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		0fi	⊢ ∅ ∈ 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	sselid	⊢ ( ( ( 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	bilani	⊢ ( ( ( 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 ) ) ) ) ) )
229		simplll	⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑦 = ∅ ) → 𝜑 )
230		eldifi	⊢ ( 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) → 𝑧 ∈ 𝑋 )
231	230	adantl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) → 𝑧 ∈ 𝑋 )
232	231	adantrl	⊢ ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) → 𝑧 ∈ 𝑋 )
233	232	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑦 = ∅ ) → 𝑧 ∈ 𝑋 )
234		simpl	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑦 = ∅ ) → 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
235		uneq1	⊢ ( 𝑦 = ∅ → ( 𝑦 ∪ { 𝑧 } ) = ( ∅ ∪ { 𝑧 } ) )
236		0un	⊢ ( ∅ ∪ { 𝑧 } ) = { 𝑧 }
237	236	a1i	⊢ ( 𝑦 = ∅ → ( ∅ ∪ { 𝑧 } ) = { 𝑧 } )
238	235 237	eqtr2d	⊢ ( 𝑦 = ∅ → { 𝑧 } = ( 𝑦 ∪ { 𝑧 } ) )
239	238	eqcomd	⊢ ( 𝑦 = ∅ → ( 𝑦 ∪ { 𝑧 } ) = { 𝑧 } )
240	239	oveq2d	⊢ ( 𝑦 = ∅ → ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) = ( ℝ ↑m { 𝑧 } ) )
241	240	adantl	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑦 = ∅ ) → ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) = ( ℝ ↑m { 𝑧 } ) )
242	234 241	eleqtrd	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑦 = ∅ ) → 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) )
243	242	adantll	⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑦 = ∅ ) → 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) )
244	229 233 243	jca31	⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) ) ) ∧ 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ 𝑦 = ∅ ) → ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) )
245	244	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 } ) ) )
246	245	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 } ) ) )
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 } ) ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑦 = ∅ ) → ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) )
248		simpl	⊢ ( ( 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑦 = ∅ ) → 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
249	240	adantl	⊢ ( ( 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑦 = ∅ ) → ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) = ( ℝ ↑m { 𝑧 } ) )
250	248 249	eleqtrd	⊢ ( ( 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑦 = ∅ ) → 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) )
251	250	adantlr	⊢ ( ( ( 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ) ∧ 𝑦 = ∅ ) → 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) )
252	251	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 } ) )
253		simpl	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ 𝑦 = ∅ ) → 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) )
254	240	oveq1d	⊢ ( 𝑦 = ∅ → ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) = ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
255	254	adantl	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ 𝑦 = ∅ ) → ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) = ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
256	253 255	eleqtrd	⊢ ( ( 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ 𝑦 = ∅ ) → 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
257	256	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 ) )
258	247 252 257	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 ) ) )
259	258	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 ) ) )
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 ) ) ∧ 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 ) ) )
261		simpl	⊢ ( ( 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ 𝑦 = ∅ ) → 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) )
262	254	adantl	⊢ ( ( 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ 𝑦 = ∅ ) → ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) = ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
263	261 262	eleqtrd	⊢ ( ( 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) ∧ 𝑦 = ∅ ) → 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) )
264	263	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 ) )
265	264	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 ) )
