Theorem prdsval 15938

Description: Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)

Hypotheses

Ref	Expression
prdsval.p	⊢ 𝑃 = (𝑆Xs𝑅)
prdsval.k	⊢ 𝐾 = (Base‘𝑆)
prdsval.i	⊢ (𝜑 → dom 𝑅 = 𝐼)
prdsval.b	⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
prdsval.a	⊢ (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
prdsval.t	⊢ (𝜑 → × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
prdsval.m	⊢ (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
prdsval.j	⊢ (𝜑 → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))
prdsval.o	⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
prdsval.l	⊢ (𝜑 → ≤ = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
prdsval.d	⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )))
prdsval.h	⊢ (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))
prdsval.x	⊢ (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd ‘𝑎)), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))
prdsval.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
prdsval.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)

Assertion

Ref	Expression
prdsval	⊢ (𝜑 → 𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})))

Distinct variable groups:   𝑎,𝑐,𝑑,𝑒,𝑓,𝑔,𝐵   𝐻,𝑎,𝑐,𝑑,𝑒   𝑥,𝑎,𝜑,𝑐,𝑑,𝑒,𝑓,𝑔   𝑥,𝐼   𝑅,𝑎,𝑐,𝑑,𝑒,𝑓,𝑔,𝑥   𝑆,𝑎,𝑐,𝑑,𝑒,𝑓,𝑔,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝑃(𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   + (𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   ∙ (𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   · (𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   × (𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝐻(𝑥,𝑓,𝑔)   , (𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝐼(𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝐾(𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   ≤ (𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝑂(𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝑊(𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝑍(𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)

