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Theorem prdsbaslemss 14122

Description: Lemma for prdsbas 14124 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 2235	. 2 ⊢ (𝜑 → 𝑃 = 𝑃)
2		prdsbaslemss.p	. . . 4 ⊢ 𝑃 = (𝑆X_s𝑅)
3		eqid 2234	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
4		eqidd 2235	. . . 4 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
5		eqidd 2235	. . . 4 ⊢ (𝜑 → X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) = X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)))
6		eqidd 2235	. . . 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 2235	. . . 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 2235	. . . 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 2235	. . . 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 2235	. . . 4 ⊢ (𝜑 → (∏_t‘(TopOpen ∘ 𝑅)) = (∏_t‘(TopOpen ∘ 𝑅)))
11		eqidd 2235	. . . 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 2235	. . . 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 2235	. . . 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 2235	. . . 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 14121	. . 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 5028	. . . . . 6 ⊢ (𝑅 ∈ 𝑊 → dom 𝑅 ∈ V)
19	16, 18	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝑅 ∈ V)
20		basfn 13361	. . . . . . 7 ⊢ Base Fn V
21		vex 2818	. . . . . . . 8 ⊢ 𝑥 ∈ V
22		fvexg 5696	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑥 ∈ V) → (𝑅‘𝑥) ∈ V)
23	16, 21, 22	sylancl 413	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝑥) ∈ V)
24		funfvex 5694	. . . . . . . 8 ⊢ ((Fun Base ∧ (𝑅‘𝑥) ∈ dom Base) → (Base‘(𝑅‘𝑥)) ∈ V)
25	24	funfni 5465	. . . . . . 7 ⊢ ((Base Fn V ∧ (𝑅‘𝑥) ∈ V) → (Base‘(𝑅‘𝑥)) ∈ V)
26	20, 23, 25	sylancr 414	. . . . . 6 ⊢ (𝜑 → (Base‘(𝑅‘𝑥)) ∈ V)
27	26	ralrimivw 2618	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V)
28		ixpexgg 6972	. . . . 5 ⊢ ((dom 𝑅 ∈ V ∧ ∀𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V) → X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V)
29	19, 27, 28	syl2anc 411	. . . 4 ⊢ (𝜑 → X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V)
30		mpoexga 6423	. . . . 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 411	. . . 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 6423	. . . . 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 411	. . . 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 2829	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
35		funfvex 5694	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
36	35	funfni 5465	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
37	20, 34, 36	sylancr 414	. . . . 5 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
38		mpoexga 6423	. . . . 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 411	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) ∈ V)
40		mpoexga 6423	. . . . 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 411	. . . 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 13547	. . . . . . 7 ⊢ TopOpen Fn V
43		fnfun 5460	. . . . . . 7 ⊢ (TopOpen Fn V → Fun TopOpen)
44	42, 43	ax-mp 5	. . . . . 6 ⊢ Fun TopOpen
45		cofunexg 6313	. . . . . 6 ⊢ ((Fun TopOpen ∧ 𝑅 ∈ 𝑊) → (TopOpen ∘ 𝑅) ∈ V)
46	44, 16, 45	sylancr 414	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅) ∈ V)
47		ptex 13567	. . . . 5 ⊢ ((TopOpen ∘ 𝑅) ∈ V → (∏_t‘(TopOpen ∘ 𝑅)) ∈ V)
48	46, 47	syl 14	. . . 4 ⊢ (𝜑 → (∏_t‘(TopOpen ∘ 𝑅)) ∈ V)
49		vex 2818	. . . . . . . 8 ⊢ 𝑓 ∈ V
50		vex 2818	. . . . . . . 8 ⊢ 𝑔 ∈ V
51	49, 50	prss 3856	. . . . . . 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 458	. . . . . 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 4183	. . . . 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 4871	. . . . . . 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 411	. . . . . 6 ⊢ (𝜑 → (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))) ∈ V)
56		opabssxp 4831	. . . . . . 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 4256	. . . . 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 2322	. . . 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 6423	. . . . 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 411	. . . 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 6423	. . . . 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 411	. . . 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 6423	. . . . 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 411	. . . 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 13569	. . 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 4137	. 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 13327	. 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 13348	1 ⊢ (𝜑 → 𝐴 = 𝑇)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1398 ∈ wcel 2205 ∀wral 2522 Vcvv 2815 ∪ cun 3212 ⊆ wss 3214 {csn 3695 {cpr 3696 {ctp 3697 ⟨cop 3698 class class class wbr 4115 {copab 4176 ↦ cmpt 4177 × cxp 4754 dom cdm 4756 ran crn 4757 ∘ ccom 4760 Fun wfun 5353 Fn wfn 5354 ‘cfv 5359 (class class class)co 6060 ∈ cmpo 6062 1^st c1st 6347 2^nd c2nd 6348 Xcixp 6948 supcsup 7288 0cc0 8145 1c1 8146 ℝ^*cxr 8325 < clt 8326 ℕcn 9259 5c5 9313 cdc 9732 Struct cstr 13298 ndxcnx 13299 Slot cslot 13301 Basecbs 13302 +_gcplusg 13380 ._rcmulr 13381 Scalarcsca 13383 ·_𝑠 cvsca 13384 ·_𝑖cip 13385 TopSetcts 13386 lecple 13387 distcds 13389 Hom chom 13391 compcco 13392 TopOpenctopn 13543 ∏_tcpt 13558 Σ_g cgsu 13560 X_scprds 14117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2208 ax-ext 2216 ax-coll 4231 ax-sep 4234 ax-pow 4293 ax-pr 4328 ax-un 4560 ax-setind 4666 ax-cnex 8236 ax-resscn 8237 ax-1cn 8238 ax-1re 8239 ax-icn 8240 ax-addcl 8241 ax-addrcl 8242 ax-mulcl 8243 ax-addcom 8245 ax-mulcom 8246 ax-addass 8247 ax-mulass 8248 ax-distr 8249 ax-i2m1 8250 ax-0lt1 8251 ax-1rid 8252 ax-0id 8253 ax-rnegex 8254 ax-cnre 8256 ax-pre-ltirr 8257 ax-pre-ltwlin 8258 ax-pre-lttrn 8259 ax-pre-apti 8260 ax-pre-ltadd 8261

