Theorem hoidmvle

Description: The dimensional volume of a n-dimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Lemma 115B of Fremlin1 p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020)

	Ref	Expression
Hypotheses	hoidmvle.l	⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) )
	hoidmvle.x	⊢ ( 𝜑 → 𝑋 ∈ Fin )
	hoidmvle.a	⊢ ( 𝜑 → 𝐴 : 𝑋 ⟶ ℝ )
	hoidmvle.b	⊢ ( 𝜑 → 𝐵 : 𝑋 ⟶ ℝ )
	hoidmvle.c	⊢ ( 𝜑 → 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑋 ) )
	hoidmvle.d	⊢ ( 𝜑 → 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑋 ) )
	hoidmvle.s	⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
Assertion	hoidmvle	⊢ ( 𝜑 → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		hoidmvle.l	⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) )
2		hoidmvle.x	⊢ ( 𝜑 → 𝑋 ∈ Fin )
3		hoidmvle.a	⊢ ( 𝜑 → 𝐴 : 𝑋 ⟶ ℝ )
4		hoidmvle.b	⊢ ( 𝜑 → 𝐵 : 𝑋 ⟶ ℝ )
5		hoidmvle.c	⊢ ( 𝜑 → 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑋 ) )
6		hoidmvle.d	⊢ ( 𝜑 → 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑋 ) )
7		hoidmvle.s	⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
8		ovex	⊢ ( ℝ ↑m 𝑋 ) ∈ V
9		nnex	⊢ ℕ ∈ V
10	8 9	pm3.2i	⊢ ( ( ℝ ↑m 𝑋 ) ∈ V ∧ ℕ ∈ V )
11	10	a1i	⊢ ( 𝜑 → ( ( ℝ ↑m 𝑋 ) ∈ V ∧ ℕ ∈ V ) )
12		elmapg	⊢ ( ( ( ℝ ↑m 𝑋 ) ∈ V ∧ ℕ ∈ V ) → ( 𝐷 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ↔ 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) )
13	11 12	syl	⊢ ( 𝜑 → ( 𝐷 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ↔ 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) )
14	6 13	mpbird	⊢ ( 𝜑 → 𝐷 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) )
15		elmapg	⊢ ( ( ( ℝ ↑m 𝑋 ) ∈ V ∧ ℕ ∈ V ) → ( 𝐶 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ↔ 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) )
16	11 15	syl	⊢ ( 𝜑 → ( 𝐶 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ↔ 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) )
17	5 16	mpbird	⊢ ( 𝜑 → 𝐶 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) )
18		reex	⊢ ℝ ∈ V
19	18	a1i	⊢ ( 𝜑 → ℝ ∈ V )
20	19 2	jca	⊢ ( 𝜑 → ( ℝ ∈ V ∧ 𝑋 ∈ Fin ) )
21		elmapg	⊢ ( ( ℝ ∈ V ∧ 𝑋 ∈ Fin ) → ( 𝐵 ∈ ( ℝ ↑m 𝑋 ) ↔ 𝐵 : 𝑋 ⟶ ℝ ) )
22	20 21	syl	⊢ ( 𝜑 → ( 𝐵 ∈ ( ℝ ↑m 𝑋 ) ↔ 𝐵 : 𝑋 ⟶ ℝ ) )
23	4 22	mpbird	⊢ ( 𝜑 → 𝐵 ∈ ( ℝ ↑m 𝑋 ) )
24		elmapg	⊢ ( ( ℝ ∈ V ∧ 𝑋 ∈ Fin ) → ( 𝐴 ∈ ( ℝ ↑m 𝑋 ) ↔ 𝐴 : 𝑋 ⟶ ℝ ) )
25	20 24	syl	⊢ ( 𝜑 → ( 𝐴 ∈ ( ℝ ↑m 𝑋 ) ↔ 𝐴 : 𝑋 ⟶ ℝ ) )
26	3 25	mpbird	⊢ ( 𝜑 → 𝐴 ∈ ( ℝ ↑m 𝑋 ) )
27		oveq2	⊢ ( 𝑥 = ∅ → ( ℝ ↑m 𝑥 ) = ( ℝ ↑m ∅ ) )
28	27	eleq2d	⊢ ( 𝑥 = ∅ → ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) ↔ 𝑎 ∈ ( ℝ ↑m ∅ ) ) )
29	27	eleq2d	⊢ ( 𝑥 = ∅ → ( 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↔ 𝑏 ∈ ( ℝ ↑m ∅ ) ) )
30	27	oveq1d	⊢ ( 𝑥 = ∅ → ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) = ( ( ℝ ↑m ∅ ) ↑m ℕ ) )
31	30	eleq2d	⊢ ( 𝑥 = ∅ → ( 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ↔ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) )
32	30	eleq2d	⊢ ( 𝑥 = ∅ → ( 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ↔ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) )
33		ixpeq1	⊢ ( 𝑥 = ∅ → X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
34		ixpeq1	⊢ ( 𝑥 = ∅ → X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
35	34	iuneq2d	⊢ ( 𝑥 = ∅ → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
36	33 35	sseq12d	⊢ ( 𝑥 = ∅ → ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
37		fveq2	⊢ ( 𝑥 = ∅ → ( 𝐿 ‘ 𝑥 ) = ( 𝐿 ‘ ∅ ) )
38	37	oveqd	⊢ ( 𝑥 = ∅ → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) = ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) )
39	37	oveqd	⊢ ( 𝑥 = ∅ → ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) = ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) )
40	39	mpteq2dv	⊢ ( 𝑥 = ∅ → ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) )
41	40	fveq2d	⊢ ( 𝑥 = ∅ → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
42	38 41	breq12d	⊢ ( 𝑥 = ∅ → ( ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
43	36 42	imbi12d	⊢ ( 𝑥 = ∅ → ( ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
44	32 43	imbi12d	⊢ ( 𝑥 = ∅ → ( ( 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) → ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) → ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
45	44	ralbidv2	⊢ ( 𝑥 = ∅ → ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
46	31 45	imbi12d	⊢ ( 𝑥 = ∅ → ( ( 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) → ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) → ∀ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
47	46	ralbidv2	⊢ ( 𝑥 = ∅ → ( ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
48	29 47	imbi12d	⊢ ( 𝑥 = ∅ → ( ( 𝑏 ∈ ( ℝ ↑m 𝑥 ) → ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑏 ∈ ( ℝ ↑m ∅ ) → ∀ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
49	48	ralbidv2	⊢ ( 𝑥 = ∅ → ( ∀ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑏 ∈ ( ℝ ↑m ∅ ) ∀ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
50	28 49	imbi12d	⊢ ( 𝑥 = ∅ → ( ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) → ∀ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑎 ∈ ( ℝ ↑m ∅ ) → ∀ 𝑏 ∈ ( ℝ ↑m ∅ ) ∀ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
51	50	ralbidv2	⊢ ( 𝑥 = ∅ → ( ∀ 𝑎 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑎 ∈ ( ℝ ↑m ∅ ) ∀ 𝑏 ∈ ( ℝ ↑m ∅ ) ∀ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
52		oveq2	⊢ ( 𝑥 = 𝑦 → ( ℝ ↑m 𝑥 ) = ( ℝ ↑m 𝑦 ) )
53	52	eleq2d	⊢ ( 𝑥 = 𝑦 → ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) ↔ 𝑎 ∈ ( ℝ ↑m 𝑦 ) ) )
54	52	eleq2d	⊢ ( 𝑥 = 𝑦 → ( 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↔ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ) )
55	52	oveq1d	⊢ ( 𝑥 = 𝑦 → ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) = ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) )
56	55	eleq2d	⊢ ( 𝑥 = 𝑦 → ( 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ↔ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ) )
57	55	eleq2d	⊢ ( 𝑥 = 𝑦 → ( 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ↔ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ) )
58		ixpeq1	⊢ ( 𝑥 = 𝑦 → X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
59		ixpeq1	⊢ ( 𝑥 = 𝑦 → X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
60	59	iuneq2d	⊢ ( 𝑥 = 𝑦 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
61	58 60	sseq12d	⊢ ( 𝑥 = 𝑦 → ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
62		fveq2	⊢ ( 𝑥 = 𝑦 → ( 𝐿 ‘ 𝑥 ) = ( 𝐿 ‘ 𝑦 ) )
63	62	oveqd	⊢ ( 𝑥 = 𝑦 → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) = ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) )
64	62	oveqd	⊢ ( 𝑥 = 𝑦 → ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) = ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) )
65	64	mpteq2dv	⊢ ( 𝑥 = 𝑦 → ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) )
66	65	fveq2d	⊢ ( 𝑥 = 𝑦 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
67	63 66	breq12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
68	61 67	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
69	57 68	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) → ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) → ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