Proof of Theorem prdsval

Dummy variables ℎ 𝑟 𝑠 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsval.p	. 2 ⊢ 𝑃 = (𝑆Xs𝑅)
2		df-prds 15931	. . . 4 ⊢ Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
3	2	a1i 11	. . 3 ⊢ (𝜑 → Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}))))
4		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑟 ∈ V
5	4	rnex 6992	. . . . . . . . . . 11 ⊢ ran 𝑟 ∈ V
6	5	uniex 6851	. . . . . . . . . 10 ⊢ ∪ ran 𝑟 ∈ V
7	6	rnex 6992	. . . . . . . . 9 ⊢ ran ∪ ran 𝑟 ∈ V
8	7	uniex 6851	. . . . . . . 8 ⊢ ∪ ran ∪ ran 𝑟 ∈ V
9		baseid 15747	. . . . . . . . . . . 12 ⊢ Base = Slot (Base‘ndx)
10	9	strfvss 15713	. . . . . . . . . . 11 ⊢ (Base‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥)
11		fvssunirn 6127	. . . . . . . . . . . 12 ⊢ (𝑟‘𝑥) ⊆ ∪ ran 𝑟
12		rnss 5275	. . . . . . . . . . . 12 ⊢ ((𝑟‘𝑥) ⊆ ∪ ran 𝑟 → ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟)
13		uniss 4394	. . . . . . . . . . . 12 ⊢ (ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟 → ∪ ran (𝑟‘𝑥) ⊆ ∪ ran ∪ ran 𝑟)
14	11, 12, 13	mp2b 10	. . . . . . . . . . 11 ⊢ ∪ ran (𝑟‘𝑥) ⊆ ∪ ran ∪ ran 𝑟
15	10, 14	sstri 3577	. . . . . . . . . 10 ⊢ (Base‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟
16	15	rgenw 2908	. . . . . . . . 9 ⊢ ∀𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟
17		iunss 4497	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 ↔ ∀𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟)
18	16, 17	mpbir 220	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟
19	8, 18	ssexi 4731	. . . . . . 7 ⊢ ∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∈ V
20		ixpssmap2g 7823	. . . . . . 7 ⊢ (∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∈ V → X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ (∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ↑𝑚 dom 𝑟))
21	19, 20	ax-mp 5	. . . . . 6 ⊢ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ (∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ↑𝑚 dom 𝑟)
22		ovex 6577	. . . . . . 7 ⊢ (∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ↑𝑚 dom 𝑟) ∈ V
23	22	ssex 4730	. . . . . 6 ⊢ (X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ (∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ↑𝑚 dom 𝑟) → X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∈ V)
24	21, 23	mp1i 13	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∈ V)
25		simpr 476	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
26	25	fveq1d 6105	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑟‘𝑥) = (𝑅‘𝑥))
27	26	fveq2d 6107	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (Base‘(𝑟‘𝑥)) = (Base‘(𝑅‘𝑥)))
28	27	ixpeq2dv 7810	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ 𝐼 (Base‘(𝑟‘𝑥)) = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
29	25	dmeqd 5248	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → dom 𝑟 = dom 𝑅)
30		prdsval.i	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑅 = 𝐼)
31	30	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → dom 𝑅 = 𝐼)
32	29, 31	eqtrd 2644	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → dom 𝑟 = 𝐼)
33	32	ixpeq1d 7806	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) = X𝑥 ∈ 𝐼 (Base‘(𝑟‘𝑥)))
34		prdsval.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
35	34	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
36	28, 33, 35	3eqtr4d 2654	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) = 𝐵)
37		ovssunirn 6579	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (Hom ‘(𝑟‘𝑥))
38		df-hom 15793	. . . . . . . . . . . . . . . . . 18 ⊢ Hom = Slot ;14
39	38	strfvss 15713	. . . . . . . . . . . . . . . . 17 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥)
40	39, 14	sstri 3577	. . . . . . . . . . . . . . . 16 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟
41		rnss 5275	. . . . . . . . . . . . . . . 16 ⊢ ((Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 → ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟)
42		uniss 4394	. . . . . . . . . . . . . . . 16 ⊢ (ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟 → ∪ ran (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟)
43	40, 41, 42	mp2b 10	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
44	37, 43	sstri 3577	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
45	44	rgenw 2908	. . . . . . . . . . . . 13 ⊢ ∀𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
46		ss2ixp 7807	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟 → X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟)
47	45, 46	ax-mp 5	. . . . . . . . . . . 12 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟
48	4	dmex 6991	. . . . . . . . . . . . 13 ⊢ dom 𝑟 ∈ V
49	8	rnex 6992	. . . . . . . . . . . . . 14 ⊢ ran ∪ ran ∪ ran 𝑟 ∈ V
50	49	uniex 6851	. . . . . . . . . . . . 13 ⊢ ∪ ran ∪ ran ∪ ran 𝑟 ∈ V
51	48, 50	ixpconst 7804	. . . . . . . . . . . 12 ⊢ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟 = (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
52	47, 51	sseqtri 3600	. . . . . . . . . . 11 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
53		ovex 6577	. . . . . . . . . . . 12 ⊢ (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V
54	53	elpw2 4755	. . . . . . . . . . 11 ⊢ (X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ↔ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟))
55	52, 54	mpbir 220	. . . . . . . . . 10 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
56	55	rgen2w 2909	. . . . . . . . 9 ⊢ ∀𝑓 ∈ 𝑣 ∀𝑔 ∈ 𝑣 X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
57		eqid 2610	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)))
58	57	fmpt2 7126	. . . . . . . . 9 ⊢ (∀𝑓 ∈ 𝑣 ∀𝑔 ∈ 𝑣 X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ↔ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))):(𝑣 × 𝑣)⟶𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟))
59	56, 58	mpbi 219	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))):(𝑣 × 𝑣)⟶𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
60		vex 3176	. . . . . . . . 9 ⊢ 𝑣 ∈ V
61	60, 60	xpex 6860	. . . . . . . 8 ⊢ (𝑣 × 𝑣) ∈ V
62	53	pwex 4774	. . . . . . . 8 ⊢ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V
63		fex2 7014	. . . . . . . 8 ⊢ (((𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))):(𝑣 × 𝑣)⟶𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∧ (𝑣 × 𝑣) ∈ V ∧ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V)
64	59, 61, 62, 63	mp3an 1416	. . . . . . 7 ⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V
65	64	a1i 11	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V)
66		simpr 476	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
67	32	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → dom 𝑟 = 𝐼)
68	67	ixpeq1d 7806	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) = X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)))