266		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 ) ) )
267	238	ixpeq1d	⊢ ( 𝑦 = ∅ → X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
268	267	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 ) ) )
269	238	ixpeq1d	⊢ ( 𝑦 = ∅ → X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) )
270	269	adantr	⊢ ( ( 𝑦 = ∅ ∧ 𝑖 ∈ ℕ ) → X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) )
271	270	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 ) ) )
272		fveq2	⊢ ( 𝑖 = 𝑗 → ( 𝑐 ‘ 𝑖 ) = ( 𝑐 ‘ 𝑗 ) )
273	272	fveq1d	⊢ ( 𝑖 = 𝑗 → ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) = ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) )
274		fveq2	⊢ ( 𝑖 = 𝑗 → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑗 ) )
275	274	fveq1d	⊢ ( 𝑖 = 𝑗 → ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) = ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) )
276	273 275	oveq12d	⊢ ( 𝑖 = 𝑗 → ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
277	276	ixpeq2dv	⊢ ( 𝑖 = 𝑗 → X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
278	277	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 ) )
279	278	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 ) ) )
280	271 279	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 ) ) )
281	280	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 ) ) )
282	268 281	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 ) ) ) )
283	266 282	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 ) ) )
284	283	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 ) ) )
285	260 265 284	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 ) ) ) )
286		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 = ∅ )
287		fveq1	⊢ ( 𝑎 = 𝑢 → ( 𝑎 ‘ 𝑘 ) = ( 𝑢 ‘ 𝑘 ) )
288	287	oveq1d	⊢ ( 𝑎 = 𝑢 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
289	288	fveq2d	⊢ ( 𝑎 = 𝑢 → ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) )
290	289	prodeq2ad	⊢ ( 𝑎 = 𝑢 → ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) )
291	290	ifeq2d	⊢ ( 𝑎 = 𝑢 → if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) = if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) )
292		fveq2	⊢ ( 𝑘 = 𝑙 → ( 𝑢 ‘ 𝑘 ) = ( 𝑢 ‘ 𝑙 ) )
293		fveq2	⊢ ( 𝑘 = 𝑙 → ( 𝑏 ‘ 𝑘 ) = ( 𝑏 ‘ 𝑙 ) )
294	292 293	oveq12d	⊢ ( 𝑘 = 𝑙 → ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) )
295	294	fveq2d	⊢ ( 𝑘 = 𝑙 → ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) )
296	295	cbvprodv	⊢ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) )
297	296	a1i	⊢ ( 𝑏 = 𝑣 → ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) )
298		fveq1	⊢ ( 𝑏 = 𝑣 → ( 𝑏 ‘ 𝑙 ) = ( 𝑣 ‘ 𝑙 ) )
299	298	oveq2d	⊢ ( 𝑏 = 𝑣 → ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) )
300	299	fveq2d	⊢ ( 𝑏 = 𝑣 → ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) = ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) ) )
301	300	prodeq2ad	⊢ ( 𝑏 = 𝑣 → ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) = ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) ) )
302	297 301	eqtrd	⊢ ( 𝑏 = 𝑣 → ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) = ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) ) )
303	302	ifeq2d	⊢ ( 𝑏 = 𝑣 → if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) = if ( 𝑥 = ∅ , 0 , ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) ) ) )
304	291 303	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 ) ) ) ) )
305	304	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 ) ) ) ) ) )
306	1 305	eqtri	⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑢 ∈ ( ℝ ↑m 𝑥 ) , 𝑣 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑙 ∈ 𝑥 ( vol ‘ ( ( 𝑢 ‘ 𝑙 ) [,) ( 𝑣 ‘ 𝑙 ) ) ) ) ) )
307		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 )
308		eqid	⊢ { 𝑧 } = { 𝑧 }
309		elmapi	⊢ ( 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) → 𝑎 : { 𝑧 } ⟶ ℝ )
310	309	ad2antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) → 𝑎 : { 𝑧 } ⟶ ℝ )
311	310	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 )
312		elmapi	⊢ ( 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) → 𝑏 : { 𝑧 } ⟶ ℝ )
313	312	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) → 𝑏 : { 𝑧 } ⟶ ℝ )
314	313	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 )
315		elmapi	⊢ ( 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) → 𝑐 : ℕ ⟶ ( ℝ ↑m { 𝑧 } ) )