70	69	ralbidv2	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
71	56 70	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) → ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) → ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
72	71	ralbidv2	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
73	54 72	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑏 ∈ ( ℝ ↑m 𝑥 ) → ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑏 ∈ ( ℝ ↑m 𝑦 ) → ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
74	73	ralbidv2	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
75	53 74	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) → ∀ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑎 ∈ ( ℝ ↑m 𝑦 ) → ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
76	75	ralbidv2	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑎 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑎 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
77		oveq2	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ℝ ↑m 𝑥 ) = ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
78	77	eleq2d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) ↔ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) )
79	77	eleq2d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↔ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) )
80	77	oveq1d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) = ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) )
81	80	eleq2d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ↔ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) )
82	80	eleq2d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ↔ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) )
83		ixpeq1	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
84		ixpeq1	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
85	84	iuneq2d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
86	83 85	sseq12d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
87		fveq2	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝐿 ‘ 𝑥 ) = ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) )
88	87	oveqd	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) = ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) )
89	87	oveqd	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) = ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) )
90	89	mpteq2dv	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) )
91	90	fveq2d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
92	88 91	breq12d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
93	86 92	imbi12d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
94	82 93	imbi12d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) → ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) → ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
95	94	ralbidv2	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
96	81 95	imbi12d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) → ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) → ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
97	96	ralbidv2	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
98	79 97	imbi12d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝑏 ∈ ( ℝ ↑m 𝑥 ) → ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) → ∀ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
99	98	ralbidv2	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ∀ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∀ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
100	78 99	imbi12d	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) → ∀ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) → ∀ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∀ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
101	100	ralbidv2	⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ∀ 𝑎 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∀ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∀ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
102		oveq2	⊢ ( 𝑥 = 𝑋 → ( ℝ ↑m 𝑥 ) = ( ℝ ↑m 𝑋 ) )
103	102	eleq2d	⊢ ( 𝑥 = 𝑋 → ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) ↔ 𝑎 ∈ ( ℝ ↑m 𝑋 ) ) )
104	102	eleq2d	⊢ ( 𝑥 = 𝑋 → ( 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↔ 𝑏 ∈ ( ℝ ↑m 𝑋 ) ) )
105	102	oveq1d	⊢ ( 𝑥 = 𝑋 → ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) = ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) )
106	105	eleq2d	⊢ ( 𝑥 = 𝑋 → ( 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ↔ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ) )
107	105	eleq2d	⊢ ( 𝑥 = 𝑋 → ( 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ↔ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ) )
108		ixpeq1	⊢ ( 𝑥 = 𝑋 → X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
109		ixpeq1	⊢ ( 𝑥 = 𝑋 → X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
110	109	iuneq2d	⊢ ( 𝑥 = 𝑋 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
111	108 110	sseq12d	⊢ ( 𝑥 = 𝑋 → ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
112		fveq2	⊢ ( 𝑥 = 𝑋 → ( 𝐿 ‘ 𝑥 ) = ( 𝐿 ‘ 𝑋 ) )
113	112	oveqd	⊢ ( 𝑥 = 𝑋 → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) = ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) )
114	112	oveqd	⊢ ( 𝑥 = 𝑋 → ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) = ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) )
115	114	mpteq2dv	⊢ ( 𝑥 = 𝑋 → ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) )
116	115	fveq2d	⊢ ( 𝑥 = 𝑋 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
117	113 116	breq12d	⊢ ( 𝑥 = 𝑋 → ( ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
118	111 117	imbi12d	⊢ ( 𝑥 = 𝑋 → ( ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
119	107 118	imbi12d	⊢ ( 𝑥 = 𝑋 → ( ( 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) → ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) → ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
120	119	ralbidv2	⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
121	106 120	imbi12d	⊢ ( 𝑥 = 𝑋 → ( ( 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) → ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) → ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
122	121	ralbidv2	⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
123	104 122	imbi12d	⊢ ( 𝑥 = 𝑋 → ( ( 𝑏 ∈ ( ℝ ↑m 𝑥 ) → ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑏 ∈ ( ℝ ↑m 𝑋 ) → ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
124	123	ralbidv2	⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑏 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
125	103 124	imbi12d	⊢ ( 𝑥 = 𝑋 → ( ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) → ∀ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝑎 ∈ ( ℝ ↑m 𝑋 ) → ∀ 𝑏 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) ) )
