Theorem hoidmvle 47043

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.)

Hypotheses

Ref	Expression
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 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))

Assertion

Ref	Expression
hoidmvle	⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑘   𝐵,𝑏,𝑘   𝐶,𝑗,𝑘   𝐷,𝑗,𝑘   𝐿,𝑎,𝑏,𝑗,𝑥   𝑋,𝑎,𝑏,𝑗,𝑘,𝑥   𝜑,𝑎,𝑏,𝑗,𝑥

Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑥,𝑗)   𝐵(𝑥,𝑗,𝑎)   𝐶(𝑥,𝑎,𝑏)   𝐷(𝑥,𝑎,𝑏)   𝐿(𝑘)

Proof of Theorem hoidmvle

Dummy variables 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 𝑙 𝑜 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hoidmvle.s	. 2 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
2		hoidmvle.d	. . . 4 ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑m 𝑋))
3		ovex 7389	. . . . . . 7 ⊢ (ℝ ↑m 𝑋) ∈ V
4		nnex 12171	. . . . . . 7 ⊢ ℕ ∈ V
5	3, 4	pm3.2i 471	. . . . . 6 ⊢ ((ℝ ↑m 𝑋) ∈ V ∧ ℕ ∈ V)
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → ((ℝ ↑m 𝑋) ∈ V ∧ ℕ ∈ V))
7		elmapg 8776	. . . . 5 ⊢ (((ℝ ↑m 𝑋) ∈ V ∧ ℕ ∈ V) → (𝐷 ∈ ((ℝ ↑m 𝑋) ↑m ℕ) ↔ 𝐷:ℕ⟶(ℝ ↑m 𝑋)))
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (𝐷 ∈ ((ℝ ↑m 𝑋) ↑m ℕ) ↔ 𝐷:ℕ⟶(ℝ ↑m 𝑋)))
9	2, 8	mpbird 258	. . 3 ⊢ (𝜑 → 𝐷 ∈ ((ℝ ↑m 𝑋) ↑m ℕ))
10		hoidmvle.c	. . . . 5 ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑m 𝑋))
11		elmapg 8776	. . . . . 6 ⊢ (((ℝ ↑m 𝑋) ∈ V ∧ ℕ ∈ V) → (𝐶 ∈ ((ℝ ↑m 𝑋) ↑m ℕ) ↔ 𝐶:ℕ⟶(ℝ ↑m 𝑋)))
12	6, 11	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ ((ℝ ↑m 𝑋) ↑m ℕ) ↔ 𝐶:ℕ⟶(ℝ ↑m 𝑋)))
13	10, 12	mpbird 258	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ((ℝ ↑m 𝑋) ↑m ℕ))
14		hoidmvle.b	. . . . . 6 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
15		reex 11120	. . . . . . . . 9 ⊢ ℝ ∈ V
16	15	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
17		hoidmvle.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
18	16, 17	jca 516	. . . . . . 7 ⊢ (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ Fin))
19		elmapg 8776	. . . . . . 7 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
20	18, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
21	14, 20	mpbird 258	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m 𝑋))
22		hoidmvle.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
23		elmapg 8776	. . . . . . . 8 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → (𝐴 ∈ (ℝ ↑m 𝑋) ↔ 𝐴:𝑋⟶ℝ))
24	18, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ (ℝ ↑m 𝑋) ↔ 𝐴:𝑋⟶ℝ))
25	22, 24	mpbird 258	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋))
26		oveq2 7364	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (ℝ ↑m 𝑥) = (ℝ ↑m ∅))
27	26	eleq2d 2825	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑎 ∈ (ℝ ↑m 𝑥) ↔ 𝑎 ∈ (ℝ ↑m ∅)))
28	26	eleq2d 2825	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑏 ∈ (ℝ ↑m 𝑥) ↔ 𝑏 ∈ (ℝ ↑m ∅)))
29	26	oveq1d 7371	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → ((ℝ ↑m 𝑥) ↑m ℕ) = ((ℝ ↑m ∅) ↑m ℕ))
30	29	eleq2d 2825	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)))
31	29	eleq2d 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → (𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)))
32		ixpeq1 8846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)))
33		ixpeq1 8846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
34	33	iuneq2d 4952	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
35	32, 34	sseq12d 3948	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))))
36		fveq2 6827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (𝐿‘𝑥) = (𝐿‘∅))
37	36	oveqd 7373	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (𝑎(𝐿‘𝑥)𝑏) = (𝑎(𝐿‘∅)𝑏))
38	36	oveqd 7373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ∅ → ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))
39	38	mpteq2dv 5166	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))
40	39	fveq2d 6831	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))
41	37, 40	breq12d 5085	. . . . . . . . . . . . . . . 16 ⊢ (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))))))
42	35, 41	imbi12d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → ((X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))))
43	31, 42	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → ((𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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) → (X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))))
44	43	ralbidv2 3158	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))))
45	30, 44	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ((𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → ∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))))
46	45	ralbidv2 3158	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))))
47	28, 46	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑏 ∈ (ℝ ↑m 𝑥) → ∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))))
48	47	ralbidv2 3158	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))))
49	27, 48	imbi12d 345	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝑎 ∈ (ℝ ↑m 𝑥) → ∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))))
50	49	ralbidv2 3158	. . . . . . 7 ⊢ (𝑥 = ∅ → (∀𝑎 ∈ (ℝ ↑m 𝑥)∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))))
51		oveq2 7364	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑦))
52	51	eleq2d 2825	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑎 ∈ (ℝ ↑m 𝑥) ↔ 𝑎 ∈ (ℝ ↑m 𝑦)))
53	51	eleq2d 2825	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑏 ∈ (ℝ ↑m 𝑥) ↔ 𝑏 ∈ (ℝ ↑m 𝑦)))
54	51	oveq1d 7371	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((ℝ ↑m 𝑥) ↑m ℕ) = ((ℝ ↑m 𝑦) ↑m ℕ))
55	54	eleq2d 2825	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)))
56	54	eleq2d 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)))
57		ixpeq1 8846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)))
58		ixpeq1 8846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
59	58	iuneq2d 4952	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
60	57, 59	sseq12d 3948	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))))
61		fveq2 6827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝐿‘𝑥) = (𝐿‘𝑦))
62	61	oveqd 7373	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑎(𝐿‘𝑥)𝑏) = (𝑎(𝐿‘𝑦)𝑏))
63	61	oveqd 7373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))
64	63	mpteq2dv 5166	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))
65	64	fveq2d 6831	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))
66	62, 65	breq12d 5085	. . . . . . . . . . . . . . . 16 ⊢ (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))))))
67	60, 66	imbi12d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ((X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
68	56, 67	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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) → (X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))))
69	68	ralbidv2 3158	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
70	55, 69	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → ∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))))
71	70	ralbidv2 3158	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
