Theorem prdsmulr 17639

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 = (𝑆Xs𝑅)
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 = (𝑆Xs𝑅)
2		eqid 2760	. . 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 17637	. . 3 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
8		eqid 2760	. . . 4 ⊢ (+g‘𝑃) = (+g‘𝑃)
9	1, 4, 5, 6, 3, 8	prdsplusg 17638	. . 3 ⊢ (𝜑 → (+g‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
10		eqidd 2761	. . 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 2761	. . 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 2761	. . 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))))))
13		eqidd 2761	. . 3 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
14		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))})
15		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*, < )))
16		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))))
17		eqidd 2761	. . 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))))))
18	1, 2, 3, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 4, 5	prdsval 17635	. 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 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)))))⟩})))
19		prdsmulr.t	. 2 ⊢ · = (.r‘𝑃)
20		mulridx 17475	. 2 ⊢ .r = Slot (.r‘ndx)
21		ovssunirn 7460	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (.r‘(𝑅‘𝑥))
22	20	strfvss 17374	. . . . . . . . . . . . 13 ⊢ (.r‘(𝑅‘𝑥)) ⊆ ∪ ran (𝑅‘𝑥)
23		fvssunirn 6921	. . . . . . . . . . . . . 14 ⊢ (𝑅‘𝑥) ⊆ ∪ ran 𝑅
24		rnss 5922	. . . . . . . . . . . . . 14 ⊢ ((𝑅‘𝑥) ⊆ ∪ ran 𝑅 → ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅)
25		uniss 4874	. . . . . . . . . . . . . 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 3918	. . . . . . . . . . . 12 ⊢ (.r‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅
28		rnss 5922	. . . . . . . . . . . 12 ⊢ ((.r‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅 → ran (.r‘(𝑅‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑅)
29		uniss 4874	. . . . . . . . . . . 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 3918	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
32		ovex 7457	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)) ∈ V
33	32	elpw 4560	. . . . . . . . . 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 234	. . . . . . . . 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 7120	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))):𝐼⟶𝒫 ∪ ran ∪ ran ∪ ran 𝑅)
37		rnexg 7918	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑊 → ran 𝑅 ∈ V)
38		uniexg 7761	. . . . . . . . . . . 12 ⊢ (ran 𝑅 ∈ V → ∪ ran 𝑅 ∈ V)
39	5, 37, 38	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ ran 𝑅 ∈ V)
40		rnexg 7918	. . . . . . . . . . 11 ⊢ (∪ ran 𝑅 ∈ V → ran ∪ ran 𝑅 ∈ V)
41		uniexg 7761	. . . . . . . . . . 11 ⊢ (ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran 𝑅 ∈ V)
42	39, 40, 41	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran 𝑅 ∈ V)
43		rnexg 7918	. . . . . . . . . 10 ⊢ (∪ ran ∪ ran 𝑅 ∈ V → ran ∪ ran ∪ ran 𝑅 ∈ V)
44		uniexg 7761	. . . . . . . . . 10 ⊢ (ran ∪ ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
45	42, 43, 44	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
46	45	pwexd 5341	. . . . . . . 8 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ∈ V)
47	5	dmexd 7919	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑅 ∈ V)
48	3, 47	eqeltrrd 2861	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V)
49	46, 48	elmapd 8868	. . . . . . 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 260	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼))
51	50	ralrimivw 3158	. . . . 5 ⊢ (𝜑 → ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼))
52	51	ralrimivw 3158	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼))
53		eqid 2760	. . . . 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 8084	. . . 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 221	. . 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 7771	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
58		ovex 7457	. . . 4 ⊢ (𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼) ∈ V
59		fex2 7952	. . . 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 1482	. . 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 18	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
62		snsstp3 4773	. . . 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 4175	. . . 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 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩})
64	62, 63	sstri 3918	. . 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 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩})
65		ssun1 4175	. . 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 Σg (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 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	64, 65	sstri 3918	. 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 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)))))⟩}))
67	18, 19, 20, 61, 66	prdsbaslem 17633	1 ⊢ (𝜑 → · = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3896   ∪ cun 4148  𝒫 cpw 4556  {csn 4584  {cpr 4586  {ctp 4588  ⟨cop 4834  ∪ cuni 4866   class class class wbr 5102  {copab 5166   ↦ cmpt 5185   × cxp 5649  dom cdm 5651  ran crn 5652   ∘ ccom 5655  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426  1^st c1st 8003  2^nd c2nd 8004   ↑_m cmap 8855  Xcixp 8933  supcsup 9439  0cc0 11207  ℝ^*cxr 11349   < clt 11350  ndxcnx 17380  Basecbs 17396  +_gcplusg 17437  ._rcmulr 17438  Scalarcsca 17440   ·_𝑠 cvsca 17441  ·_𝑖cip 17442  TopSetcts 17443  lecple 17444  distcds 17446  Hom chom 17448  compcco 17449  TopOpenctopn 17601  ∏_tcpt 17618   Σ_g cgsu 17620  X_scprds 17625

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  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 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-er 8725  df-map 8857  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-sup 9441  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-3 12413  df-4 12414  df-5 12415  df-6 12416  df-7 12417  df-8 12418  df-9 12419  df-n0 12614  df-z 12701  df-dec 12822  df-uz 12973  df-fz 13647  df-struct 17334  df-slot 17369  df-ndx 17381  df-base 17397  df-plusg 17450  df-mulr 17451  df-sca 17453  df-vsca 17454  df-ip 17455  df-tset 17456  df-ple 17457  df-ds 17459  df-hom 17461  df-cco 17462  df-prds 17627

This theorem is used by:  prdsvsca  17640  prdsle  17642  prdsds  17644  prdstset  17646  prdshom  17647  prdsco  17648  prdsmulrval  17655  prdsmgp  20380