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Theorem prdsbaslemss 13315

Description: Lemma for prdsbas 13317 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 2230	. 2 ⊢ (𝜑 → 𝑃 = 𝑃)
2		prdsbaslemss.p	. . . 4 ⊢ 𝑃 = (𝑆X_s𝑅)
3		eqid 2229	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
4		eqidd 2230	. . . 4 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
5		eqidd 2230	. . . 4 ⊢ (𝜑 → X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) = X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)))
6		eqidd 2230	. . . 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 2230	. . . 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 2230	. . . 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 2230	. . . 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 2230	. . . 4 ⊢ (𝜑 → (∏_t‘(TopOpen ∘ 𝑅)) = (∏_t‘(TopOpen ∘ 𝑅)))
11		eqidd 2230	. . . 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 2230	. . . 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 2230	. . . 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 2230	. . . 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 13314	. . 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 4988	. . . . . 6 ⊢ (𝑅 ∈ 𝑊 → dom 𝑅 ∈ V)
19	16, 18	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝑅 ∈ V)
20		basfn 13099	. . . . . . 7 ⊢ Base Fn V
21		vex 2802	. . . . . . . 8 ⊢ 𝑥 ∈ V
22		fvexg 5648	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑥 ∈ V) → (𝑅‘𝑥) ∈ V)
23	16, 21, 22	sylancl 413	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝑥) ∈ V)
24		funfvex 5646	. . . . . . . 8 ⊢ ((Fun Base ∧ (𝑅‘𝑥) ∈ dom Base) → (Base‘(𝑅‘𝑥)) ∈ V)
25	24	funfni 5423	. . . . . . 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 2604	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∈ V)
28		ixpexgg 6877	. . . . 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 6364	. . . . 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 6364	. . . . 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 2813	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
35		funfvex 5646	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
36	35	funfni 5423	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
37	20, 34, 36	sylancr 414	. . . . 5 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
38		mpoexga 6364	. . . . 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 6364	. . . . 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 13285	. . . . . . 7 ⊢ TopOpen Fn V
43		fnfun 5418	. . . . . . 7 ⊢ (TopOpen Fn V → Fun TopOpen)
44	42, 43	ax-mp 5	. . . . . 6 ⊢ Fun TopOpen
45		cofunexg 6260	. . . . . 6 ⊢ ((Fun TopOpen ∧ 𝑅 ∈ 𝑊) → (TopOpen ∘ 𝑅) ∈ V)
46	44, 16, 45	sylancr 414	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅) ∈ V)
47		ptex 13305	. . . . 5 ⊢ ((TopOpen ∘ 𝑅) ∈ V → (∏_t‘(TopOpen ∘ 𝑅)) ∈ V)
48	46, 47	syl 14	. . . 4 ⊢ (𝜑 → (∏_t‘(TopOpen ∘ 𝑅)) ∈ V)
49		vex 2802	. . . . . . . 8 ⊢ 𝑓 ∈ V
50		vex 2802	. . . . . . . 8 ⊢ 𝑔 ∈ V
51	49, 50	prss 3824	. . . . . . 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 4151	. . . . 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 4833	. . . . . . 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 4793	. . . . . . 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 4224	. . . . 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 2317	. . . 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 6364	. . . . 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 6364	. . . . 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 6364	. . . . 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 13312	. . 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 4105	. 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 13065	. 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 13086	1 ⊢ (𝜑 → 𝐴 = 𝑇)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 ∀wral 2508 Vcvv 2799 ∪ cun 3195 ⊆ wss 3197 {csn 3666 {cpr 3667 {ctp 3668 ⟨cop 3669 class class class wbr 4083 {copab 4144 ↦ cmpt 4145 × cxp 4717 dom cdm 4719 ran crn 4720 ∘ ccom 4723 Fun wfun 5312 Fn wfn 5313 ‘cfv 5318 (class class class)co 6007 ∈ cmpo 6009 1^st c1st 6290 2^nd c2nd 6291 Xcixp 6853 supcsup 7157 0cc0 8007 1c1 8008 ℝ^*cxr 8188 < clt 8189 ℕcn 9118 5c5 9172 cdc 9586 Struct cstr 13036 ndxcnx 13037 Slot cslot 13039 Basecbs 13040 +_gcplusg 13118 ._rcmulr 13119 Scalarcsca 13121 ·_𝑠 cvsca 13122 ·_𝑖cip 13123 TopSetcts 13124 lecple 13125 distcds 13127 Hom chom 13129 compcco 13130 TopOpenctopn 13281 ∏_tcpt 13296 Σ_g cgsu 13298 X_scprds 13306

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-tp 3674 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-map 6805 df-ixp 6854 df-sup 7159 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-5 9180 df-6 9181 df-7 9182 df-8 9183 df-9 9184 df-n0 9378 df-z 9455 df-dec 9587 df-uz 9731 df-fz 10213 df-struct 13042 df-ndx 13043 df-slot 13044 df-base 13046 df-plusg 13131 df-mulr 13132 df-sca 13134 df-vsca 13135 df-ip 13136 df-tset 13137 df-ple 13138 df-ds 13140 df-hom 13142 df-cco 13143 df-rest 13282 df-topn 13283 df-topgen 13301 df-pt 13302 df-prds 13308

This theorem is referenced by: prdssca 13316 prdsbas 13317 prdsplusg 13318 prdsmulr 13319

W3C validator