72	53, 71	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑏 ∈ (ℝ ↑m 𝑥) → ∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))))
73	72	ralbidv2 3158	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
74	52, 73	imbi12d 345	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑎 ∈ (ℝ ↑m 𝑥) → ∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))))
75	74	ralbidv2 3158	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∀𝑎 ∈ (ℝ ↑m 𝑥)∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
76		oveq2 7364	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (ℝ ↑m 𝑥) = (ℝ ↑m (𝑦 ∪ {𝑧})))
77	76	eleq2d 2825	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑎 ∈ (ℝ ↑m 𝑥) ↔ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))))
78	76	eleq2d 2825	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑏 ∈ (ℝ ↑m 𝑥) ↔ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))))
79	76	oveq1d 7371	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((ℝ ↑m 𝑥) ↑m ℕ) = ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ))
80	79	eleq2d 2825	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)))
81	79	eleq2d 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)))
82		ixpeq1 8846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)))
83		ixpeq1 8846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
84	83	iuneq2d 4952	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
85	82, 84	sseq12d 3948	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))))
86		fveq2 6827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐿‘𝑥) = (𝐿‘(𝑦 ∪ {𝑧})))
87	86	oveqd 7373	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑎(𝐿‘𝑥)𝑏) = (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏))
88	86	oveqd 7373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))
89	88	mpteq2dv 5166	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))
90	89	fveq2d 6831	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))
91	87, 90	breq12d 5085	. . . . . . . . . . . . . . . 16 ⊢ (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))))))
92	85, 91	imbi12d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))))
93	81, 92	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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) → (X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))))))
94	93	ralbidv2 3158	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))))
95	80, 94	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → ∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))))))
96	95	ralbidv2 3158	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))))
97	78, 96	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑏 ∈ (ℝ ↑m 𝑥) → ∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))))))
98	97	ralbidv2 3158	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))))
99	77, 98	imbi12d 345	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑎 ∈ (ℝ ↑m 𝑥) → ∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))))))
100	99	ralbidv2 3158	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑎 ∈ (ℝ ↑m 𝑥)∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))))
101		oveq2 7364	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑋))
102	101	eleq2d 2825	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑎 ∈ (ℝ ↑m 𝑥) ↔ 𝑎 ∈ (ℝ ↑m 𝑋)))
103	101	eleq2d 2825	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑏 ∈ (ℝ ↑m 𝑥) ↔ 𝑏 ∈ (ℝ ↑m 𝑋)))
104	101	oveq1d 7371	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → ((ℝ ↑m 𝑥) ↑m ℕ) = ((ℝ ↑m 𝑋) ↑m ℕ))
105	104	eleq2d 2825	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)))
106	104	eleq2d 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → (𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)))
107		ixpeq1 8846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)))
108		ixpeq1 8846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑋 → X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
109	108	iuneq2d 4952	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
110	107, 109	sseq12d 3948	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑋 → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))))
111		fveq2 6827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑋 → (𝐿‘𝑥) = (𝐿‘𝑋))
112	111	oveqd 7373	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → (𝑎(𝐿‘𝑥)𝑏) = (𝑎(𝐿‘𝑋)𝑏))
113	111	oveqd 7373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑋 → ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))
114	113	mpteq2dv 5166	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑋 → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))
115	114	fveq2d 6831	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))
116	112, 115	breq12d 5085	. . . . . . . . . . . . . . . 16 ⊢ (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))))))
117	110, 116	imbi12d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → ((X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
118	106, 117	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → ((𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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) → (X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))))
119	118	ralbidv2 3158	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
120	105, 119	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → ((𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → ∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))))
121	120	ralbidv2 3158	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
122	103, 121	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝑏 ∈ (ℝ ↑m 𝑥) → ∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))))
123	122	ralbidv2 3158	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
124	102, 123	imbi12d 345	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑎 ∈ (ℝ ↑m 𝑥) → ∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))))
125	124	ralbidv2 3158	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (∀𝑎 ∈ (ℝ ↑m 𝑥)∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
126		hoidmvle.l	. . . . . . . . . . . . . . . 16 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
127		fveq1 6826	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑒 → (𝑎‘𝑘) = (𝑒‘𝑘))
128	127	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑒 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑒‘𝑘)[,)(𝑏‘𝑘)))
129	128	fveq2d 6831	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑒 → (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘))))
130	129	prodeq2ad 46037	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑒 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘))))
131	130	ifeq2d 4475	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑒 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘)))))
132		fveq1 6826	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝑓 → (𝑏‘𝑘) = (𝑓‘𝑘))
133	132	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = 𝑓 → ((𝑒‘𝑘)[,)(𝑏‘𝑘)) = ((𝑒‘𝑘)[,)(𝑓‘𝑘)))
134	133	fveq2d 6831	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑓 → (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑒‘𝑘)[,)(𝑓‘𝑘))))