316	315	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑎 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m { 𝑧 } ) ) ∧ 𝑐 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) ) → 𝑐 : ℕ ⟶ ( ℝ ↑m { 𝑧 } ) )
317	316	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 } ) )
318		elmapi	⊢ ( 𝑑 ∈ ( ( ℝ ↑m { 𝑧 } ) ↑m ℕ ) → 𝑑 : ℕ ⟶ ( ℝ ↑m { 𝑧 } ) )
319	318	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 } ) )
320		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 ) ) )
321		fveq2	⊢ ( 𝑘 = 𝑙 → ( 𝑎 ‘ 𝑘 ) = ( 𝑎 ‘ 𝑙 ) )
322	321 293	oveq12d	⊢ ( 𝑘 = 𝑙 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) )
323		eqcom	⊢ ( 𝑘 = 𝑙 ↔ 𝑙 = 𝑘 )
324	323	imbi1i	⊢ ( ( 𝑘 = 𝑙 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ↔ ( 𝑙 = 𝑘 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) )
325		eqcom	⊢ ( ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ↔ ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
326	325	imbi2i	⊢ ( ( 𝑙 = 𝑘 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ↔ ( 𝑙 = 𝑘 → ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) )
327	324 326	bitri	⊢ ( ( 𝑘 = 𝑙 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) ) ↔ ( 𝑙 = 𝑘 → ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) )
328	322 327	mpbi	⊢ ( 𝑙 = 𝑘 → ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
329	328	cbvixpv	⊢ X 𝑙 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) ) = X 𝑘 ∈ { 𝑧 } ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) )
330	329	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 ) ) )
331	276	ixpeq2dv	⊢ ( 𝑖 = 𝑗 → X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
332	331	cbviunv	⊢ ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) )
333		fveq2	⊢ ( 𝑘 = 𝑙 → ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) )
334		fveq2	⊢ ( 𝑘 = 𝑙 → ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) )
335	333 334	oveq12d	⊢ ( 𝑘 = 𝑙 → ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) )
336	335	cbvixpv	⊢ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑙 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) )
337	336	a1i	⊢ ( 𝑗 ∈ ℕ → X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑙 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) )
338	337	iuneq2i	⊢ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑙 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) )
339	332 338	eqtr2i	⊢ ∪ 𝑗 ∈ ℕ X 𝑙 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) = ∪ 𝑖 ∈ ℕ X 𝑘 ∈ { 𝑧 } ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑘 ) )
340	339	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 ) ) )
341	330 340	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 ) ) ) )
342	320 341	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 ) ) )
343	342	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 ) ) )
344	306 307 308 311 314 317 319 343	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 ) ) ) ) )
345	344	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 ) ) ) ) )
346	235 237	eqtrd	⊢ ( 𝑦 = ∅ → ( 𝑦 ∪ { 𝑧 } ) = { 𝑧 } )
347	346	fveq2d	⊢ ( 𝑦 = ∅ → ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) = ( 𝐿 ‘ { 𝑧 } ) )
348	347	oveqd	⊢ ( 𝑦 = ∅ → ( 𝑎 ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) 𝑏 ) = ( 𝑎 ( 𝐿 ‘ { 𝑧 } ) 𝑏 ) )
349	348	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 ) )
350	347	oveqd	⊢ ( 𝑦 = ∅ → ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) = ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ { 𝑧 } ) ( 𝑑 ‘ 𝑗 ) ) )
351	350	mpteq2dv	⊢ ( 𝑦 = ∅ → ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ { 𝑧 } ) ( 𝑑 ‘ 𝑗 ) ) ) )
352	351	fveq2d	⊢ ( 𝑦 = ∅ → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ { 𝑧 } ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
353	352	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 ) ) ) ) )
354	349 353	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 ) ) ) ) ) )
355	345 354	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 ) ) ) ) )
356	285 286 355	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 ) ) ) ) )
357	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 )
358		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 )
359	358	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 ) ) → 𝑦 ⊆ 𝑋 )
360	359	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 = ∅ ) → 𝑦 ⊆ 𝑋 )
361		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 ) )
