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Theorem prdsbaslemss 14119

Description: Lemma for prdsbas 14121 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 14118	. . 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 5026	. . . . . 6 ⊢ (𝑅 ∈ 𝑊 → dom 𝑅 ∈ V)
19	16, 18	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝑅 ∈ V)
20		basfn 13358	. . . . . . 7 ⊢ Base Fn V
21		vex 2818	. . . . . . . 8 ⊢ 𝑥 ∈ V
22		fvexg 5694	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑥 ∈ V) → (𝑅‘𝑥) ∈ V)
23	16, 21, 22	sylancl 413	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝑥) ∈ V)
24		funfvex 5692	. . . . . . . 8 ⊢ ((Fun Base ∧ (𝑅‘𝑥) ∈ dom Base) → (Base‘(𝑅‘𝑥)) ∈ V)
25	24	funfni 5463	. . . . . . 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 6970	. . . . 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 6421	. . . . 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 6421	. . . . 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 5692	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
36	35	funfni 5463	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
37	20, 34, 36	sylancr 414	. . . . 5 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
38		mpoexga 6421	. . . . 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 6421	. . . . 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 13544	. . . . . . 7 ⊢ TopOpen Fn V
43		fnfun 5458	. . . . . . 7 ⊢ (TopOpen Fn V → Fun TopOpen)
44	42, 43	ax-mp 5	. . . . . 6 ⊢ Fun TopOpen
45		cofunexg 6311	. . . . . 6 ⊢ ((Fun TopOpen ∧ 𝑅 ∈ 𝑊) → (TopOpen ∘ 𝑅) ∈ V)
46	44, 16, 45	sylancr 414	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅) ∈ V)
47		ptex 13564	. . . . 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 3855	. . . . . . 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 4182	. . . . 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 4869	. . . . . . 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 4829	. . . . . . 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 4255	. . . . 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 6421	. . . . 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 6421	. . . . 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 6421	. . . . 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 13566	. . 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 4136	. 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 13324	. 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 13345	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 3694 {cpr 3695 {ctp 3696 ⟨cop 3697 class class class wbr 4114 {copab 4175 ↦ cmpt 4176 × cxp 4752 dom cdm 4754 ran crn 4755 ∘ ccom 4758 Fun wfun 5351 Fn wfn 5352 ‘cfv 5357 (class class class)co 6058 ∈ cmpo 6060 1^st c1st 6345 2^nd c2nd 6346 Xcixp 6946 supcsup 7286 0cc0 8143 1c1 8144 ℝ^*cxr 8323 < clt 8324 ℕcn 9257 5c5 9311 cdc 9730 Struct cstr 13295 ndxcnx 13296 Slot cslot 13298 Basecbs 13299 +_gcplusg 13377 ._rcmulr 13378 Scalarcsca 13380 ·_𝑠 cvsca 13381 ·_𝑖cip 13382 TopSetcts 13383 lecple 13384 distcds 13386 Hom chom 13388 compcco 13389 TopOpenctopn 13540 ∏_tcpt 13555 Σ_g cgsu 13557 X_scprds 14114

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-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259

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 3676 df-sn 3700 df-pr 3701 df-tp 3702 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-map 6897 df-ixp 6947 df-sup 7288 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-5 9319 df-6 9320 df-7 9321 df-8 9322 df-9 9323 df-n0 9517 df-z 9598 df-dec 9731 df-uz 9875 df-fz 10365 df-struct 13301 df-ndx 13302 df-slot 13303 df-base 13305 df-plusg 13390 df-mulr 13391 df-sca 13393 df-vsca 13394 df-ip 13395 df-tset 13396 df-ple 13397 df-ds 13399 df-hom 13401 df-cco 13402 df-rest 13541 df-topn 13542 df-topgen 13560 df-pt 13561 df-prds 14115

This theorem is referenced by: prdssca 14120 prdsbas 14121 prdsplusg 14122 prdsmulr 14123

W3C validator