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-pw 3677 df-sn 3701 df-pr 3702 df-tp 3703 df-op 3704 df-uni 3921 df-int 3956 df-iun 3999 df-br 4116 df-opab 4178 df-mpt 4179 df-id 4420 df-xp 4762 df-rel 4763 df-cnv 4764 df-co 4765 df-dm 4766 df-rn 4767 df-res 4768 df-ima 4769 df-iota 5319 df-fun 5361 df-fn 5362 df-f 5363 df-f1 5364 df-fo 5365 df-f1o 5366 df-fv 5367 df-riota 6013 df-ov 6063 df-oprab 6064 df-mpo 6065 df-1st 6349 df-2nd 6350 df-map 6899 df-ixp 6949 df-sup 7290 df-pnf 8328 df-mnf 8329 df-xr 8330 df-ltxr 8331 df-le 8332 df-sub 8465 df-neg 8466 df-inn 9260 df-2 9318 df-3 9319 df-4 9320 df-5 9321 df-6 9322 df-7 9323 df-8 9324 df-9 9325 df-n0 9519 df-z 9600 df-dec 9733 df-uz 9877 df-fz 10367 df-struct 13304 df-ndx 13305 df-slot 13306 df-base 13308 df-plusg 13393 df-mulr 13394 df-sca 13396 df-vsca 13397 df-ip 13398 df-tset 13399 df-ple 13400 df-ds 13402 df-hom 13404 df-cco 13405 df-rest 13544 df-topn 13545 df-topgen 13563 df-pt 13564 df-prds 14118

This theorem is referenced by: prdssca 14123 prdsbas 14124 prdsplusg 14125 prdsmulr 14126

W3C validator