362	361	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 ) ) → 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) )
363	362	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 = ∅ ) → 𝑧 ∈ ( 𝑋 ∖ 𝑦 ) )
364		eqid	⊢ ( 𝑦 ∪ { 𝑧 } ) = ( 𝑦 ∪ { 𝑧 } )
365		elmapi	⊢ ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) → 𝑎 : ( 𝑦 ∪ { 𝑧 } ) ⟶ ℝ )
366	365	adantr	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) → 𝑎 : ( 𝑦 ∪ { 𝑧 } ) ⟶ ℝ )
367	366	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 )
368	367	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 )
369		elmapi	⊢ ( 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) → 𝑏 : ( 𝑦 ∪ { 𝑧 } ) ⟶ ℝ )
370	369	adantl	⊢ ( ( 𝑎 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ∧ 𝑏 ∈ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ) → 𝑏 : ( 𝑦 ∪ { 𝑧 } ) ⟶ ℝ )
371	370	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 )
372	371	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 )
373		elmapi	⊢ ( 𝑐 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) → 𝑐 : ℕ ⟶ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
374	373	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 } ) ) )
375	374	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 } ) ) )
376		elmapi	⊢ ( 𝑑 ∈ ( ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) ↑m ℕ ) → 𝑑 : ℕ ⟶ ( ℝ ↑m ( 𝑦 ∪ { 𝑧 } ) ) )
377	376	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 } ) ) )
378	377	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 } ) ) )
379		fveq2	⊢ ( 𝑘 = 𝑙 → ( 𝑒 ‘ 𝑘 ) = ( 𝑒 ‘ 𝑙 ) )
380		fveq2	⊢ ( 𝑘 = 𝑙 → ( 𝑓 ‘ 𝑘 ) = ( 𝑓 ‘ 𝑙 ) )
381	379 380	oveq12d	⊢ ( 𝑘 = 𝑙 → ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) )
382	381	cbvixpv	⊢ X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) )
383	382	a1i	⊢ ( ℎ = 𝑜 → X 𝑘 ∈ 𝑦 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = X 𝑙 ∈ 𝑦 ( ( 𝑒 ‘ 𝑙 ) [,) ( 𝑓 ‘ 𝑙 ) ) )
384		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝑔 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑖 ) )
385	384	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) )
386		fveq2	⊢ ( 𝑗 = 𝑖 → ( ℎ ‘ 𝑗 ) = ( ℎ ‘ 𝑖 ) )
387	386	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) )
388	385 387	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) )
389	388	ixpeq2dv	⊢ ( 𝑗 = 𝑖 → X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) )
390	389	cbviunv	⊢ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑖 ∈ ℕ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) )
391	390	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 ) ) )
392		fveq2	⊢ ( 𝑘 = 𝑙 → ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) = ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) )
393		fveq2	⊢ ( 𝑘 = 𝑙 → ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) = ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) )
394	392 393	oveq12d	⊢ ( 𝑘 = 𝑙 → ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) = ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) ) )
395	394	cbvixpv	⊢ X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) )
396	395	a1i	⊢ ( ℎ = 𝑜 → X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) ) )
397		fveq1	⊢ ( ℎ = 𝑜 → ( ℎ ‘ 𝑖 ) = ( 𝑜 ‘ 𝑖 ) )
398	397	fveq1d	⊢ ( ℎ = 𝑜 → ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) = ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) )
399	398	oveq2d	⊢ ( ℎ = 𝑜 → ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) ) = ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) )
400	399	ixpeq2dv	⊢ ( ℎ = 𝑜 → X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑙 ) ) = X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) )
401	396 400	eqtrd	⊢ ( ℎ = 𝑜 → X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) )
402	401	adantr	⊢ ( ( ℎ = 𝑜 ∧ 𝑖 ∈ ℕ ) → X 𝑘 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑖 ) ‘ 𝑘 ) ) = X 𝑙 ∈ 𝑦 ( ( ( 𝑔 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑜 ‘ 𝑖 ) ‘ 𝑙 ) ) )
403	402	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 ) ) )
404	391 403	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 ) ) )
405	383 404	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 ) ) ) )
406	384 386	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) = ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑖 ) ) )
407	406	cbvmptv	⊢ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑖 ) ) )
408	407	a1i	⊢ ( ℎ = 𝑜 → ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑖 ) ) ) )
409	397	oveq2d	⊢ ( ℎ = 𝑜 → ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑖 ) ) = ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) )
410	409	mpteq2dv	⊢ ( ℎ = 𝑜 → ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑖 ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) )
411	408 410	eqtrd	⊢ ( ℎ = 𝑜 → ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) )
412	411	fveq2d	⊢ ( ℎ = 𝑜 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑦 ) ( ℎ ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑖 ) ( 𝐿 ‘ 𝑦 ) ( 𝑜 ‘ 𝑖 ) ) ) ) )
413	412	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 ) ) ) ) ) )
414	405 413	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 ) ) ) ) ) ) )
415	414	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 ) ) ) ) ) )
416	415	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 ) ) ) ) ) )
417	416	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 ) ) ) ) ) )
418	417	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 ) ) ) ) ) )
419	418	bilani	⊢ ( ( ( 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 ) ) ) ) ) )
420	419	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 ) ) ) ) ) )
421	322	cbvixpv	⊢ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( 𝑎 ‘ 𝑙 ) [,) ( 𝑏 ‘ 𝑙 ) )
422	335	cbvixpv	⊢ X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) )
423	422	a1i	⊢ ( 𝑗 = 𝑖 → X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) )
424		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝑐 ‘ 𝑗 ) = ( 𝑐 ‘ 𝑖 ) )
425	424	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) = ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑙 ) )
426		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝑑 ‘ 𝑗 ) = ( 𝑑 ‘ 𝑖 ) )
427	426	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) = ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑙 ) )
428	425 427	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) = ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑙 ) ) )
429	428	ixpeq2dv	⊢ ( 𝑗 = 𝑖 → X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑙 ) ) = X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑙 ) ) )
430	423 429	eqtrd	⊢ ( 𝑗 = 𝑖 → X 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑙 ∈ ( 𝑦 ∪ { 𝑧 } ) ( ( ( 𝑐 ‘ 𝑖 ) ‘ 𝑙 ) [,) ( ( 𝑑 ‘ 𝑖 ) ‘ 𝑙 ) ) )
431	430	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 ) )
432	421 431	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 ) ) )
433	432	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 ) ) )
434	433	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 ) ) )
435		neqne	⊢ ( ¬ 𝑦 = ∅ → 𝑦 ≠ ∅ )
436	435	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 ≠ ∅ )
437	306 357 360 363 364 368 372 375 378 420 434 436	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 ) ) ) ) )
438	272 274	oveq12d	⊢ ( 𝑖 = 𝑗 → ( ( 𝑐 ‘ 𝑖 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑖 ) ) = ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) )
439	438	cbvmptv	⊢ ( 𝑖 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑖 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑖 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) )
440	439	fveq2i	⊢ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑖 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑖 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ ( 𝑦 ∪ { 𝑧 } ) ) ( 𝑑 ‘ 𝑗 ) ) ) )
441	440	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 ) ) ) ) )
442	437 441	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 ) ) ) ) )
443	356 442	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 ) ) ) ) )
444	443	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 ) ) ) ) ) )
445	444	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 ) ) ) ) ) )
446	445	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 ) ) ) ) ) )
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	177 228 448	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 ) ) ) ) ) )
450	449	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 ) ) ) ) ) ) )
451	51 76 101 126 176 450 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 ) ) ) ) ) )
452		fveq1	⊢ ( 𝑎 = 𝐴 → ( 𝑎 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) )
453	452	oveq1d	⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