126	125	ralbidv2	⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑎 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑥 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑥 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑥 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑥 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑥 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑎 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
127		fveq1	⊢ ( 𝑎 = 𝑒 → ( 𝑎 ‘ 𝑘 ) = ( 𝑒 ‘ 𝑘 ) )
128	127	oveq1d	⊢ ( 𝑎 = 𝑒 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
129	128	fveq2d	⊢ ( 𝑎 = 𝑒 → ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) )
130	129	prodeq2ad	⊢ ( 𝑎 = 𝑒 → ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) )
131	130	ifeq2d	⊢ ( 𝑎 = 𝑒 → if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) = if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) )
132		fveq1	⊢ ( 𝑏 = 𝑓 → ( 𝑏 ‘ 𝑘 ) = ( 𝑓 ‘ 𝑘 ) )
133	132	oveq2d	⊢ ( 𝑏 = 𝑓 → ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) )
134	133	fveq2d	⊢ ( 𝑏 = 𝑓 → ( vol ‘ ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ) )
135	134	prodeq2ad	⊢ ( 𝑏 = 𝑓 → ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ) )
136	135	ifeq2d	⊢ ( 𝑏 = 𝑓 → if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) = if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ) ) )
137	131 136	cbvmpov	⊢ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) = ( 𝑒 ∈ ( ℝ ↑m 𝑥 ) , 𝑓 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ) ) )
138	137	mpteq2i	⊢ ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) ) = ( 𝑥 ∈ Fin ↦ ( 𝑒 ∈ ( ℝ ↑m 𝑥 ) , 𝑓 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ) ) ) )
139	1 138	eqtri	⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑒 ∈ ( ℝ ↑m 𝑥 ) , 𝑓 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ) ) ) )
140		elmapi	⊢ ( 𝑎 ∈ ( ℝ ↑m ∅ ) → 𝑎 : ∅ ⟶ ℝ )
141	140	adantr	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ∅ ) ∧ 𝑏 ∈ ( ℝ ↑m ∅ ) ) → 𝑎 : ∅ ⟶ ℝ )
142		elmapi	⊢ ( 𝑏 ∈ ( ℝ ↑m ∅ ) → 𝑏 : ∅ ⟶ ℝ )
143	142	adantl	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ∅ ) ∧ 𝑏 ∈ ( ℝ ↑m ∅ ) ) → 𝑏 : ∅ ⟶ ℝ )
144	139 141 143	hoidmv0val	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ∅ ) ∧ 𝑏 ∈ ( ℝ ↑m ∅ ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) = 0 )
145	144	ad5ant23	⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( ℝ ↑m ∅ ) ) ∧ 𝑏 ∈ ( ℝ ↑m ∅ ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) = 0 )
146		nfv	⊢ Ⅎ 𝑗 ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) )
147	9	a1i	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) → ℕ ∈ V )
148		icossicc	⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ )
149		0fin	⊢ ∅ ∈ Fin
150	149	a1i	⊢ ( ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) ∧ 𝑗 ∈ ℕ ) → ∅ ∈ Fin )
151		ovexd	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑗 ∈ ℕ ) → ( ℝ ↑m ∅ ) ∈ V )
152	9	a1i	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑗 ∈ ℕ ) → ℕ ∈ V )
153		simpl	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑗 ∈ ℕ ) → 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) )
154		simpr	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℕ )
155	151 152 153 154	fvmap	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑗 ∈ ℕ ) → ( 𝑐 ‘ 𝑗 ) ∈ ( ℝ ↑m ∅ ) )
156		elmapi	⊢ ( ( 𝑐 ‘ 𝑗 ) ∈ ( ℝ ↑m ∅ ) → ( 𝑐 ‘ 𝑗 ) : ∅ ⟶ ℝ )
157	155 156	syl	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑗 ∈ ℕ ) → ( 𝑐 ‘ 𝑗 ) : ∅ ⟶ ℝ )
158	157	adantlr	⊢ ( ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) ∧ 𝑗 ∈ ℕ ) → ( 𝑐 ‘ 𝑗 ) : ∅ ⟶ ℝ )
159		ovexd	⊢ ( ( 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑗 ∈ ℕ ) → ( ℝ ↑m ∅ ) ∈ V )
160	9	a1i	⊢ ( ( 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑗 ∈ ℕ ) → ℕ ∈ V )
161		simpl	⊢ ( ( 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑗 ∈ ℕ ) → 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) )
162		simpr	⊢ ( ( 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℕ )
163	159 160 161 162	fvmap	⊢ ( ( 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑗 ∈ ℕ ) → ( 𝑑 ‘ 𝑗 ) ∈ ( ℝ ↑m ∅ ) )
164		elmapi	⊢ ( ( 𝑑 ‘ 𝑗 ) ∈ ( ℝ ↑m ∅ ) → ( 𝑑 ‘ 𝑗 ) : ∅ ⟶ ℝ )
165	163 164	syl	⊢ ( ( 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑗 ∈ ℕ ) → ( 𝑑 ‘ 𝑗 ) : ∅ ⟶ ℝ )
166	165	adantll	⊢ ( ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) ∧ 𝑗 ∈ ℕ ) → ( 𝑑 ‘ 𝑗 ) : ∅ ⟶ ℝ )
167	1 150 158 166	hoidmvcl	⊢ ( ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ∈ ( 0 [,) +∞ ) )
168	148 167	sseldi	⊢ ( ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ∈ ( 0 [,] +∞ ) )
169	146 147 168	sge0ge0mpt	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∧ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) → 0 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
170	169	adantll	⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( ℝ ↑m ∅ ) ) ∧ 𝑏 ∈ ( ℝ ↑m ∅ ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) → 0 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
171	145 170	eqbrtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( ℝ ↑m ∅ ) ) ∧ 𝑏 ∈ ( ℝ ↑m ∅ ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
172	171	a1d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( ℝ ↑m ∅ ) ) ∧ 𝑏 ∈ ( ℝ ↑m ∅ ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) → ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
173	172	ralrimiva	⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( ℝ ↑m ∅ ) ) ∧ 𝑏 ∈ ( ℝ ↑m ∅ ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ) → ∀ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
174	173	ralrimiva	⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( ℝ ↑m ∅ ) ) ∧ 𝑏 ∈ ( ℝ ↑m ∅ ) ) → ∀ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
175	174	ralrimiva	⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( ℝ ↑m ∅ ) ) → ∀ 𝑏 ∈ ( ℝ ↑m ∅ ) ∀ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
176	175	ralrimiva	⊢ ( 𝜑 → ∀ 𝑎 ∈ ( ℝ ↑m ∅ ) ∀ 𝑏 ∈ ( ℝ ↑m ∅ ) ∀ 𝑐 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ∅ ) ↑m ℕ ) ( X 𝑘 ∈ ∅ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ∅ ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ∅ ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
177		simpl	⊢ ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑎 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) → ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) )
178	128	ixpeq2dv	⊢ ( 𝑎 = 𝑒 → X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
179	178	sseq1d	⊢ ( 𝑎 = 𝑒 → ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
180		oveq1	⊢ ( 𝑎 = 𝑒 → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) = ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) )
181	180	breq1d	⊢ ( 𝑎 = 𝑒 → ( ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
182	179 181	imbi12d	⊢ ( 𝑎 = 𝑒 → ( ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
183	182	ralbidv	⊢ ( 𝑎 = 𝑒 → ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
184	183	ralbidv	⊢ ( 𝑎 = 𝑒 → ( ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
185	184	ralbidv	⊢ ( 𝑎 = 𝑒 → ( ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
186	133	ixpeq2dv	⊢ ( 𝑏 = 𝑓 → X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) )
187	186	sseq1d	⊢ ( 𝑏 = 𝑓 → ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
188		oveq2	⊢ ( 𝑏 = 𝑓 → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) = ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) )
