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Theorem prdsvsca 17088

Description: Scalar multiplication in a structure product. (Contributed by Stefan O'Rear, 5-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)
prdsvsca.k	⊢ 𝐾 = (Base‘𝑆)
prdsvsca.m	⊢ · = ( ·_𝑠 ‘𝑃)

**Assertion**
Ref	Expression
prdsvsca	⊢ (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))

Distinct variable groups: 𝑓,𝑔,𝑥,𝐵 𝑓,𝐾,𝑔 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) 𝐾(𝑥) 𝑉(𝑥,𝑓,𝑔) 𝑊(𝑥,𝑓,𝑔)

Proof of Theorem prdsvsca 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		prdsvsca.k	. . 3 ⊢ 𝐾 = (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 17085	. . 3 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
8		eqid 2738	. . . 4 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
9	1, 4, 5, 6, 3, 8	prdsplusg 17086	. . 3 ⊢ (𝜑 → (+_g‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+_g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
10		eqid 2738	. . . 4 ⊢ (._r‘𝑃) = (._r‘𝑃)
11	1, 4, 5, 6, 3, 10	prdsmulr 17087	. . 3 ⊢ (𝜑 → (._r‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
12		eqidd 2739	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
13		eqidd 2739	. . 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))))))
14		eqidd 2739	. . 3 ⊢ (𝜑 → (∏_t‘(TopOpen ∘ 𝑅)) = (∏_t‘(TopOpen ∘ 𝑅)))
15		eqidd 2739	. . 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))})
16		eqidd 2739	. . 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^, < )))
17		eqidd 2739	. . 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))))
18		eqidd 2739	. . 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))))))
19	1, 2, 3, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 4, 5	prdsval 17083	. 2 ⊢ (𝜑 → 𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), (+_g‘𝑃)⟩, ⟨(._r‘ndx), (._r‘𝑃)⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ 𝐾, 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)))))⟩})))
20		prdsvsca.m	. 2 ⊢ · = ( ·_𝑠 ‘𝑃)
21		vscaid 16956	. 2 ⊢ ·_𝑠 = Slot ( ·_𝑠 ‘ndx)
22		ovssunirn 7291	. . . . . . . . . . 11 ⊢ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ( ·_𝑠 ‘(𝑅‘𝑥))
23	21	strfvss 16816	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘(𝑅‘𝑥)) ⊆ ∪ ran (𝑅‘𝑥)
24		fvssunirn 6785	. . . . . . . . . . . . . 14 ⊢ (𝑅‘𝑥) ⊆ ∪ ran 𝑅
25		rnss 5837	. . . . . . . . . . . . . 14 ⊢ ((𝑅‘𝑥) ⊆ ∪ ran 𝑅 → ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅)
26		uniss 4844	. . . . . . . . . . . . . 14 ⊢ (ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅 → ∪ ran (𝑅‘𝑥) ⊆ ∪ ran ∪ ran 𝑅)
27	24, 25, 26	mp2b 10	. . . . . . . . . . . . 13 ⊢ ∪ ran (𝑅‘𝑥) ⊆ ∪ ran ∪ ran 𝑅
28	23, 27	sstri 3926	. . . . . . . . . . . 12 ⊢ ( ·_𝑠 ‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅
29		rnss 5837	. . . . . . . . . . . 12 ⊢ (( ·_𝑠 ‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅 → ran ( ·_𝑠 ‘(𝑅‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑅)
30		uniss 4844	. . . . . . . . . . . 12 ⊢ (ran ( ·_𝑠 ‘(𝑅‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑅 → ∪ ran ( ·_𝑠 ‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅)
31	28, 29, 30	mp2b 10	. . . . . . . . . . 11 ⊢ ∪ ran ( ·_𝑠 ‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
32	22, 31	sstri 3926	. . . . . . . . . 10 ⊢ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
33		ovex 7288	. . . . . . . . . . 11 ⊢ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ V
34	33	elpw 4534	. . . . . . . . . 10 ⊢ ((𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↔ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅)
35	32, 34	mpbir 230	. . . . . . . . 9 ⊢ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅)
37	36	fmpttd 6971	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))):𝐼⟶𝒫 ∪ ran ∪ ran ∪ ran 𝑅)
38		rnexg 7725	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑊 → ran 𝑅 ∈ V)
39		uniexg 7571	. . . . . . . . . . 11 ⊢ (ran 𝑅 ∈ V → ∪ ran 𝑅 ∈ V)
40	5, 38, 39	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran 𝑅 ∈ V)
41		rnexg 7725	. . . . . . . . . 10 ⊢ (∪ ran 𝑅 ∈ V → ran ∪ ran 𝑅 ∈ V)
42		uniexg 7571	. . . . . . . . . 10 ⊢ (ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran 𝑅 ∈ V)
43	40, 41, 42	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ∪ ran 𝑅 ∈ V)
44		rnexg 7725	. . . . . . . . 9 ⊢ (∪ ran ∪ ran 𝑅 ∈ V → ran ∪ ran ∪ ran 𝑅 ∈ V)
