩, ⟨(._r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·_𝑖‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏_t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))⟩} ∪ {⟨(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)))))⟩})))
19		prdsmulr.t	. 2 ⊢ · = (._r‘𝑃)
20		mulridx 17272	. 2 ⊢ ._r = Slot (._r‘ndx)
21		ovssunirn 7450	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (._r‘(𝑅‘𝑥))
22	20	strfvss 17153	. . . . . . . . . . . . 13 ⊢ (._r‘(𝑅‘𝑥)) ⊆ ∪ ran (𝑅‘𝑥)
23		fvssunirn 6923	. . . . . . . . . . . . . 14 ⊢ (𝑅‘𝑥) ⊆ ∪ ran 𝑅
24		rnss 5933	. . . . . . . . . . . . . 14 ⊢ ((𝑅‘𝑥) ⊆ ∪ ran 𝑅 → ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅)
25		uniss 4909	. . . . . . . . . . . . . 14 ⊢ (ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅 → ∪ ran (𝑅‘𝑥) ⊆ ∪ ran ∪ ran 𝑅)
26	23, 24, 25	mp2b 10	. . . . . . . . . . . . 13 ⊢ ∪ ran (𝑅‘𝑥) ⊆ ∪ ran ∪ ran 𝑅
27	22, 26	sstri 3981	. . . . . . . . . . . 12 ⊢ (._r‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅
28		rnss 5933	. . . . . . . . . . . 12 ⊢ ((._r‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅 → ran (._r‘(𝑅‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑅)
29		uniss 4909	. . . . . . . . . . . 12 ⊢ (ran (._r‘(𝑅‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑅 → ∪ ran (._r‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅)
30	27, 28, 29	mp2b 10	. . . . . . . . . . 11 ⊢ ∪ ran (._r‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
31	21, 30	sstri 3981	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
32		ovex 7447	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ V
33	32	elpw 4600	. . . . . . . . . 10 ⊢ (((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↔ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅)
34	31, 33	mpbir 230	. . . . . . . . 9 ⊢ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅
35	34	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅)
36	35	fmpttd 7118	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))):𝐼⟶𝒫 ∪ ran ∪ ran ∪ ran 𝑅)
37		rnexg 7906	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑊 → ran 𝑅 ∈ V)
38		uniexg 7741	. . . . . . . . . . . 12 ⊢ (ran 𝑅 ∈ V → ∪ ran 𝑅 ∈ V)
39	5, 37, 38	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ ran 𝑅 ∈ V)
40		rnexg 7906	. . . . . . . . . . 11 ⊢ (∪ ran 𝑅 ∈ V → ran ∪ ran 𝑅 ∈ V)
41		uniexg 7741	. . . . . . . . . . 11 ⊢ (ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran 𝑅 ∈ V)
42	39, 40, 41	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran 𝑅 ∈ V)
43		rnexg 7906	. . . . . . . . . 10 ⊢ (∪ ran ∪ ran 𝑅 ∈ V → ran ∪ ran ∪ ran 𝑅 ∈ V)
44		uniexg 7741	. . . . . . . . . 10 ⊢ (ran ∪ ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
45	42, 43, 44	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
46	45	pwexd 5371	. . . . . . . 8 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
47	5	dmexd 7907	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑅 ∈ V)
48	3, 47	eqeltrrd 2826	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V)
49	46, 48	elmapd 8855	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ↔ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))):𝐼⟶𝒫 ∪ ran ∪ ran ∪ ran 𝑅))
50	36, 49	mpbird 256	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
51	50	ralrimivw 3140	. . . . 5 ⊢ (𝜑 → ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
52	51	ralrimivw 3140	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
53		eqid 2725	. . . . 5 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))
54	53	fmpo 8068	. . . 4 ⊢ (∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ↔ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))):(𝐵 × 𝐵)⟶(𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
55	52, 54	sylib 217	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))):(𝐵 × 𝐵)⟶(𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
56	6	fvexi 6904	. . . . 5 ⊢ 𝐵 ∈ V
57	56, 56	xpex 7751	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
58		ovex 7447	. . . 4 ⊢ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ∈ V
59		fex2 7937	. . . 4 ⊢ (((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(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 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
60	57, 58, 59	mp3an23 1449	. . 3 ⊢ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))):(𝐵 × 𝐵)⟶(𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
61	55, 60	syl 17	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
62		snsstp3 4815	. . . 4 ⊢ {⟨(._r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), (+_g‘𝑃)⟩, ⟨(._r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩}
63		ssun1 4164	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), (+_g‘𝑃)⟩, ⟨(._r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), (+_g‘𝑃)⟩, ⟨(._r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·_𝑖‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩})
64	62, 63	sstri 3981	. . 3 ⊢ {⟨(._r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), (+_g‘𝑃)⟩, ⟨(._r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·_𝑖‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩})
65		ssun1 4164	. . 3 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), (+_g‘𝑃)⟩, ⟨(._r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·_𝑖‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), (+_g‘𝑃)⟩, ⟨(._r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·_𝑖‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏_t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))⟩} ∪ {⟨(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)))))⟩}))
66	64, 65	sstri 3981	. 2 ⊢ {⟨(._r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), (+_g‘𝑃)⟩, ⟨(._r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·_𝑖‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏_t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))⟩} ∪ {⟨(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)))))⟩}))
67	18, 19, 20, 61, 66	prdsbaslem 17432	1 ⊢ (𝜑 → · = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 Vcvv 3463 ∪ cun 3937 ⊆ wss 3939 𝒫 cpw 4596 {csn 4622 {cpr 4624 {ctp 4626 ⟨cop 4628 ∪ cuni 4901 class class class wbr 5141 {copab 5203 ↦ cmpt 5224 × cxp 5668 dom cdm 5670 ran crn 5671 ∘ ccom 5674 ⟶wf 6537 ‘cfv 6541 (class class class)co 7414 ∈ cmpo 7416 1^st c1st 7987 2^nd c2nd 7988 ↑_m cmap 8841 Xcixp 8912 supcsup 9461 0cc0 11136 ℝ^*cxr 11275 < clt 11276 ndxcnx 17159 Basecbs 17177 +_gcplusg 17230 ._rcmulr 17231 Scalarcsca 17233 ·_𝑠 cvsca 17234 ·_𝑖cip 17235 TopSetcts 17236 lecple 17237 distcds 17239 Hom chom 17241 compcco 17242 TopOpenctopn 17400 ∏_tcpt 17417 Σ_g cgsu 17419 X_scprds 17424

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-slot 17148 df-ndx 17160 df-base 17178 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-prds 17426

This theorem is referenced by: prdsvsca 17439 prdsle 17441 prdsds 17443 prdstset 17445 prdshom 17446 prdsco 17447 prdsmulrval 17454 prdsmgp 20093

W3C validator