189	188	breq1d	⊢ ( 𝑏 = 𝑓 → ( ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
190	187 189	imbi12d	⊢ ( 𝑏 = 𝑓 → ( ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
191	190	ralbidv	⊢ ( 𝑏 = 𝑓 → ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
192	191	ralbidv	⊢ ( 𝑏 = 𝑓 → ( ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
193		fveq1	⊢ ( 𝑐 = 𝑔 → ( 𝑐 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑗 ) )
194	193	fveq1d	⊢ ( 𝑐 = 𝑔 → ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) )
195	194	oveq1d	⊢ ( 𝑐 = 𝑔 → ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
196	195	ixpeq2dv	⊢ ( 𝑐 = 𝑔 → X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
197	196	adantr	⊢ ( ( 𝑐 = 𝑔 ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
198	197	iuneq2dv	⊢ ( 𝑐 = 𝑔 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
199	198	sseq2d	⊢ ( 𝑐 = 𝑔 → ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
200	193	oveq1d	⊢ ( 𝑐 = 𝑔 → ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) = ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) )
201	200	mpteq2dv	⊢ ( 𝑐 = 𝑔 → ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) )
202	201	fveq2d	⊢ ( 𝑐 = 𝑔 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
203	202	breq2d	⊢ ( 𝑐 = 𝑔 → ( ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
204	199 203	imbi12d	⊢ ( 𝑐 = 𝑔 → ( ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
205	204	ralbidv	⊢ ( 𝑐 = 𝑔 → ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
206		fveq1	⊢ ( 𝑑 = ℎ → ( 𝑑 ‘ 𝑗 ) = ( ℎ ‘ 𝑗 ) )
207	206	fveq1d	⊢ ( 𝑑 = ℎ → ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) )
208	207	oveq2d	⊢ ( 𝑑 = ℎ → ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) )
209	208	ixpeq2dv	⊢ ( 𝑑 = ℎ → X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) )
210	209	adantr	⊢ ( ( 𝑑 = ℎ ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) )
211	210	iuneq2dv	⊢ ( 𝑑 = ℎ → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) )
212	211	sseq2d	⊢ ( 𝑑 = ℎ → ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
213	206	oveq2d	⊢ ( 𝑑 = ℎ → ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) = ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) )
214	213	mpteq2dv	⊢ ( 𝑑 = ℎ → ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) )
215	214	fveq2d	⊢ ( 𝑑 = ℎ → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) )
216	215	breq2d	⊢ ( 𝑑 = ℎ → ( ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
217	212 216	imbi12d	⊢ ( 𝑑 = ℎ → ( ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
218	217	cbvralvw	⊢ ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
219	218	a1i	⊢ ( 𝑐 = 𝑔 → ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
220	205 219	bitrd	⊢ ( 𝑐 = 𝑔 → ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
221	220	cbvralvw	⊢ ( ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
222	221	a1i	⊢ ( 𝑏 = 𝑓 → ( ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
223	192 222	bitrd	⊢ ( 𝑏 = 𝑓 → ( ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
224	223	cbvralvw	⊢ ( ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
225	224	a1i	⊢ ( 𝑎 = 𝑒 → ( ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
226	185 225	bitrd	⊢ ( 𝑎 = 𝑒 → ( ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
227	226	cbvralvw	⊢ ( ∀ 𝑎 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
228	227	biimpi	⊢ ( ∀ 𝑎 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) → ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
229	228	adantl	⊢ ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑎 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) → ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
230		simplll	⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑦 = ∅ ) → 𝜑 )
231		eldifi	⊢ ( 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) → 𝑧 ∈ 𝑋 )
232	231	adantl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) → 𝑧 ∈ 𝑋 )
233	232	adantrl	⊢ ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) → 𝑧 ∈ 𝑋 )
234	233	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑦 = ∅ ) → 𝑧 ∈ 𝑋 )
235		simpl	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑦 = ∅ ) → 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
236		uneq1	⊢ ( 𝑦 = ∅ → ( 𝑦 ∪ { 𝑧 } ) = ( ∅ ∪ { 𝑧 } ) )
237		0un	⊢ ( ∅ ∪ { 𝑧 } ) = { 𝑧 }
238	237	a1i	⊢ ( 𝑦 = ∅ → ( ∅ ∪ { 𝑧 } ) = { 𝑧 } )
239	236 238	eqtr2d	⊢ ( 𝑦 = ∅ → { 𝑧 } = ( 𝑦 ∪ { 𝑧 } ) )
240	239	eqcomd	⊢ ( 𝑦 = ∅ → ( 𝑦 ∪ { 𝑧 } ) = { 𝑧 } )
241	240	oveq2d	⊢ ( 𝑦 = ∅ → ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) = ( ℝ ↑m { 𝑧 } ) )
242	241	adantl	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑦 = ∅ ) → ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) = ( ℝ ↑m { 𝑧 } ) )
243	235 242	eleqtrd	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑦 = ∅ ) → 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) )
244	243	adantll	⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑦 = ∅ ) → 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) )
245	230 234 244	jca31	⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑦 = ∅ ) → ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) )
246	245	adantllr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑦 = ∅ ) → ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) )
247	246	adantlr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑦 = ∅ ) → ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) )
248	247	adantlr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑦 = ∅ ) → ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) )
249		simpl	⊢ ( ( 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑦 = ∅ ) → 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
250	241	adantl	⊢ ( ( 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑦 = ∅ ) → ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) = ( ℝ ↑m { 𝑧 } ) )
251	249 250	eleqtrd	⊢ ( ( 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑦 = ∅ ) → 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) )
252	251	adantlr	⊢ ( ( ( 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑦 = ∅ ) → 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) )
253	252	adantlll	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑦 = ∅ ) → 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) )
254		simpl	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ 𝑦 = ∅ ) → 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) )
255	241	oveq1d	⊢ ( 𝑦 = ∅ → ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) = ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
256	255	adantl	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ 𝑦 = ∅ ) → ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) = ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
257	254 256	eleqtrd	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ 𝑦 = ∅ ) → 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
258	257	adantll	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑦 = ∅ ) → 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