135	134	prodeq2ad 46037	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑓 → ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑓‘𝑘))))
136	135	ifeq2d 4475	. . . . . . . . . . . . . . . . . 18 ⊢ (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 7451	. . . . . . . . . . . . . . . . 17 ⊢ (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 5168	. . . . . . . . . . . . . . . 16 ⊢ (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	126, 138	eqtri 2762	. . . . . . . . . . . . . . 15 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑒 ∈ (ℝ ↑m 𝑥), 𝑓 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑓‘𝑘))))))
140		elmapi 8786	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (ℝ ↑m ∅) → 𝑎:∅⟶ℝ)
141	140	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ (ℝ ↑m ∅) ∧ 𝑏 ∈ (ℝ ↑m ∅)) → 𝑎:∅⟶ℝ)
142		elmapi 8786	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ (ℝ ↑m ∅) → 𝑏:∅⟶ℝ)
143	142	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ (ℝ ↑m ∅) ∧ 𝑏 ∈ (ℝ ↑m ∅)) → 𝑏:∅⟶ℝ)
144	139, 141, 143	hoidmv0val 47026	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ (ℝ ↑m ∅) ∧ 𝑏 ∈ (ℝ ↑m ∅)) → (𝑎(𝐿‘∅)𝑏) = 0)
145	144	ad5ant23 765	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (ℝ ↑m ∅)) ∧ 𝑏 ∈ (ℝ ↑m ∅)) ∧ 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → (𝑎(𝐿‘∅)𝑏) = 0)
146		nfv 1921	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ))
147	4	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → ℕ ∈ V)
148		icossicc 13380	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
149		0fi 8979	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ Fin
150	149	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → ∅ ∈ Fin)
151		ovexd 7391	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (ℝ ↑m ∅) ∈ V)
152	4	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → ℕ ∈ V)
153		simpl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ))
154		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
155	151, 152, 153, 154	fvmap 45644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑐‘𝑗) ∈ (ℝ ↑m ∅))
156		elmapi 8786	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐‘𝑗) ∈ (ℝ ↑m ∅) → (𝑐‘𝑗):∅⟶ℝ)
157	155, 156	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑐‘𝑗):∅⟶ℝ)
158	157	adantlr 721	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → (𝑐‘𝑗):∅⟶ℝ)
159		ovexd 7391	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (ℝ ↑m ∅) ∈ V)
160	4	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → ℕ ∈ V)
161		simpl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ))
162		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
163	159, 160, 161, 162	fvmap 45644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑑‘𝑗) ∈ (ℝ ↑m ∅))
164		elmapi 8786	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑‘𝑗) ∈ (ℝ ↑m ∅) → (𝑑‘𝑗):∅⟶ℝ)
165	163, 164	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑑‘𝑗):∅⟶ℝ)
166	165	adantll 720	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → (𝑑‘𝑗):∅⟶ℝ)
167	126, 150, 158, 166	hoidmvcl 47025	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)) ∈ (0[,)+∞))
168	148, 167	sselid 3913	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)) ∈ (0[,]+∞))
169	146, 147, 168	sge0ge0mpt 46881	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))
170	169	adantll 720	. . . . . . . . . . . . 13 ⊢ (((((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 5094	. . . . . . . . . . . 12 ⊢ (((((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 25	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ (ℝ ↑m ∅)) ∧ 𝑏 ∈ (ℝ ↑m ∅)) ∧ 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → (X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))
173	172	ralrimiva 3131	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (ℝ ↑m ∅)) ∧ 𝑏 ∈ (ℝ ↑m ∅)) ∧ 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → ∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))
174	173	ralrimiva 3131	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (ℝ ↑m ∅)) ∧ 𝑏 ∈ (ℝ ↑m ∅)) → ∀𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))
175	174	ralrimiva 3131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℝ ↑m ∅)) → ∀𝑏 ∈ (ℝ ↑m ∅)∀𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))
176	175	ralrimiva 3131	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ (ℝ ↑m ∅)∀𝑏 ∈ (ℝ ↑m ∅)∀𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))
177		simpl 483	. . . . . . . . 9 ⊢ (((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)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 8851	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑒 → X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)))
179	178	sseq1d 3946	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑒 → (X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))))
180		oveq1 7363	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑒 → (𝑎(𝐿‘𝑦)𝑏) = (𝑒(𝐿‘𝑦)𝑏))
181	180	breq1d 5082	. . . . . . . . . . . . . . . 16 ⊢ (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 345	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑒 → ((X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
183	182	ralbidv 3162	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑒 → (∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
184	183	ralbidv 3162	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑒 → (∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
185	184	ralbidv 3162	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑒 → (∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
186	133	ixpeq2dv 8851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑓 → X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)))
187	186	sseq1d 3946	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑓 → (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))))
188		oveq2 7364	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑓 → (𝑒(𝐿‘𝑦)𝑏) = (𝑒(𝐿‘𝑦)𝑓))
189	188	breq1d 5082	. . . . . . . . . . . . . . . . . 18 ⊢ (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 345	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑓 → ((X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
191	190	ralbidv 3162	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑓 → (∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
192	191	ralbidv 3162	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑓 → (∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
193		fveq1 6826	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 = 𝑔 → (𝑐‘𝑗) = (𝑔‘𝑗))
194	193	fveq1d 6829	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = 𝑔 → ((𝑐‘𝑗)‘𝑘) = ((𝑔‘𝑗)‘𝑘))
195	194	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = 𝑔 → (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
196	195	ixpeq2dv 8851	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = 𝑔 → X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
197	196	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑐 = 𝑔 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
198	197	iuneq2dv 4946	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑔 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
