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Theorem prdsbaslemss 14176

Description: Lemma for prdsbas 14178 and similar theorems. (Contributed by Jim Kingdon, 10-Nov-2025.)

**Hypotheses**
Ref	Expression
prdsbaslemss.p	⊢ 𝑃 = (𝑆X_s𝑅)
prdsbaslemss.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdsbaslemss.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)
prdsbaslem.1	⊢ 𝐴 = (𝐸‘𝑃)
prdsbaslem.2	⊢ 𝐸 = Slot (𝐸‘ndx)
prdsbaslemss.e	⊢ (𝐸‘ndx) ∈ ℕ
prdsbaslem.3	⊢ (𝜑 → 𝑇 ∈ 𝑋)
prdsbaslemss.ss	⊢ (𝜑 → {⟨(𝐸‘ndx), 𝑇⟩} ⊆ 𝑃)

**Assertion**
Ref	Expression
prdsbaslemss	⊢ (𝜑 → 𝐴 = 𝑇)

Proof of Theorem prdsbaslemss Dummy variables 𝑎 𝑐 𝑑 𝑒 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2239	. 2 ⊢ (𝜑 → 𝑃 = 𝑃)
2		prdsbaslemss.p	. . . 4 ⊢ 𝑃 = (𝑆X_s𝑅)
3		eqid 2238	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
4		eqidd 2239	. . . 4 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
5		eqidd 2239	. . . 4 ⊢ (𝜑 → X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) = X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)))
6		eqidd 2239	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+_g‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+_g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
7		eqidd 2239	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
8		eqidd 2239	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
9		eqidd 2239	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σ_g (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))) = (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σ_g (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))
10		eqidd 2239	. . . 4 ⊢ (𝜑 → (∏_t‘(TopOpen ∘ 𝑅)) = (∏_t‘(TopOpen ∘ 𝑅)))
11		eqidd 2239	. . . 4 ⊢ (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
12		eqidd 2239	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^, < )) = (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^, < )))
13		eqidd 2239	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) = (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))
14		eqidd 2239	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) = (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))
15		prdsbaslemss.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
16		prdsbaslemss.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
17	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	prdsval 14175	. . 3 ⊢ (𝜑 → 𝑃 = (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))⟩, ⟨(+_g‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+_g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(._r‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·_𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σ_g (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏_t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
18		dmexg 5046	. . . . . 6 ⊢ (𝑅 ∈ 𝑊 → dom 𝑅 ∈ V)
19	16, 18	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝑅 ∈ V)
20		basfn 13413	. . . . . . 7 ⊢ Base Fn V
21		vex 2824	. . . . . . . 8 ⊢ 𝑥 ∈ V
22		fvexg 5714	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑥 ∈ V) → (𝑅‘𝑥) ∈ V)
23	16, 21, 22	sylancl 417	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝑥) ∈ V)
24		funfvex 5712	. . . . . . . 8 ⊢ ((Fun Base ∧ (𝑅‘𝑥) ∈ dom Base) → (Base‘(𝑅‘𝑥)) ∈ V)
25	24	funfni 5483	. . . . . . 7 ⊢ ((Base Fn V ∧ (𝑅‘𝑥) ∈ V) → (Base‘(𝑅‘𝑥)) ∈ V)
26	20, 23, 25	sylancr 418	. . . . . 6 ⊢ (𝜑 → (Base‘(𝑅‘𝑥)) ∈ V)
27	26	ralrimivw 2624	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V)
28		ixpexgg 7004	. . . . 5 ⊢ ((dom 𝑅 ∈ V ∧ ∀𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V) → X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V)
29	19, 27, 28	syl2anc 415	. . . 4 ⊢ (𝜑 → X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V)
30		mpoexga 6448	. . . . 5 ⊢ ((X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V ∧ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V) → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+_g‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
31	29, 29, 30	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+_g‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
32		mpoexga 6448	. . . . 5 ⊢ ((X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V ∧ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V) → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
33	29, 29, 32	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
34	15	elexd 2835	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
35		funfvex 5712	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
36	35	funfni 5483	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
37	20, 34, 36	sylancr 418	. . . . 5 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
38		mpoexga 6448	. . . . 5 ⊢ (((Base‘𝑆) ∈ V ∧ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V) → (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
39	37, 29, 38	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
40		mpoexga 6448	. . . . 5 ⊢ ((X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V ∧ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V) → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σ_g (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))) ∈ V)