259	248 253 258	jca31	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑦 = ∅ ) → ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) )
260	259	adantlr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑦 = ∅ ) → ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) )
261	260	adantlr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑦 = ∅ ) → ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) )
262		simpl	⊢ ( ( 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ 𝑦 = ∅ ) → 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) )
263	255	adantl	⊢ ( ( 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ 𝑦 = ∅ ) → ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) = ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
264	262 263	eleqtrd	⊢ ( ( 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ 𝑦 = ∅ ) → 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
265	264	adantlr	⊢ ( ( ( 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑦 = ∅ ) → 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
266	265	adantlll	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑦 = ∅ ) → 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
267		simpl	⊢ ( ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑦 = ∅ ) → X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
268	239	ixpeq1d	⊢ ( 𝑦 = ∅ → X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
269	268	adantl	⊢ ( ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑦 = ∅ ) → X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
270	239	ixpeq1d	⊢ ( 𝑦 = ∅ → X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) )
271	270	adantr	⊢ ( ( 𝑦 = ∅ ∧ 𝑖 ∈ ℕ ) → X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) )
272	271	iuneq2dv	⊢ ( 𝑦 = ∅ → ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = ∪ 𝑖 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) )
273		fveq2	⊢ ( 𝑖 = 𝑗 → ( 𝑐 ‘ 𝑖 ) = ( 𝑐 ‘ 𝑗 ) )
274	273	fveq1d	⊢ ( 𝑖 = 𝑗 → ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) = ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) )
275		fveq2	⊢ ( 𝑖 = 𝑗 → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑗 ) )
276	275	fveq1d	⊢ ( 𝑖 = 𝑗 → ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) = ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) )
277	274 276	oveq12d	⊢ ( 𝑖 = 𝑗 → ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
278	277	ixpeq2dv	⊢ ( 𝑖 = 𝑗 → X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
279	278	cbviunv	⊢ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) )
280	279	a1i	⊢ ( 𝑦 = ∅ → ∪ 𝑖 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
281	272 280	eqtrd	⊢ ( 𝑦 = ∅ → ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
282	281	adantl	⊢ ( ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑦 = ∅ ) → ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
283	269 282	sseq12d	⊢ ( ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑦 = ∅ ) → ( X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
284	267 283	mpbird	⊢ ( ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑦 = ∅ ) → X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) )
285	284	adantll	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑦 = ∅ ) → X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) )
286	261 266 285	jca31	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑦 = ∅ ) → ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) )
287		simpr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑦 = ∅ ) → 𝑦 = ∅ )
288		fveq1	⊢ ( 𝑎 = 𝑢 → ( 𝑎 ‘ 𝑘 ) = ( 𝑢 ‘ 𝑘 ) )
289	288	oveq1d	⊢ ( 𝑎 = 𝑢 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
290	289	fveq2d	⊢ ( 𝑎 = 𝑢 → ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) )
291	290	prodeq2ad	⊢ ( 𝑎 = 𝑢 → ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) )
292	291	ifeq2d	⊢ ( 𝑎 = 𝑢 → if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) = if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) )
293		fveq2	⊢ ( 𝑘 = 𝑙 → ( 𝑢 ‘ 𝑘 ) = ( 𝑢 ‘ 𝑙 ) )
294		fveq2	⊢ ( 𝑘 = 𝑙 → ( 𝑏 ‘ 𝑘 ) = ( 𝑏 ‘ 𝑙 ) )
295	293 294	oveq12d	⊢ ( 𝑘 = 𝑙 → ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) )
296	295	fveq2d	⊢ ( 𝑘 = 𝑙 → ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) )
297	296	cbvprodv	⊢ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) )
298	297	a1i	⊢ ( 𝑏 = 𝑣 → ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) )
299		fveq1	⊢ ( 𝑏 = 𝑣 → ( 𝑏 ‘ 𝑙 ) = ( 𝑣 ‘ 𝑙 ) )
300	299	oveq2d	⊢ ( 𝑏 = 𝑣 → ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) )
301	300	fveq2d	⊢ ( 𝑏 = 𝑣 → ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) = ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) ) )
302	301	prodeq2ad	⊢ ( 𝑏 = 𝑣 → ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) = ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) ) )
303	298 302	eqtrd	⊢ ( 𝑏 = 𝑣 → ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) ) )
304	303	ifeq2d	⊢ ( 𝑏 = 𝑣 → if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) = if ( 𝑥 = ∅ , 0 , ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) ) ) )
305	292 304	cbvmpov	⊢ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) = ( 𝑢 ∈ ( ℝ ↑m 𝑥 ) , 𝑣 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) ) ) )
306	305	mpteq2i	⊢ ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) ) = ( 𝑥 ∈ Fin ↦ ( 𝑢 ∈ ( ℝ ↑m 𝑥 ) , 𝑣 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) ) ) ) )
307	1 306	eqtri	⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑢 ∈ ( ℝ ↑m 𝑥 ) , 𝑣 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) ) ) ) )
308		simp-6r	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) → 𝑧 ∈ 𝑋 )
309		eqid	⊢ { 𝑧 } = { 𝑧 }
310		elmapi	⊢ ( 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) → 𝑎 : { 𝑧 } ⟶ ℝ )
311	310	ad2antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) → 𝑎 : { 𝑧 } ⟶ ℝ )
312	311	ad3antrrr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) → 𝑎 : { 𝑧 } ⟶ ℝ )
313		elmapi	⊢ ( 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) → 𝑏 : { 𝑧 } ⟶ ℝ )
314	313	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) → 𝑏 : { 𝑧 } ⟶ ℝ )
315	314	ad3antrrr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) → 𝑏 : { 𝑧 } ⟶ ℝ )
316		elmapi	⊢ ( 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) → 𝑐 : ℕ ⟶ ( ℝ ↑m { 𝑧 } ) )
317	316	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) → 𝑐 : ℕ ⟶ ( ℝ ↑m { 𝑧 } ) )
318	317	ad2antrr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) → 𝑐 : ℕ ⟶ ( ℝ ↑m { 𝑧 } ) )
319		elmapi	⊢ ( 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) → 𝑑 : ℕ ⟶ ( ℝ ↑m { 𝑧 } ) )
320	319	ad2antlr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) → 𝑑 : ℕ ⟶ ( ℝ ↑m { 𝑧 } ) )
321		id	⊢ ( X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) → X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) )
322		fveq2	⊢ ( 𝑘 = 𝑙 → ( 𝑎 ‘ 𝑘 ) = ( 𝑎 ‘ 𝑙 ) )
323	322 294	oveq12d	⊢ ( 𝑘 = 𝑙 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) )
324		eqcom	⊢ ( 𝑘 = 𝑙 ↔ 𝑙 = 𝑘 )
325	324	imbi1i	⊢ ( ( 𝑘 = 𝑙 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ↔ ( 𝑙 = 𝑘 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) )