199	198	sseq2d 3947	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = 𝑔 → (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))))
200	193	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = 𝑔 → ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)) = ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))
201	200	mpteq2dv 5166	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = 𝑔 → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))
202	201	fveq2d 6831	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑔 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))
203	202	breq2d 5084	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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 345	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝑔 → ((X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
205	204	ralbidv 3162	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝑔 → (∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))
206		fveq1 6826	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑑 = ℎ → (𝑑‘𝑗) = (ℎ‘𝑗))
207	206	fveq1d 6829	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑑 = ℎ → ((𝑑‘𝑗)‘𝑘) = ((ℎ‘𝑗)‘𝑘))
208	207	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑑 = ℎ → (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
209	208	ixpeq2dv 8851	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑑 = ℎ → X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
210	209	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 = ℎ ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
211	210	iuneq2dv 4946	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 = ℎ → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
212	211	sseq2d 3947	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 = ℎ → (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))))
213	206	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑑 = ℎ → ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)) = ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))
214	213	mpteq2dv 5166	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 = ℎ → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))))
215	214	fveq2d 6831	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 = ℎ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))))
216	215	breq2d 5084	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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 345	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 = ℎ → ((X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))))))
218	217	cbvralvw 3217	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))))))
219	218	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝑔 → (∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))))))
220	205, 219	bitrd 280	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑔 → (∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))))))
221	220	cbvralvw 3217	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))))))
222	221	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑓 → (∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))))))
223	192, 222	bitrd 280	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑓 → (∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))))))
224	223	cbvralvw 3217	. . . . . . . . . . . . 13 ⊢ (∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))))))
225	224	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑒 → (∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))))))
226	185, 225	bitrd 280	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑒 → (∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))))))
227	226	cbvralvw 3217	. . . . . . . . . 10 ⊢ (∀𝑎 ∈ (ℝ ↑m 𝑦)∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))))))
228	227	bilani 505	. . . . . . . . 9 ⊢ (((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)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))))))
229		simplll 780	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝜑)
230		eldifi 4061	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ (𝑋 ∖ 𝑦) → 𝑧 ∈ 𝑋)
231	230	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦)) → 𝑧 ∈ 𝑋)
232	231	adantrl 722	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) → 𝑧 ∈ 𝑋)
233	232	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝑧 ∈ 𝑋)
234		simpl 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})))
235		uneq1 4091	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = (∅ ∪ {𝑧}))
236		0un 4324	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∅ ∪ {𝑧}) = {𝑧}
237	236	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → (∅ ∪ {𝑧}) = {𝑧})
238	235, 237	eqtr2d 2775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = ∅ → {𝑧} = (𝑦 ∪ {𝑧}))
239	238	eqcomd 2745	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = {𝑧})
240	239	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = ∅ → (ℝ ↑m (𝑦 ∪ {𝑧})) = (ℝ ↑m {𝑧}))
241	240	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → (ℝ ↑m (𝑦 ∪ {𝑧})) = (ℝ ↑m {𝑧}))
242	234, 241	eleqtrd 2841	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑m {𝑧}))
243	242	adantll 720	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑m {𝑧}))
244	229, 233, 243	jca31 519	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → ((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})))
245	244	adantllr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 721	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 721	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})))
249	240	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → (ℝ ↑m (𝑦 ∪ {𝑧})) = (ℝ ↑m {𝑧}))
250	248, 249	eleqtrd 2841	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑m {𝑧}))
251	250	adantlr 721	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑m {𝑧}))
252	251	adantlll 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ))
254	240	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = ∅ → ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) = ((ℝ ↑m {𝑧}) ↑m ℕ))
255	254	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) = ((ℝ ↑m {𝑧}) ↑m ℕ))
256	253, 255	eleqtrd 2841	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ))
257	256	adantll 720	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 519	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 721	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 721	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ))
262	254	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) = ((ℝ ↑m {𝑧}) ↑m ℕ))
263	261, 262	eleqtrd 2841	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ))
264	263	adantlr 721	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ))
265	264	adantlll 724	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ))
266		simpl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ∧ 𝑦 = ∅) → X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
267	238	ixpeq1d 8847	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = ∅ → X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)))