41	29, 29, 40	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σ_g (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))) ∈ V)
42		topnfn 13600	. . . . . . 7 ⊢ TopOpen Fn V
43		fnfun 5478	. . . . . . 7 ⊢ (TopOpen Fn V → Fun TopOpen)
44	42, 43	ax-mp 5	. . . . . 6 ⊢ Fun TopOpen
45		cofunexg 6338	. . . . . 6 ⊢ ((Fun TopOpen ∧ 𝑅 ∈ 𝑊) → (TopOpen ∘ 𝑅) ∈ V)
46	44, 16, 45	sylancr 418	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅) ∈ V)
47		ptex 13620	. . . . 5 ⊢ ((TopOpen ∘ 𝑅) ∈ V → (∏_t‘(TopOpen ∘ 𝑅)) ∈ V)
48	46, 47	syl 14	. . . 4 ⊢ (𝜑 → (∏_t‘(TopOpen ∘ 𝑅)) ∈ V)
49		vex 2824	. . . . . . . 8 ⊢ 𝑓 ∈ V
50		vex 2824	. . . . . . . 8 ⊢ 𝑔 ∈ V
51	49, 50	prss 3871	. . . . . . 7 ⊢ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))) ↔ {𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)))
52	51	anbi1i 462	. . . . . 6 ⊢ (((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥)) ↔ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥)))
53	52	opabbii 4198	. . . . 5 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}
54		xpexg 4889	. . . . . . 7 ⊢ ((X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V ∧ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V) → (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))) ∈ V)
55	29, 29, 54	syl2anc 415	. . . . . 6 ⊢ (𝜑 → (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))) ∈ V)
56		opabssxp 4849	. . . . . . 7 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} ⊆ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)))
57	56	a1i 9	. . . . . 6 ⊢ (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} ⊆ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))))
58	55, 57	ssexd 4273	. . . . 5 ⊢ (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} ∈ V)
59	53, 58	eqeltrrid 2326	. . . 4 ⊢ (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} ∈ V)
60		mpoexga 6448	. . . . 5 ⊢ ((X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V ∧ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V) → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < )) ∈ V)
61	29, 29, 60	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < )) ∈ V)
62		mpoexga 6448	. . . . 5 ⊢ ((X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V ∧ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V) → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ V)
63	29, 29, 62	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) ∈ V)
64		mpoexga 6448	. . . . 5 ⊢ (((X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))) ∈ V ∧ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V) → (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) ∈ V)
65	55, 29, 64	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) ∈ V)
66	29, 31, 33, 15, 39, 41, 48, 59, 61, 63, 65	prdsvalstrd 13622	. . 3 ⊢ (𝜑 → (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))⟩, ⟨(+_g‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+_g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(._r‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·_𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σ_g (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏_t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2^nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})) Struct ⟨1, ;15⟩)
67	17, 66	eqbrtrd 4152	. 2 ⊢ (𝜑 → 𝑃 Struct ⟨1, ;15⟩)
68		prdsbaslem.2	. . 3 ⊢ 𝐸 = Slot (𝐸‘ndx)
69		prdsbaslemss.e	. . 3 ⊢ (𝐸‘ndx) ∈ ℕ
70	68, 69	ndxslid 13379	. 2 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
71		prdsbaslemss.ss	. 2 ⊢ (𝜑 → {⟨(𝐸‘ndx), 𝑇⟩} ⊆ 𝑃)
72		prdsbaslem.3	. 2 ⊢ (𝜑 → 𝑇 ∈ 𝑋)
73		prdsbaslem.1	. 2 ⊢ 𝐴 = (𝐸‘𝑃)
74	1, 67, 70, 71, 72, 73	strslfv3 13400	1 ⊢ (𝜑 → 𝐴 = 𝑇)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 ∀wral 2528 Vcvv 2821 ∪ cun 3218 ⊆ wss 3220 {csn 3709 {cpr 3710 {ctp 3711 ⟨cop 3712 class class class wbr 4130 {copab 4191 ↦ cmpt 4192 × cxp 4772 dom cdm 4774 ran crn 4775 ∘ ccom 4778 Fun wfun 5371 Fn wfn 5372 ‘cfv 5377 (class class class)co 6085 ∈ cmpo 6087 1^st c1st 6372 2^nd c2nd 6373 Xcixp 6980 supcsup 7322 0cc0 8179 1c1 8180 ℝ^*cxr 8359 < clt 8360 ℕcn 9305 5c5 9359 cdc 9779 Struct cstr 13350 ndxcnx 13351 Slot cslot 13353 Basecbs 13354 +_gcplusg 13433 ._rcmulr 13434 Scalarcsca 13436 ·_𝑠 cvsca 13437 ·_𝑖cip 13438 TopSetcts 13439 lecple 13440 distcds 13442 Hom chom 13444 compcco 13445 TopOpenctopn 13596 ∏_tcpt 13611 Σ_g cgsu 14152 X_scprds 14171

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295

This proof depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-tp 3717 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-map 6924 df-ixp 6981 df-sup 7324 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-inn 9306 df-2 9364 df-3 9365 df-4 9366 df-5 9367 df-6 9368 df-7 9369 df-8 9370 df-9 9371 df-n0 9566 df-z 9647 df-dec 9780 df-uz 9924 df-fz 10414 df-struct 13356 df-ndx 13357 df-slot 13358 df-base 13360 df-plusg 13446 df-mulr 13447 df-sca 13449 df-vsca 13450 df-ip 13451 df-tset 13452 df-ple 13453 df-ds 13455 df-hom 13457 df-cco 13458 df-rest 13597 df-topn 13598 df-topgen 13616 df-pt 13617 df-prds 14172

This theorem is used by: prdssca 14177 prdsbas 14178 prdsplusg 14179 prdsmulr 14180

W3C validator