326		eqcom	⊢ ( ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ↔ ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
327	326	imbi2i	⊢ ( ( 𝑙 = 𝑘 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ↔ ( 𝑙 = 𝑘 → ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) )
328	325 327	bitri	⊢ ( ( 𝑘 = 𝑙 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ↔ ( 𝑙 = 𝑘 → ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) )
329	323 328	mpbi	⊢ ( 𝑙 = 𝑘 → ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
330	329	cbvixpv	⊢ X 𝑙 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) )
331	330	a1i	⊢ ( X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) → X 𝑙 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
332	277	ixpeq2dv	⊢ ( 𝑖 = 𝑗 → X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
333	332	cbviunv	⊢ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) )
334		fveq2	⊢ ( 𝑘 = 𝑙 → ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) )
335		fveq2	⊢ ( 𝑘 = 𝑙 → ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) )
336	334 335	oveq12d	⊢ ( 𝑘 = 𝑙 → ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) )
337	336	cbvixpv	⊢ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑙 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) )
338	337	a1i	⊢ ( 𝑗 ∈ ℕ → X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑙 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) )
339	338	iuneq2i	⊢ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑙 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) )
340	333 339	eqtr2i	⊢ ∪ 𝑗 ∈ ℕ X 𝑙 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) = ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) )
341	340	a1i	⊢ ( X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) → ∪ 𝑗 ∈ ℕ X 𝑙 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) = ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) )
342	331 341	sseq12d	⊢ ( X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) → ( X 𝑙 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑙 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) ↔ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) )
343	321 342	mpbird	⊢ ( X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) → X 𝑙 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑙 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) )
344	343	adantl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) → X 𝑙 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑙 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) )
345	307 308 309 312 315 318 320 344	hoidmv1le	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) → ( 𝑎 ( 𝐿 ‘ { 𝑧 } ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ { 𝑧 } ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
346	345	adantr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) ∧ 𝑦 = ∅ ) → ( 𝑎 ( 𝐿 ‘ { 𝑧 } ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ { 𝑧 } ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
347	236 238	eqtrd	⊢ ( 𝑦 = ∅ → ( 𝑦 ∪ { 𝑧 } ) = { 𝑧 } )
348	347	fveq2d	⊢ ( 𝑦 = ∅ → ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) = ( 𝐿 ‘ { 𝑧 } ) )
349	348	oveqd	⊢ ( 𝑦 = ∅ → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) = ( 𝑎 ( 𝐿 ‘ { 𝑧 } ) 𝑏 ) )
350	349	adantl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) ∧ 𝑦 = ∅ ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) = ( 𝑎 ( 𝐿 ‘ { 𝑧 } ) 𝑏 ) )
351	348	oveqd	⊢ ( 𝑦 = ∅ → ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) = ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ { 𝑧 } ) ( 𝑑 ‘ 𝑗 ) ) )
352	351	mpteq2dv	⊢ ( 𝑦 = ∅ → ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ { 𝑧 } ) ( 𝑑 ‘ 𝑗 ) ) ) )
353	352	fveq2d	⊢ ( 𝑦 = ∅ → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ { 𝑧 } ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
354	353	adantl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) ∧ 𝑦 = ∅ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ { 𝑧 } ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
355	350 354	breq12d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) ∧ 𝑦 = ∅ ) → ( ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝑎 ( 𝐿 ‘ { 𝑧 } ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ { 𝑧 } ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
356	346 355	mpbird	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) ∧ X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) ) ∧ 𝑦 = ∅ ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
357	286 287 356	syl2anc	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑦 = ∅ ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
358	2	ad8antr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → 𝑋 ∈ Fin )
359		simplrl	⊢ ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) → 𝑦 ⊆ 𝑋 )
360	359	ad5ant12	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) → 𝑦 ⊆ 𝑋 )
361	360	ad5ant12	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → 𝑦 ⊆ 𝑋 )
362		simplrr	⊢ ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) → 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) )
363	362	ad5ant12	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) → 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) )
364	363	ad5ant12	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) )
365		eqid	⊢ ( 𝑦 ∪ { 𝑧 } ) = ( 𝑦 ∪ { 𝑧 } )
366		elmapi	⊢ ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) → 𝑎 : ( 𝑦 ∪ { 𝑧 } ) ⟶ ℝ )
367	366	adantr	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) → 𝑎 : ( 𝑦 ∪ { 𝑧 } ) ⟶ ℝ )
368	367	ad4ant23	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) → 𝑎 : ( 𝑦 ∪ { 𝑧 } ) ⟶ ℝ )
369	368	ad3antrrr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → 𝑎 : ( 𝑦 ∪ { 𝑧 } ) ⟶ ℝ )
370		elmapi	⊢ ( 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) → 𝑏 : ( 𝑦 ∪ { 𝑧 } ) ⟶ ℝ )
371	370	adantl	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) → 𝑏 : ( 𝑦 ∪ { 𝑧 } ) ⟶ ℝ )
372	371	ad4ant23	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) → 𝑏 : ( 𝑦 ∪ { 𝑧 } ) ⟶ ℝ )
373	372	ad3antrrr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → 𝑏 : ( 𝑦 ∪ { 𝑧 } ) ⟶ ℝ )
374		elmapi	⊢ ( 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) → 𝑐 : ℕ ⟶ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
375	374	adantl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) → 𝑐 : ℕ ⟶ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
376	375	ad3antrrr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → 𝑐 : ℕ ⟶ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
377		elmapi	⊢ ( 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) → 𝑑 : ℕ ⟶ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
378	377	ad2antrr	⊢ ( ( ( 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → 𝑑 : ℕ ⟶ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
379	378	adantlll	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → 𝑑 : ℕ ⟶ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
380		fveq2	⊢ ( 𝑘 = 𝑙 → ( 𝑒 ‘ 𝑘 ) = ( 𝑒 ‘ 𝑙 ) )
381		fveq2	⊢ ( 𝑘 = 𝑙 → ( 𝑓 ‘ 𝑘 ) = ( 𝑓 ‘ 𝑙 ) )
382	380 381	oveq12d	⊢ ( 𝑘 = 𝑙 → ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) )
383	382	cbvixpv	⊢ X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) )