268	267	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ∧ 𝑦 = ∅) → X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)))
269	238	ixpeq1d 8847	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = ∅ → X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)))
270	269	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 = ∅ ∧ 𝑖 ∈ ℕ) → X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)))
271	270	iuneq2dv 4946	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = ∅ → ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪ 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)))
272		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝑗 → (𝑐‘𝑖) = (𝑐‘𝑗))
273	272	fveq1d 6829	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖)‘𝑘) = ((𝑐‘𝑗)‘𝑘))
274		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝑗 → (𝑑‘𝑖) = (𝑑‘𝑗))
275	274	fveq1d 6829	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = 𝑗 → ((𝑑‘𝑖)‘𝑘) = ((𝑑‘𝑗)‘𝑘))
276	273, 275	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑗 → (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
277	276	ixpeq2dv 8851	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 𝑗 → X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
278	277	cbviunv 4968	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∪ 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))
279	278	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = ∅ → ∪ 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
280	271, 279	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = ∅ → ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
281	280	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ∧ 𝑦 = ∅) → ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
282	268, 281	sseq12d 3948	. . . . . . . . . . . . . . . . . . 19 ⊢ ((X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ∧ 𝑦 = ∅) → (X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) ↔ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))))
283	266, 282	mpbird 258	. . . . . . . . . . . . . . . . . 18 ⊢ ((X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ∧ 𝑦 = ∅) → X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)))
284	283	adantll 720	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ 𝑦 = ∅) → X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)))
285	260, 265, 284	jca31 519	. . . . . . . . . . . . . . . 16 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))))
286		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ 𝑦 = ∅) → 𝑦 = ∅)
287		fveq1 6826	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 𝑢 → (𝑎‘𝑘) = (𝑢‘𝑘))
288	287	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑢 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑢‘𝑘)[,)(𝑏‘𝑘)))
289	288	fveq2d 6831	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑢 → (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))))
290	289	prodeq2ad 46037	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑢 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))))
291	290	ifeq2d 4475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑢 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘)))))
292		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑙 → (𝑢‘𝑘) = (𝑢‘𝑙))
293		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑙 → (𝑏‘𝑘) = (𝑏‘𝑙))
294	292, 293	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑙 → ((𝑢‘𝑘)[,)(𝑏‘𝑘)) = ((𝑢‘𝑙)[,)(𝑏‘𝑙)))
295	294	fveq2d 6831	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 = 𝑙 → (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙))))
296	295	cbvprodv 15870	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙)))
297	296	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = 𝑣 → ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙))))
298		fveq1 6826	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑏 = 𝑣 → (𝑏‘𝑙) = (𝑣‘𝑙))
299	298	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 = 𝑣 → ((𝑢‘𝑙)[,)(𝑏‘𝑙)) = ((𝑢‘𝑙)[,)(𝑣‘𝑙)))
300	299	fveq2d 6831	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = 𝑣 → (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙))) = (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙))))
301	300	prodeq2ad 46037	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = 𝑣 → ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙))) = ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙))))
302	297, 301	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = 𝑣 → ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙))))
303	302	ifeq2d 4475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (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 7451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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 5168	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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	126, 305	eqtri 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑢 ∈ (ℝ ↑m 𝑥), 𝑣 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙))))))
307		simp-6r 793	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → 𝑧 ∈ 𝑋)
308		eqid 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑧} = {𝑧}
309		elmapi 8786	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ (ℝ ↑m {𝑧}) → 𝑎:{𝑧}⟶ℝ)
310	309	ad2antlr 733	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) → 𝑎:{𝑧}⟶ℝ)
311	310	ad3antrrr 736	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → 𝑎:{𝑧}⟶ℝ)
312		elmapi 8786	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 ∈ (ℝ ↑m {𝑧}) → 𝑏:{𝑧}⟶ℝ)
313	312	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) → 𝑏:{𝑧}⟶ℝ)
314	313	ad3antrrr 736	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → 𝑏:{𝑧}⟶ℝ)
315		elmapi 8786	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ) → 𝑐:ℕ⟶(ℝ ↑m {𝑧}))
316	315	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((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 732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → 𝑐:ℕ⟶(ℝ ↑m {𝑧}))
318		elmapi 8786	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ) → 𝑑:ℕ⟶(ℝ ↑m {𝑧}))
319	318	ad2antlr 733	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → 𝑑:ℕ⟶(ℝ ↑m {𝑧}))
320		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) → X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)))
321		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 = 𝑙 → (𝑎‘𝑘) = (𝑎‘𝑙))
322	321, 293	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 𝑙 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙)))
323		eqcom 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑙 ↔ 𝑙 = 𝑘)
324	323	imbi1i 350	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 = 𝑙 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙))))
325		eqcom 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙)) ↔ ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘)))
326	325	imbi2i 337	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑙 = 𝑘 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘))))
327	324, 326	bitri 276	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 = 𝑙 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘))))