69	26	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (Hom ‘(𝑟‘𝑥)) = (Hom ‘(𝑅‘𝑥)))
70	69	oveqd 6566	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))
71	70	ixpeq2dv 7810	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) = X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))
72	71	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) = X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))
73	68, 72	eqtrd 2644	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) = X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))
74	66, 66, 73	mpt2eq123dv 6615	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))
75		prdsval.h	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))
76	75	ad3antrrr 762	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))
77	74, 76	eqtr4d 2647	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) = 𝐻)
78		simplr 788	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 𝑣 = 𝐵)
79	78	opeq2d 4347	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ⟨(Base‘ndx), 𝑣⟩ = ⟨(Base‘ndx), 𝐵⟩)
80	26	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (+g‘(𝑟‘𝑥)) = (+g‘(𝑅‘𝑥)))
81	80	oveqd 6566	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))
82	32, 81	mpteq12dv 4663	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))
83	82	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))
84	66, 66, 83	mpt2eq123dv 6615	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
85	84	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
86		prdsval.a	. . . . . . . . . . . 12 ⊢ (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
87	86	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
88	85, 87	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥)))) = + )
89	88	opeq2d 4347	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩ = ⟨(+g‘ndx), + ⟩)
90	26	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (.r‘(𝑟‘𝑥)) = (.r‘(𝑅‘𝑥)))
91	90	oveqd 6566	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))
92	32, 91	mpteq12dv 4663	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))
93	92	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))
94	66, 66, 93	mpt2eq123dv 6615	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
95	94	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
96		prdsval.t	. . . . . . . . . . . 12 ⊢ (𝜑 → × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
97	96	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
98	95, 97	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥)))) = × )
99	98	opeq2d 4347	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩ = ⟨(.r‘ndx), × ⟩)
100	79, 89, 99	tpeq123d 4227	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → {⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩})
101		simp-4r 803	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 𝑠 = 𝑆)
102	101	opeq2d 4347	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ⟨(Scalar‘ndx), 𝑠⟩ = ⟨(Scalar‘ndx), 𝑆⟩)
103		simpllr 795	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → 𝑠 = 𝑆)
104	103	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (Base‘𝑠) = (Base‘𝑆))
105		prdsval.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (Base‘𝑆)
106	104, 105	syl6eqr 2662	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (Base‘𝑠) = 𝐾)
107	26	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ( ·𝑠 ‘(𝑟‘𝑥)) = ( ·𝑠 ‘(𝑅‘𝑥)))
108	107	oveqd 6566	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥)) = (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))
109	32, 108	mpteq12dv 4663	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))
110	109	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))
111	106, 66, 110	mpt2eq123dv 6615	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
112	111	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
113		prdsval.m	. . . . . . . . . . . 12 ⊢ (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
114	113	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
115	112, 114	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥)))) = · )
116	115	opeq2d 4347	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
117	26	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (·𝑖‘(𝑟‘𝑥)) = (·𝑖‘(𝑅‘𝑥)))
118	117	oveqd 6566	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))
119	32, 118	mpteq12dv 4663	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))
120	119	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))
121	103, 120	oveq12d 6567	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))
122	66, 66, 121	mpt2eq123dv 6615	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥))))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))
123	122	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥))))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))
124		prdsval.j	. . . . . . . . . . . 12 ⊢ (𝜑 → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))
125	124	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))
126	123, 125	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥))))) = , )
127	126	opeq2d 4347	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩ = ⟨(·𝑖‘ndx), , ⟩)
128	102, 116, 127	tpeq123d 4227	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩} = {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})
129	100, 128	uneq12d 3730	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
130		simpllr 795	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 𝑟 = 𝑅)
131	130	coeq2d 5206	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (TopOpen ∘ 𝑟) = (TopOpen ∘ 𝑅))
132	131	fveq2d 6107	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (∏t‘(TopOpen ∘ 𝑟)) = (∏t‘(TopOpen ∘ 𝑅)))
133		prdsval.o	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
134	133	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
135	132, 134	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (∏t‘(TopOpen ∘ 𝑟)) = 𝑂)
136	135	opeq2d 4347	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩ = ⟨(TopSet‘ndx), 𝑂⟩)
137	66	sseq2d 3596	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → ({𝑓, 𝑔} ⊆ 𝑣 ↔ {𝑓, 𝑔} ⊆ 𝐵))
138	26	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (le‘(𝑟‘𝑥)) = (le‘(𝑅‘𝑥)))
139	138	breqd 4594	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥) ↔ (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥)))
140	32, 139	raleqbidv 3129	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥) ↔ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥)))
141	140	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥) ↔ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥)))
142	137, 141	anbi12d 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥)) ↔ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))))
