Theorem prdsplusg 17609

Description: Addition in a structure product. (Contributed by Stefan O'Rear, 3-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 = (𝑆Xs𝑅)
prdsbas.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdsbas.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)
prdsbas.b	⊢ 𝐵 = (Base‘𝑃)
prdsbas.i	⊢ (𝜑 → dom 𝑅 = 𝐼)
prdsplusg.b	⊢ + = (+g‘𝑃)

Assertion

Ref	Expression
prdsplusg	⊢ (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(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 prdsplusg

Dummy variables 𝑎 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsbas.p	. . 3 ⊢ 𝑃 = (𝑆Xs𝑅)
2		eqid 2761	. . 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 17608	. . 3 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
8		eqidd 2762	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
9		eqidd 2762	. . 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)))))
10		eqidd 2762	. . 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)))))
11		eqidd 2762	. . 3 ⊢ (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))))))
12		eqidd 2762	. . 3 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
13		eqidd 2762	. . 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))})
14		eqidd 2762	. . 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*, < )))
15		eqidd 2762	. . 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))))
16		eqidd 2762	. . 3 ⊢ (𝜑 → (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘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)(⟨((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 ↦ 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)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))
17	1, 2, 3, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 4, 5	prdsval 17606	. 2 ⊢ (𝜑 → 𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.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 Σg (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 ∈ ((2nd ‘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)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
18		prdsplusg.b	. 2 ⊢ + = (+g‘𝑃)
19		plusgid 17435	. 2 ⊢ +g = Slot (+g‘ndx)
20		ovssunirn 7448	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (+g‘(𝑅‘𝑥))
21	19	strfvss 17345	. . . . . . . . . . . . 13 ⊢ (+g‘(𝑅‘𝑥)) ⊆ ∪ ran (𝑅‘𝑥)
22		fvssunirn 6908	. . . . . . . . . . . . . 14 ⊢ (𝑅‘𝑥) ⊆ ∪ ran 𝑅
23		rnss 5921	. . . . . . . . . . . . . 14 ⊢ ((𝑅‘𝑥) ⊆ ∪ ran 𝑅 → ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅)
24		uniss 4875	. . . . . . . . . . . . . 14 ⊢ (ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅 → ∪ ran (𝑅‘𝑥) ⊆ ∪ ran ∪ ran 𝑅)
25	22, 23, 24	mp2b 10	. . . . . . . . . . . . 13 ⊢ ∪ ran (𝑅‘𝑥) ⊆ ∪ ran ∪ ran 𝑅
26	21, 25	sstri 3940	. . . . . . . . . . . 12 ⊢ (+g‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅
27		rnss 5921	. . . . . . . . . . . 12 ⊢ ((+g‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅 → ran (+g‘(𝑅‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑅)
28		uniss 4875	. . . . . . . . . . . 12 ⊢ (ran (+g‘(𝑅‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑅 → ∪ ran (+g‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅)
29	26, 27, 28	mp2b 10	. . . . . . . . . . 11 ⊢ ∪ ran (+g‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
30	20, 29	sstri 3940	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
31		ovex 7445	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ V
32	31	elpw 4561	. . . . . . . . . 10 ⊢ (((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↔ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅)
33	30, 32	mpbir 234	. . . . . . . . 9 ⊢ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅)
35	34	fmpttd 7107	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))):𝐼⟶𝒫 ∪ ran ∪ ran ∪ ran 𝑅)
36		rnexg 7903	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑊 → ran 𝑅 ∈ V)
37		uniexg 7746	. . . . . . . . . . 11 ⊢ (ran 𝑅 ∈ V → ∪ ran 𝑅 ∈ V)
38	5, 36, 37	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran 𝑅 ∈ V)
39		rnexg 7903	. . . . . . . . . 10 ⊢ (∪ ran 𝑅 ∈ V → ran ∪ ran 𝑅 ∈ V)
40		uniexg 7746	. . . . . . . . . 10 ⊢ (ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran 𝑅 ∈ V)
41	38, 39, 40	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ∪ ran 𝑅 ∈ V)
42		rnexg 7903	. . . . . . . . 9 ⊢ (∪ ran ∪ ran 𝑅 ∈ V → ran ∪ ran ∪ ran 𝑅 ∈ V)
43		uniexg 7746	. . . . . . . . 9 ⊢ (ran ∪ ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