384	383	a1i	⊢ ( ℎ = 𝑜 → X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) )
385		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝑔 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑖 ) )
386	385	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) )
387		fveq2	⊢ ( 𝑗 = 𝑖 → ( ℎ ‘ 𝑗 ) = ( ℎ ‘ 𝑖 ) )
388	387	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) )
389	386 388	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) )
390	389	ixpeq2dv	⊢ ( 𝑗 = 𝑖 → X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) )
391	390	cbviunv	⊢ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑖 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) )
392	391	a1i	⊢ ( ℎ = 𝑜 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑖 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) )
393		fveq2	⊢ ( 𝑘 = 𝑙 → ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) = ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) )
394		fveq2	⊢ ( 𝑘 = 𝑙 → ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) = ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) )
395	393 394	oveq12d	⊢ ( 𝑘 = 𝑙 → ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) = ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) ) )
396	395	cbvixpv	⊢ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) )
397	396	a1i	⊢ ( ℎ = 𝑜 → X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) ) )
398		fveq1	⊢ ( ℎ = 𝑜 → ( ℎ ‘ 𝑖 ) = ( 𝑜 ‘ 𝑖 ) )
399	398	fveq1d	⊢ ( ℎ = 𝑜 → ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) = ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) )
400	399	oveq2d	⊢ ( ℎ = 𝑜 → ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) ) = ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) )
401	400	ixpeq2dv	⊢ ( ℎ = 𝑜 → X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) ) = X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) )
402	397 401	eqtrd	⊢ ( ℎ = 𝑜 → X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) )
403	402	adantr	⊢ ( ( ℎ = 𝑜 ∧ 𝑖 ∈ ℕ ) → X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) )
404	403	iuneq2dv	⊢ ( ℎ = 𝑜 → ∪ 𝑖 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) = ∪ 𝑖 ∈ ℕ X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) )
405	392 404	eqtrd	⊢ ( ℎ = 𝑜 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑖 ∈ ℕ X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) )
406	384 405	sseq12d	⊢ ( ℎ = 𝑜 → ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) ) )
407	385 387	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) = ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑖 ) ) )
408	407	cbvmptv	⊢ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑖 ) ) )
409	408	a1i	⊢ ( ℎ = 𝑜 → ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑖 ) ) ) )
410	398	oveq2d	⊢ ( ℎ = 𝑜 → ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑖 ) ) = ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) )
411	410	mpteq2dv	⊢ ( ℎ = 𝑜 → ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑖 ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) )
412	409 411	eqtrd	⊢ ( ℎ = 𝑜 → ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) )
413	412	fveq2d	⊢ ( ℎ = 𝑜 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) ) )
414	413	breq2d	⊢ ( ℎ = 𝑜 → ( ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ↔ ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) ) ) )
415	406 414	imbi12d	⊢ ( ℎ = 𝑜 → ( ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) ) ) ) )
416	415	cbvralvw	⊢ ( ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑜 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) ) ) )
417	416	ralbii	⊢ ( ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑜 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) ) ) )
418	417	ralbii	⊢ ( ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑜 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) ) ) )
419	418	ralbii	⊢ ( ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑜 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) ) ) )
420	419	biimpi	⊢ ( ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) → ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑜 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) ) ) )
421	420	adantl	⊢ ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) → ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑜 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) ) ) )
422	421	ad6antr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑜 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) ) ) )
423	323	cbvixpv	⊢ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) )
424	336	cbvixpv	⊢ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) )
425	424	a1i	⊢ ( 𝑗 = 𝑖 → X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) )
426		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝑐 ‘ 𝑗 ) = ( 𝑐 ‘ 𝑖 ) )
427	426	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) = ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑙 ) )
428		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝑑 ‘ 𝑗 ) = ( 𝑑 ‘ 𝑖 ) )
429	428	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) = ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑙 ) )
430	427 429	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) = ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑙 ) ) )
431	430	ixpeq2dv	⊢ ( 𝑗 = 𝑖 → X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) = X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑙 ) ) )
432	425 431	eqtrd	⊢ ( 𝑗 = 𝑖 → X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑙 ) ) )
433	432	cbviunv	⊢ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑖 ∈ ℕ X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑙 ) )
434	423 433	sseq12i	⊢ ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑙 ) ) )
435	434	biimpi	⊢ ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑙 ) ) )
436	435	ad2antlr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ⊆ ∪ 𝑖 ∈ ℕ X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑙 ) ) )
437		neqne	⊢ ( ¬ 𝑦 = ∅ → 𝑦 ≠ ∅ )
438	437	adantl	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → 𝑦 ≠ ∅ )
439	307 358 361 364 365 369 373 376 379 422 436 438	hoidmvlelem5	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑖 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑖 ) ) ) ) )
440	273 275	oveq12d	⊢ ( 𝑖 = 𝑗 → ( ( 𝑐 ‘ 𝑖 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑖 ) ) = ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) )
441	440	cbvmptv	⊢ ( 𝑖 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑖 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑖 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) )
442	441	fveq2i	⊢ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑖 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑖 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) )
443	442	breq2i	⊢ ( ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑖 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑖 ) ) ) ) ↔ ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