328	322, 327	mpbi 231	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 = 𝑘 → ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘)))
329	328	cbvixpv 8853	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ X𝑙 ∈ {𝑧} ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘))
330	329	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) → X𝑙 ∈ {𝑧} ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)))
331	276	ixpeq2dv 8851	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑗 → X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = X𝑘 ∈ {𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
332	331	cbviunv 4968	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))
333		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑙 → ((𝑐‘𝑗)‘𝑘) = ((𝑐‘𝑗)‘𝑙))
334		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑙 → ((𝑑‘𝑗)‘𝑘) = ((𝑑‘𝑗)‘𝑙))
335	333, 334	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑙 → (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)))
336	335	cbvixpv 8853	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ X𝑘 ∈ {𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙))
337	336	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ ℕ → X𝑘 ∈ {𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)))
338	337	iuneq2i 4943	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙))
339	332, 338	eqtr2i 2763	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∪ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) = ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))
340	339	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) → ∪ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) = ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)))
341	330, 340	sseq12d 3948	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) → (X𝑙 ∈ {𝑧} ((𝑎‘𝑙)[,)(𝑏‘𝑙)) ⊆ ∪ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) ↔ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))))
342	320, 341	mpbird 258	. . . . . . . . . . . . . . . . . . . 20 ⊢ (X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) → X𝑙 ∈ {𝑧} ((𝑎‘𝑙)[,)(𝑏‘𝑙)) ⊆ ∪ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)))
343	342	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → X𝑙 ∈ {𝑧} ((𝑎‘𝑙)[,)(𝑏‘𝑙)) ⊆ ∪ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)))
344	306, 307, 308, 311, 314, 317, 319, 343	hoidmv1le 47037	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → (𝑎(𝐿‘{𝑧})𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗)))))
345	344	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {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 2774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = {𝑧})
347	346	fveq2d 6831	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ∅ → (𝐿‘(𝑦 ∪ {𝑧})) = (𝐿‘{𝑧}))
348	347	oveqd 7373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ∅ → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) = (𝑎(𝐿‘{𝑧})𝑏))
349	348	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) ∧ 𝑦 = ∅) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) = (𝑎(𝐿‘{𝑧})𝑏))
350	347	oveqd 7373	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = ∅ → ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗)))
351	350	mpteq2dv 5166	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ∅ → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗))))
352	351	fveq2d 6831	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ∅ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗)))))
353	352	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {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 5085	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {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 258	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {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 590	. . . . . . . . . . . . . . 15 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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	17	ad8antr 746	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑋 ∈ Fin)
358		simplrl 782	. . . . . . . . . . . . . . . . . . 19 ⊢ (((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 736	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑦 ⊆ 𝑋)
361		simplrr 783	. . . . . . . . . . . . . . . . . . 19 ⊢ (((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 736	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑧 ∈ (𝑋 ∖ 𝑦))
364		eqid 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧})
365		elmapi 8786	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ)
366	365	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ)
367	366	ad4ant23 759	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ)
369		elmapi 8786	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ)
370	369	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ)
371	370	ad4ant23 759	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ)
373		elmapi 8786	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) → 𝑐:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
374	373	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑐:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
376		elmapi 8786	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) → 𝑑:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
377	376	ad2antrr 732	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑑:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
378	377	adantlll 724	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑑:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
379		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑙 → (𝑒‘𝑘) = (𝑒‘𝑙))
380		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑙 → (𝑓‘𝑘) = (𝑓‘𝑙))
381	379, 380	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑙 → ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = ((𝑒‘𝑙)[,)(𝑓‘𝑙)))
382	381	cbvixpv 8853	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙))
383	382	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ = 𝑜 → X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)))
384		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 = 𝑖 → (𝑔‘𝑗) = (𝑔‘𝑖))
385	384	fveq1d 6829	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 = 𝑖 → ((𝑔‘𝑗)‘𝑘) = ((𝑔‘𝑖)‘𝑘))
386		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 = 𝑖 → (ℎ‘𝑗) = (ℎ‘𝑖))
387	386	fveq1d 6829	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 = 𝑖 → ((ℎ‘𝑗)‘𝑘) = ((ℎ‘𝑖)‘𝑘))
388	385, 387	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = 𝑖 → (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)))
389	388	ixpeq2dv 8851	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 = 𝑖 → X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)))
390	389	cbviunv 4968	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪ 𝑖 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘))
391	390	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ = 𝑜 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪ 𝑖 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)))
392		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑙 → ((𝑔‘𝑖)‘𝑘) = ((𝑔‘𝑖)‘𝑙))