454	453	ixpeq2dv	⊢ ( 𝑎 = 𝐴 → X 𝑘 ∈ 𝑋 ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) )
455	454	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 ) ) ) )
456		oveq1	⊢ ( 𝑎 = 𝐴 → ( 𝑎 ( 𝐿 ‘ 𝑋 ) 𝑏 ) = ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) )
457	456	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 ) ) ) ) ) )
458	455 457	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 ) ) ) ) ) ) )
459	458	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 ) ) ) ) ) ) )
460	459	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 ) ) ) ) ) ) )
461	460	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 ) ) ) ) ) ) )
462	461	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 ) ) ) ) ) )
463	26 451 462	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 ) ) ) ) ) )
464		fveq1	⊢ ( 𝑏 = 𝐵 → ( 𝑏 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑘 ) )
465	464	oveq2d	⊢ ( 𝑏 = 𝐵 → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
466	465	ixpeq2dv	⊢ ( 𝑏 = 𝐵 → X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
467	466	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 ) ) ) )
468		oveq2	⊢ ( 𝑏 = 𝐵 → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝑏 ) = ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) )
469	468	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 ) ) ) ) ) )
470	467 469	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 ) ) ) ) ) ) )
471	470	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 ) ) ) ) ) ) )
472	471	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 ) ) ) ) ) ) )
473	472	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 ) ) ) ) ) )
474	23 463 473	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 ) ) ) ) ) )
475		fveq1	⊢ ( 𝑐 = 𝐶 → ( 𝑐 ‘ 𝑗 ) = ( 𝐶 ‘ 𝑗 ) )
476	475	fveq1d	⊢ ( 𝑐 = 𝐶 → ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) )
477	476	oveq1d	⊢ ( 𝑐 = 𝐶 → ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
478	477	ixpeq2dv	⊢ ( 𝑐 = 𝐶 → X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
479	478	adantr	⊢ ( ( 𝑐 = 𝐶 ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( ( 𝑐 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) )
480	479	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 ) ) )
481	480	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 ) ) ) )
482	475	oveq1d	⊢ ( 𝑐 = 𝐶 → ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) )
483	482	mpteq2dv	⊢ ( 𝑐 = 𝐶 → ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) )
484	483	fveq2d	⊢ ( 𝑐 = 𝐶 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑐 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) )
485	484	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 ) ) ) ) ) )
486	481 485	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 ) ) ) ) ) ) )
487	486	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 ) ) ) ) ) ) )
488	487	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 ) ) ) ) ) )
489	17 474 488	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 ) ) ) ) ) )
490		fveq1	⊢ ( 𝑑 = 𝐷 → ( 𝑑 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑗 ) )
491	490	fveq1d	⊢ ( 𝑑 = 𝐷 → ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
492	491	oveq2d	⊢ ( 𝑑 = 𝐷 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
493	492	ixpeq2dv	⊢ ( 𝑑 = 𝐷 → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
494	493	adantr	⊢ ( ( 𝑑 = 𝐷 ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑑 ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
495	494	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 ) ) )
496	495	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 ) ) ) )
497	490	oveq2d	⊢ ( 𝑑 = 𝐷 → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) )
498	497	mpteq2dv	⊢ ( 𝑑 = 𝐷 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) )
499	498	fveq2d	⊢ ( 𝑑 = 𝐷 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝑑 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) )
500	499	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 ) ) ) ) ) )
501	496 500	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 ) ) ) ) ) ) )
502	501	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 ) ) ) ) ) )
503	14 489 502	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 ) ) ) ) ) )
504	7 503	mpd	⊢ ( 𝜑 → ( 𝐴 ( 𝐿 ‘ 𝑋 ) 𝐵 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑋 ) ( 𝐷 ‘ 𝑗 ) ) ) ) )