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Theorem prdsvsca 17578

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 17575	. . 3 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
8		eqid 2760	. . . 4 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
9	1, 4, 5, 6, 3, 8	prdsplusg 17576	. . 3 ⊢ (𝜑 → (+_g‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+_g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
10		eqid 2760	. . . 4 ⊢ (._r‘𝑃) = (._r‘𝑃)
11	1, 4, 5, 6, 3, 10	prdsmulr 17577	. . 3 ⊢ (𝜑 → (._r‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
12		eqidd 2761	. . 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 2761	. . 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 2761	. . 3 ⊢ (𝜑 → (∏_t‘(TopOpen ∘ 𝑅)) = (∏_t‘(TopOpen ∘ 𝑅)))
15		eqidd 2761	. . 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 2761	. . 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 2761	. . 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 2761	. . 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 17573	. 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 17438	. 2 ⊢ ·_𝑠 = Slot ( ·_𝑠 ‘ndx)
22		ovssunirn 7445	. . . . . . . . . . 11 ⊢ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ( ·_𝑠 ‘(𝑅‘𝑥))
23	21	strfvss 17312	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘(𝑅‘𝑥)) ⊆ ∪ ran (𝑅‘𝑥)
24		fvssunirn 6905	. . . . . . . . . . . . . 14 ⊢ (𝑅‘𝑥) ⊆ ∪ ran 𝑅
25		rnss 5918	. . . . . . . . . . . . . 14 ⊢ ((𝑅‘𝑥) ⊆ ∪ ran 𝑅 → ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅)
26		uniss 4875	. . . . . . . . . . . . . 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 3940	. . . . . . . . . . . 12 ⊢ ( ·_𝑠 ‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅
29		rnss 5918	. . . . . . . . . . . 12 ⊢ (( ·_𝑠 ‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅 → ran ( ·_𝑠 ‘(𝑅‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑅)
30		uniss 4875	. . . . . . . . . . . 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 3940	. . . . . . . . . 10 ⊢ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
33		ovex 7442	. . . . . . . . . . 11 ⊢ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ V
34	33	elpw 4561	. . . . . . . . . 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 234	. . . . . . . . 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 7104	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))):𝐼⟶𝒫 ∪ ran ∪ ran ∪ ran 𝑅)
38		rnexg 7898	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑊 → ran 𝑅 ∈ V)
39		uniexg 7741	. . . . . . . . . . 11 ⊢ (ran 𝑅 ∈ V → ∪ ran 𝑅 ∈ V)
40	5, 38, 39	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran 𝑅 ∈ V)
41		rnexg 7898	. . . . . . . . . 10 ⊢ (∪ ran 𝑅 ∈ V → ran ∪ ran 𝑅 ∈ V)
42		uniexg 7741	. . . . . . . . . 10 ⊢ (ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran 𝑅 ∈ V)
43	40, 41, 42	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ∪ ran 𝑅 ∈ V)
44		rnexg 7898	. . . . . . . . 9 ⊢ (∪ ran ∪ ran 𝑅 ∈ V → ran ∪ ran ∪ ran 𝑅 ∈ V)
45		uniexg 7741	. . . . . . . . 9 ⊢ (ran ∪ ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
46		pwexg 5340	. . . . . . . . 9 ⊢ (∪ ran ∪ ran ∪ ran 𝑅 ∈ V → 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
47	43, 44, 45, 46	4syl 20	. . . . . . . 8 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
48	5	dmexd 7899	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑅 ∈ V)
49	3, 48	eqeltrrd 2861	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V)
50	47, 49	elmapd 8839	. . . . . . 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 260	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
52	51	ralrimivw 3158	. . . . 5 ⊢ (𝜑 → ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
53	52	ralrimivw 3158	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ 𝐾 ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼))
54		eqid 2760	. . . . 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 8063	. . . 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 221	. . 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 6888	. . . . 5 ⊢ 𝐾 ∈ V
58	6	fvexi 6888	. . . . 5 ⊢ 𝐵 ∈ V
59	57, 58	xpex 7751	. . . 4 ⊢ (𝐾 × 𝐵) ∈ V
60		ovex 7442	. . . 4 ⊢ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑_m 𝐼) ∈ V
61		fex2 7932	. . . 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 1482	. . 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 18	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
64		snsstp2 4778	. . . 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 4125	. . . 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 3940	. . 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 4124	. . 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 3940	. 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 17571	1 ⊢ (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 Vcvv 3450 ∪ cun 3897 ⊆ wss 3899 𝒫 cpw 4557 {csn 4584 {cpr 4586 {ctp 4588 ⟨cop 4590 ∪ cuni 4867 class class class wbr 5103 {copab 5167 ↦ cmpt 5186 × cxp 5646 dom cdm 5648 ran crn 5649 ∘ ccom 5652 ⟶wf 6524 ‘cfv 6528 (class class class)co 7409 ∈ cmpo 7411 1^st c1st 7983 2^nd c2nd 7984 ↑_m cmap 8826 Xcixp 8904 supcsup 9410 0cc0 11157 ℝ^*cxr 11299 < clt 11300 ndxcnx 17318 Basecbs 17334 +_gcplusg 17375 ._rcmulr 17376 Scalarcsca 17378 ·_𝑠 cvsca 17379 ·_𝑖cip 17380 TopSetcts 17381 lecple 17382 distcds 17384 Hom chom 17386 compcco 17387 TopOpenctopn 17539 ∏_tcpt 17556 Σ_g cgsu 17558 X_scprds 17563

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-cnex 11213 ax-resscn 11214 ax-1cn 11215 ax-icn 11216 ax-addcl 11217 ax-addrcl 11218 ax-mulcl 11219 ax-mulrcl 11220 ax-mulcom 11221 ax-addass 11222 ax-mulass 11223 ax-distr 11224 ax-i2m1 11225 ax-1ne0 11226 ax-1rid 11227 ax-rnegex 11228 ax-rrecex 11229 ax-cnre 11230 ax-pre-lttri 11231 ax-pre-lttrn 11232 ax-pre-ltadd 11233 ax-pre-mulgt0 11234

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6294 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-riota 7366 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8455 df-er 8696 df-map 8828 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11500 df-neg 11501 df-nn 12291 df-2 12360 df-3 12361 df-4 12362 df-5 12363 df-6 12364 df-7 12365 df-8 12366 df-9 12367 df-n0 12562 df-z 12649 df-dec 12770 df-uz 12921 df-fz 13595 df-struct 17272 df-slot 17307 df-ndx 17319 df-base 17335 df-plusg 17388 df-mulr 17389 df-sca 17391 df-vsca 17392 df-ip 17393 df-tset 17394 df-ple 17395 df-ds 17397 df-hom 17399 df-cco 17400 df-prds 17565

This theorem is used by: prdsle 17580 prdsds 17582 prdstset 17584 prdshom 17585 prdsco 17586 prdsvscaval 17597

W3C validator