143	142	opabbidv 4648	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
144	143	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
145		prdsval.l	. . . . . . . . . . . 12 ⊢ (𝜑 → ≤ = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
146	145	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ≤ = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
147	144, 146	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))} = ≤ )
148	147	opeq2d 4347	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩ = ⟨(le‘ndx), ≤ ⟩)
149	26	fveq2d 6107	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (dist‘(𝑟‘𝑥)) = (dist‘(𝑅‘𝑥)))
150	149	oveqd 6566	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥)))
151	32, 150	mpteq12dv 4663	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))))
152	151	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))))
153	152	rneqd 5274	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) = ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))))
154	153	uneq1d 3728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}) = (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}))
155	154	supeq1d 8235	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))
156	66, 66, 155	mpt2eq123dv 6615	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((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*, < )))
157	156	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((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*, < )))
158		prdsval.d	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )))
159	158	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )))
160	157, 159	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )) = 𝐷)
161	160	opeq2d 4347	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩ = ⟨(dist‘ndx), 𝐷⟩)
162	136, 148, 161	tpeq123d 4227	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → {⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} = {⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})
163		simpr 476	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ℎ = 𝐻)
164	163	opeq2d 4347	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ⟨(Hom ‘ndx), ℎ⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
165	78	sqxpeqd 5065	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑣 × 𝑣) = (𝐵 × 𝐵))
166	163	oveqd 6566	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑐ℎ(2nd ‘𝑎)) = (𝑐𝐻(2nd ‘𝑎)))
167	163	fveq1d 6105	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (ℎ‘𝑎) = (𝐻‘𝑎))
168	26	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (comp‘(𝑟‘𝑥)) = (comp‘(𝑅‘𝑥)))
169	168	oveqd 6566	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥)) = (⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥)))
170	169	oveqd 6566	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)) = ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))
171	32, 170	mpteq12dv 4663	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))
172	171	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))
173	166, 167, 172	mpt2eq123dv 6615	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))) = (𝑑 ∈ (𝑐𝐻(2nd ‘𝑎)), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))
174	165, 78, 173	mpt2eq123dv 6615	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd ‘𝑎)), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))
175		prdsval.x	. . . . . . . . . . . 12 ⊢ (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd ‘𝑎)), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))
176	175	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd ‘𝑎)), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))
177	174, 176	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) = ∙ )
178	177	opeq2d 4347	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩ = ⟨(comp‘ndx), ∙ ⟩)
179	164, 178	preq12d 4220	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩} = {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})
180	162, 179	uneq12d 3730	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}) = ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩}))
181	129, 180	uneq12d 3730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})))
182	65, 77, 181	csbied2 3527	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → ⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})))
183	24, 36, 182	csbied2 3527	. . . 4 ⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ⦋X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})))
184	183	anasss 677	. . 3 ⊢ ((𝜑 ∧ (𝑠 = 𝑆 ∧ 𝑟 = 𝑅)) → ⦋X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})))
185		prdsval.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
186		elex 3185	. . . 4 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
187	185, 186	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
188		prdsval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
189		elex 3185	. . . 4 ⊢ (𝑅 ∈ 𝑍 → 𝑅 ∈ V)
190	188, 189	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
191		tpex 6855	. . . . . 6 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∈ V
192		tpex 6855	. . . . . 6 ⊢ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩} ∈ V
193	191, 192	unex 6854	. . . . 5 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∈ V
194		tpex 6855	. . . . . 6 ⊢ {⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∈ V
195		prex 4836	. . . . . 6 ⊢ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩} ∈ V
196	194, 195	unex 6854	. . . . 5 ⊢ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩}) ∈ V
197	193, 196	unex 6854	. . . 4 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})) ∈ V
198	197	a1i 11	. . 3 ⊢ (𝜑 → (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})) ∈ V)
199	3, 184, 187, 190, 198	ovmpt2d 6686	. 2 ⊢ (𝜑 → (𝑆Xs𝑅) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})))
200	1, 199	syl5eq 2656	1 ⊢ (𝜑 → 𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⦋csb 3499   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  {cpr 4127  {ctp 4129  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  Xcixp 7794  supcsup 8229  0cc0 9815  1c1 9816  ℝ^*cxr 9952   < clt 9953  4c4 10949  ;cdc 11369  ndxcnx 15692  Basecbs 15695  +_gcplusg 15768  ._rcmulr 15769  Scalarcsca 15771   ·_𝑠 cvsca 15772  ·_𝑖cip 15773  TopSetcts 15774  lecple 15775  distcds 15777  Hom chom 15779  compcco 15780  TopOpenctopn 15905  ∏_tcpt 15922   Σ_g cgsu 15924  X_scprds 15929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-map 7746  df-ixp 7795  df-sup 8231  df-nn 10898  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-prds 15931

This theorem is referenced by:  prdssca  15939  prdsbas  15940  prdsplusg  15941  prdsmulr  15942  prdsvsca  15943  prdsip  15944  prdsle  15945  prdsds  15947  prdstset  15949  prdshom  15950  prdsco  15951