393		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑙 → ((ℎ‘𝑖)‘𝑘) = ((ℎ‘𝑖)‘𝑙))
394	392, 393	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 𝑙 → (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙)))
395	394	cbvixpv 8853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙))
396	395	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = 𝑜 → X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙)))
397		fveq1 6826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (ℎ = 𝑜 → (ℎ‘𝑖) = (𝑜‘𝑖))
398	397	fveq1d 6829	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (ℎ = 𝑜 → ((ℎ‘𝑖)‘𝑙) = ((𝑜‘𝑖)‘𝑙))
399	398	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (ℎ = 𝑜 → (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙)) = (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))
400	399	ixpeq2dv 8851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = 𝑜 → X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))
401	396, 400	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ℎ = 𝑜 → X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))
402	401	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((ℎ = 𝑜 ∧ 𝑖 ∈ ℕ) → X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))
403	402	iuneq2dv 4946	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ = 𝑜 → ∪ 𝑖 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))
404	391, 403	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ = 𝑜 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))
405	383, 404	sseq12d 3948	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = 𝑜 → (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙))))
406	384, 386	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = 𝑖 → ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)) = ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖)))
407	406	cbvmptv 5176	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖)))
408	407	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℎ = 𝑜 → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖))))
409	397	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ℎ = 𝑜 → ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖)) = ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))
410	409	mpteq2dv 5166	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℎ = 𝑜 → (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖))) = (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))
411	408, 410	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ = 𝑜 → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))
412	411	fveq2d 6831	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ = 𝑜 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))) = (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))))
413	412	breq2d 5084	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (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 345	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑜 → ((X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))))) ↔ (X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))))))
415	414	cbvralvw 3217	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀ℎ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))))
416	415	ralbii 3085	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ℎ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))))
417	416	ralbii 3085	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ℎ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))))
418	417	ralbii 3085	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ℎ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))))
419	418	bilani 505	. . . . . . . . . . . . . . . . . 18 ⊢ (((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))))
420	419	ad6antr 742	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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)(X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))))
421	322	cbvixpv 8853	. . . . . . . . . . . . . . . . . . . 20 ⊢ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑙)[,)(𝑏‘𝑙))
422	335	cbvixpv 8853	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙))
423	422	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑖 → X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)))
424		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → (𝑐‘𝑗) = (𝑐‘𝑖))
425	424	fveq1d 6829	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑖 → ((𝑐‘𝑗)‘𝑙) = ((𝑐‘𝑖)‘𝑙))
426		fveq2 6827	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → (𝑑‘𝑗) = (𝑑‘𝑖))
427	426	fveq1d 6829	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑖 → ((𝑑‘𝑗)‘𝑙) = ((𝑑‘𝑖)‘𝑙))
428	425, 427	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑖 → (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) = (((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙)))
429	428	ixpeq2dv 8851	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑖 → X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙)))
430	423, 429	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑖 → X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙)))
431	430	cbviunv 4968	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙))
432	421, 431	sseq12i 3945	. . . . . . . . . . . . . . . . . . 19 ⊢ (X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑙)[,)(𝑏‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙)))
433	432	biimpi 217	. . . . . . . . . . . . . . . . . 18 ⊢ (X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑙)[,)(𝑏‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙)))
434	433	ad2antlr 733	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑙)[,)(𝑏‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙)))
435		neqne 2942	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑦 = ∅ → 𝑦 ≠ ∅)
436	435	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑦 ≠ ∅)
437	306, 357, 360, 363, 364, 368, 372, 375, 378, 420, 434, 436	hoidmvlelem5 47042	. . . . . . . . . . . . . . . 16 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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 7374	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑖)) = ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))
439	438	cbvmptv 5176	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℕ ↦ ((𝑐‘𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑖))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))
440	439	fveq2i 6830	. . . . . . . . . . . . . . . . 17 ⊢ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑐‘𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑖)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))