44		pwexg 5340	. . . . . . . . 9 ⊢ (∪ ran ∪ ran ∪ ran 𝑅 ∈ V → 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
45	41, 42, 43, 44	4syl 20	. . . . . . . 8 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
46	5	dmexd 7904	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑅 ∈ V)
47	3, 46	eqeltrrd 2862	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V)
48	45, 47	elmapd 8844	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼) ↔ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))):𝐼⟶𝒫 ∪ ran ∪ ran ∪ ran 𝑅))
49	35, 48	mpbird 260	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼))
50	49	ralrimivw 3159	. . . . 5 ⊢ (𝜑 → ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼))
51	50	ralrimivw 3159	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼))
52		eqid 2761	. . . . 5 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))
53	52	fmpo 8068	. . . 4 ⊢ (∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼) ↔ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))):(𝐵 × 𝐵)⟶(𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼))
54	51, 53	sylib 221	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))):(𝐵 × 𝐵)⟶(𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼))
55	6	fvexi 6891	. . . . 5 ⊢ 𝐵 ∈ V
56	55, 55	xpex 7756	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
57		ovex 7445	. . . 4 ⊢ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼) ∈ V
58		fex2 7937	. . . 4 ⊢ (((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(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)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
59	56, 57, 58	mp3an23 1482	. . 3 ⊢ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))):(𝐵 × 𝐵)⟶(𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼) → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
60	54, 59	syl 18	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
61		snsstp2 4778	. . . 4 ⊢ {⟨(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩}
62		ssun1 4124	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.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 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩})
63	61, 62	sstri 3940	. . 3 ⊢ {⟨(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.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 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩})
64		ssun1 4124	. . 3 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.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 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.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 Σg (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 ∈ ((2nd ‘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)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}))
65	63, 64	sstri 3940	. 2 ⊢ {⟨(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.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 Σg (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 ∈ ((2nd ‘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)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}))
66	17, 18, 19, 60, 65	prdsbaslem 17604	1 ⊢ (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ∪ 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 5649  dom cdm 5651  ran crn 5652   ∘ ccom 5655  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7988  2^nd c2nd 7989   ↑_m cmap 8831  Xcixp 8909  supcsup 9416  0cc0 11181  ℝ^*cxr 11323   < clt 11324  ndxcnx 17351  Basecbs 17367  +_gcplusg 17408  ._rcmulr 17409  Scalarcsca 17411   ·_𝑠 cvsca 17412  ·_𝑖cip 17413  TopSetcts 17414  lecple 17415  distcds 17417  Hom chom 17419  compcco 17420  TopOpenctopn 17572  ∏_tcpt 17589   Σ_g cgsu 17591  X_scprds 17596

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 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-map 8833  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-7 12391  df-8 12392  df-9 12393  df-n0 12588  df-z 12675  df-dec 12796  df-uz 12947  df-fz 13621  df-struct 17305  df-slot 17340  df-ndx 17352  df-base 17368  df-plusg 17421  df-mulr 17422  df-sca 17424  df-vsca 17425  df-ip 17426  df-tset 17427  df-ple 17428  df-ds 17430  df-hom 17432  df-cco 17433  df-prds 17598

This theorem is used by:  prdsmulr  17610  prdsvsca  17611  prdsip  17612  prdsle  17613  prdsds  17615  prdstset  17617  prdshom  17618  prdsco  17619  prdsplusgval  17624  prdsmgp  20351