444	439 443	sylib	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ¬ 𝑦 = ∅ ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
445	357 444	pm2.61dan	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
446	445	ex	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) → ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
447	446	ralrimiva	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) → ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
448	447	ralrimiva	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) → ∀ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
449	448	ralrimiva	⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) → ∀ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∀ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
450	449	ralrimiva	⊢ ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑦 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) → ∀ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∀ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∀ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
451	177 229 450	syl2anc	⊢ ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ ∀ 𝑎 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) → ∀ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∀ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∀ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
452	451	ex	⊢ ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) → ( ∀ 𝑎 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑦 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑦 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑦 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑦 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) → ∀ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∀ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∀ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ( X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
453	51 76 101 126 176 452 2	findcard2d	⊢ ( 𝜑 → ∀ 𝑎 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
454		fveq1	⊢ ( 𝑎 = 𝐴 → ( 𝑎 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) )
455	454	oveq1d	⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
456	455	ixpeq2dv	⊢ ( 𝑎 = 𝐴 → X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
457	456	sseq1d	⊢ ( 𝑎 = 𝐴 → ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
458		oveq1	⊢ ( 𝑎 = 𝐴 → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) = ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) )
459	458	breq1d	⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
460	457 459	imbi12d	⊢ ( 𝑎 = 𝐴 → ( ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
461	460	ralbidv	⊢ ( 𝑎 = 𝐴 → ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
462	461	ralbidv	⊢ ( 𝑎 = 𝐴 → ( ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
463	462	ralbidv	⊢ ( 𝑎 = 𝐴 → ( ∀ 𝑏 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑏 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
464	463	rspcva	⊢ ( ( 𝐴 ∈ ( ℝ ↑m 𝑋 ) ∧ ∀ 𝑎 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑏 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) → ∀ 𝑏 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
465	26 453 464	syl2anc	⊢ ( 𝜑 → ∀ 𝑏 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
466		fveq1	⊢ ( 𝑏 = 𝐵 → ( 𝑏 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑘 ) )
467	466	oveq2d	⊢ ( 𝑏 = 𝐵 → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
468	467	ixpeq2dv	⊢ ( 𝑏 = 𝐵 → X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
469	468	sseq1d	⊢ ( 𝑏 = 𝐵 → ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
470		oveq2	⊢ ( 𝑏 = 𝐵 → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) = ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) )
471	470	breq1d	⊢ ( 𝑏 = 𝐵 → ( ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
472	469 471	imbi12d	⊢ ( 𝑏 = 𝐵 → ( ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
473	472	ralbidv	⊢ ( 𝑏 = 𝐵 → ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
474	473	ralbidv	⊢ ( 𝑏 = 𝐵 → ( ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
475	474	rspcva	⊢ ( ( 𝐵 ∈ ( ℝ ↑m 𝑋 ) ∧ ∀ 𝑏 ∈ ( ℝ ↑m 𝑋 ) ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) → ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
476	23 465 475	syl2anc	⊢ ( 𝜑 → ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
477		fveq1	⊢ ( 𝑐 = 𝐶 → ( 𝑐 ‘ 𝑗 ) = ( 𝐶 ‘ 𝑗 ) )
478	477	fveq1d	⊢ ( 𝑐 = 𝐶 → ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) )
479	478	oveq1d	⊢ ( 𝑐 = 𝐶 → ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
480	479	ixpeq2dv	⊢ ( 𝑐 = 𝐶 → X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
481	480	adantr	⊢ ( ( 𝑐 = 𝐶 ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
482	481	iuneq2dv	⊢ ( 𝑐 = 𝐶 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
483	482	sseq2d	⊢ ( 𝑐 = 𝐶 → ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
484	477	oveq1d	⊢ ( 𝑐 = 𝐶 → ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) )
485	484	mpteq2dv	⊢ ( 𝑐 = 𝐶 → ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) )
486	485	fveq2d	⊢ ( 𝑐 = 𝐶 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
487	486	breq2d	⊢ ( 𝑐 = 𝐶 → ( ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
488	483 487	imbi12d	⊢ ( 𝑐 = 𝐶 → ( ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
489	488	ralbidv	⊢ ( 𝑐 = 𝐶 → ( ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) )
490	489	rspcva	⊢ ( ( 𝐶 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∧ ∀ 𝑐 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) → ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
491	17 476 490	syl2anc	⊢ ( 𝜑 → ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) )
492		fveq1	⊢ ( 𝑑 = 𝐷 → ( 𝑑 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑗 ) )
493	492	fveq1d	⊢ ( 𝑑 = 𝐷 → ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
494	493	oveq2d	⊢ ( 𝑑 = 𝐷 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
495	494	ixpeq2dv	⊢ ( 𝑑 = 𝐷 → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
496	495	adantr	⊢ ( ( 𝑑 = 𝐷 ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
497	496	iuneq2dv	⊢ ( 𝑑 = 𝐷 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
498	497	sseq2d	⊢ ( 𝑑 = 𝐷 → ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
499	492	oveq2d	⊢ ( 𝑑 = 𝐷 → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) )
500	499	mpteq2dv	⊢ ( 𝑑 = 𝐷 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) )
501	500	fveq2d	⊢ ( 𝑑 = 𝐷 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) )
502	501	breq2d	⊢ ( 𝑑 = 𝐷 → ( ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ↔ ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) )
503	498 502	imbi12d	⊢ ( 𝑑 = 𝐷 → ( ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) )
504	503	rspcva	⊢ ( ( 𝐷 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ∧ ∀ 𝑑 ∈ ( ( ℝ ↑m 𝑋 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) ) ) → ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) )
505	14 491 504	syl2anc	⊢ ( 𝜑 → ( X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ) )
506	7 505	mpd	⊢ ( 𝜑 → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) )