45		uniexg 7571	. . . . . . . . 9 ⊢ (ran ∪ ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
46		pwexg 5296	. . . . . . . . 9 ⊢ (∪ ran ∪ ran ∪ ran 𝑅 ∈ V → 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
47	43, 44, 45, 46	4syl 19	. . . . . . . 8 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
48	5	dmexd 7726	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑅 ∈ V)
49	3, 48	eqeltrrd 2840	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V)
50	47, 49	elmapd 8587	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ↔ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))):𝐼⟶𝒫 ∪ ran ∪ ran ∪ ran 𝑅))
51	37, 50	mpbird 256	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
52	51	ralrimivw 3108	. . . . 5 ⊢ (𝜑 → ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
53	52	ralrimivw 3108	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ 𝐾 ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
54		eqid 2738	. . . . 5 ⊢ (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))
55	54	fmpo 7881	. . . 4 ⊢ (∀𝑓 ∈ 𝐾 ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ↔ (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))):(𝐾 × 𝐵)⟶(𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
56	53, 55	sylib 217	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))):(𝐾 × 𝐵)⟶(𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
57	2	fvexi 6770	. . . . 5 ⊢ 𝐾 ∈ V
58	6	fvexi 6770	. . . . 5 ⊢ 𝐵 ∈ V
59	57, 58	xpex 7581	. . . 4 ⊢ (𝐾 × 𝐵) ∈ V
60		ovex 7288	. . . 4 ⊢ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ∈ V
61		fex2 7754	. . . 4 ⊢ (((𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))):(𝐾 × 𝐵)⟶(𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ∧ (𝐾 × 𝐵) ∈ V ∧ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ∈ V) → (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
62	59, 60, 61	mp3an23 1451	. . 3 ⊢ ((𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))):(𝐾 × 𝐵)⟶(𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) → (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
63	56, 62	syl 17	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
64		snsstp2 4747	. . . 4 ⊢ {⟨( ·_𝑠 ‘ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ⊆ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ 𝐾, 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		ssun2 4103	. . . 4 ⊢ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ 𝐾, 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), (._r‘𝑃)⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ 𝐾, 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)))))⟩})
66	64, 65	sstri 3926	. . 3 ⊢ {⟨( ·_𝑠 ‘ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), (+_g‘𝑃)⟩, ⟨(._r‘ndx), (._r‘𝑃)⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ 𝐾, 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)))))⟩})
67		ssun1 4102	. . 3 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), (+_g‘𝑃)⟩, ⟨(._r‘ndx), (._r‘𝑃)⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ 𝐾, 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), (._r‘𝑃)⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ 𝐾, 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)))))⟩}))
68	66, 67	sstri 3926	. 2 ⊢ {⟨( ·_𝑠 ‘ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), (+_g‘𝑃)⟩, ⟨(._r‘ndx), (._r‘𝑃)⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ 𝐾, 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)))))⟩}))
69	19, 20, 21, 63, 68	prdsbaslem 17081	1 ⊢ (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 Vcvv 3422 ∪ cun 3881 ⊆ wss 3883 𝒫 cpw 4530 {csn 4558 {cpr 4560 {ctp 4562 ⟨cop 4564 ∪ cuni 4836 class class class wbr 5070 {copab 5132 ↦ cmpt 5153 × cxp 5578 dom cdm 5580 ran crn 5581 ∘ ccom 5584 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 1^st c1st 7802 2^nd c2nd 7803 ↑_m cmap 8573 Xcixp 8643 supcsup 9129 0cc0 10802 ℝ^*cxr 10939 < clt 10940 ndxcnx 16822 Basecbs 16840 +_gcplusg 16888 ._rcmulr 16889 Scalarcsca 16891 ·_𝑠 cvsca 16892 ·_𝑖cip 16893 TopSetcts 16894 lecple 16895 distcds 16897 Hom chom 16899 compcco 16900 TopOpenctopn 17049 ∏_tcpt 17066 Σ_g cgsu 17068 X_scprds 17073

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-hom 16912 df-cco 16913 df-prds 17075

This theorem is referenced by: prdsle 17090 prdsds 17092 prdstset 17094 prdshom 17095 prdsco 17096 prdsvscaval 17107

W3C validator