441	440	breq2i 5080	. . . . . . . . . . . . . . . 16 ⊢ ((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 219	. . . . . . . . . . . . . . 15 ⊢ (((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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 818	. . . . . . . . . . . . . 14 ⊢ ((((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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 413	. . . . . . . . . . . . 13 ⊢ (((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)) → (X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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 3131	. . . . . . . . . . . 12 ⊢ ((((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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 3131	. . . . . . . . . . 11 ⊢ (((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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 3131	. . . . . . . . . 10 ⊢ ((((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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 3131	. . . . . . . . 9 ⊢ (((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)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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 590	. . . . . . . 8 ⊢ (((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)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (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 413	. . . . . . 7 ⊢ ((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)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))))
451	50, 75, 100, 125, 176, 450, 17	findcard2d 9091	. . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ (ℝ ↑m 𝑋)∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))
452		fveq1 6826	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘))
453	452	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝐴‘𝑘)[,)(𝑏‘𝑘)))
454	453	ixpeq2dv 8851	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)))
455	454	sseq1d 3946	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))))
456		oveq1 7363	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (𝑎(𝐿‘𝑋)𝑏) = (𝐴(𝐿‘𝑋)𝑏))
457	456	breq1d 5082	. . . . . . . . . . 11 ⊢ (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 345	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
459	458	ralbidv 3162	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
460	459	ralbidv 3162	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
461	460	ralbidv 3162	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
462	461	rspcva 3558	. . . . . 6 ⊢ ((𝐴 ∈ (ℝ ↑m 𝑋) ∧ ∀𝑎 ∈ (ℝ ↑m 𝑋)∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))
463	25, 451, 462	syl2anc 590	. . . . 5 ⊢ (𝜑 → ∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))
464		fveq1 6826	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → (𝑏‘𝑘) = (𝐵‘𝑘))
465	464	oveq2d 7372	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → ((𝐴‘𝑘)[,)(𝑏‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
466	465	ixpeq2dv 8851	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
467	466	sseq1d 3946	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))))
468		oveq2 7364	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝐴(𝐿‘𝑋)𝑏) = (𝐴(𝐿‘𝑋)𝐵))
469	468	breq1d 5082	. . . . . . . . 9 ⊢ (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 345	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
471	470	ralbidv 3162	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
472	471	ralbidv 3162	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
473	472	rspcva 3558	. . . . 5 ⊢ ((𝐵 ∈ (ℝ ↑m 𝑋) ∧ ∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))
474	21, 463, 473	syl2anc 590	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))
475		fveq1 6826	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑗) = (𝐶‘𝑗))
476	475	fveq1d 6829	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → ((𝑐‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
477	476	oveq1d 7371	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
478	477	ixpeq2dv 8851	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
479	478	adantr 481	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
480	479	iuneq2dv 4946	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
481	480	sseq2d 3947	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))))
482	475	oveq1d 7371	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)) = ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))
483	482	mpteq2dv 5166	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))
484	483	fveq2d 6831	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))
485	484	breq2d 5084	. . . . . . 7 ⊢ (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 345	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
487	486	ralbidv 3162	. . . . 5 ⊢ (𝑐 = 𝐶 → (∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))
488	487	rspcva 3558	. . . 4 ⊢ ((𝐶 ∈ ((ℝ ↑m 𝑋) ↑m ℕ) ∧ ∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))
489	13, 474, 488	syl2anc 590	. . 3 ⊢ (𝜑 → ∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))
490		fveq1 6826	. . . . . . . . . . 11 ⊢ (𝑑 = 𝐷 → (𝑑‘𝑗) = (𝐷‘𝑗))
491	490	fveq1d 6829	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝑑‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
492	491	oveq2d 7372	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
493	492	ixpeq2dv 8851	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
494	493	adantr 481	. . . . . . 7 ⊢ ((𝑑 = 𝐷 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
495	494	iuneq2dv 4946	. . . . . 6 ⊢ (𝑑 = 𝐷 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
496	495	sseq2d 3947	. . . . 5 ⊢ (𝑑 = 𝐷 → (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
497	490	oveq2d 7372	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)) = ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))
498	497	mpteq2dv 5166	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))))
499	498	fveq2d 6831	. . . . . 6 ⊢ (𝑑 = 𝐷 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
500	499	breq2d 5084	. . . . 5 ⊢ (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 345	. . . 4 ⊢ (𝑑 = 𝐷 → ((X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))))
502	501	rspcva 3558	. . 3 ⊢ ((𝐷 ∈ ((ℝ ↑m 𝑋) ↑m ℕ) ∧ ∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) → (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))))))
503	9, 489, 502	syl2anc 590	. 2 ⊢ (𝜑 → (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))))))
504	1, 503	mpd 15	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  Vcvv 3431   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  ∅c0 4261  ifcif 4454  {csn 4555  ∪ ciun 4921   class class class wbr 5072   ↦ cmpt 5153  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358   ↑_m cmap 8763  Xcixp 8835  Fincfn 8883  ℝcr 11028  0cc0 11029  +∞cpnf 11167   ≤ cle 11171  ℕcn 12165  [,)cico 13291  [,]cicc 13292  ∏cprod 15859  volcvol 25448  Σ^{^}csumge0 46805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9314  df-sup 9345  df-inf 9346  df-oi 9415  df-dju 9816  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-q 12890  df-rp 12934  df-xneg 13054  df-xadd 13055  df-xmul 13056  df-ioo 13293  df-ico 13295  df-icc 13296  df-fz 13453  df-fzo 13600  df-fl 13742  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441  df-rlim 15442  df-sum 15640  df-prod 15860  df-rest 17376  df-topgen 17397  df-psmet 21339  df-xmet 21340  df-met 21341  df-bl 21342  df-mopn 21343  df-top 22877  df-topon 22894  df-bases 22929  df-cmp 23370  df-ovol 25449  df-vol 25450  df-sumge0 46806

This theorem is referenced by:  ovnhoilem2  47045