Theorem hoidmvle 46023

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 7447	. . . . . . 7 ⊢ (ℝ ↑m 𝑋) ∈ V
4		nnex 12246	. . . . . . 7 ⊢ ℕ ∈ V
5	3, 4	pm3.2i 469	. . . . . 6 ⊢ ((ℝ ↑m 𝑋) ∈ V ∧ ℕ ∈ V)
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → ((ℝ ↑m 𝑋) ∈ V ∧ ℕ ∈ V))
7		elmapg 8854	. . . . 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 256	. . 3 ⊢ (𝜑 → 𝐷 ∈ ((ℝ ↑m 𝑋) ↑m ℕ))
10		hoidmvle.c	. . . . 5 ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑m 𝑋))
11		elmapg 8854	. . . . . 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 256	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ((ℝ ↑m 𝑋) ↑m ℕ))
14		hoidmvle.b	. . . . . 6 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
15		reex 11227	. . . . . . . . 9 ⊢ ℝ ∈ V
16	15	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
17		hoidmvle.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
18	16, 17	jca 510	. . . . . . 7 ⊢ (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ Fin))
19		elmapg 8854	. . . . . . 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 256	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m 𝑋))
22		hoidmvle.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
23		elmapg 8854	. . . . . . . 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 256	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋))
26		oveq2 7422	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (ℝ ↑m 𝑥) = (ℝ ↑m ∅))
27	26	eleq2d 2811	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑎 ∈ (ℝ ↑m 𝑥) ↔ 𝑎 ∈ (ℝ ↑m ∅)))
28	26	eleq2d 2811	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑏 ∈ (ℝ ↑m 𝑥) ↔ 𝑏 ∈ (ℝ ↑m ∅)))
29	26	oveq1d 7429	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → ((ℝ ↑m 𝑥) ↑m ℕ) = ((ℝ ↑m ∅) ↑m ℕ))
30	29	eleq2d 2811	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)))
31	29	eleq2d 2811	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → (𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)))
32		ixpeq1 8923	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)))
33		ixpeq1 8923	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
34	33	iuneq2d 5018	. . . . . . . . . . . . . . . . 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 4005	. . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (𝐿‘𝑥) = (𝐿‘∅))
37	36	oveqd 7431	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (𝑎(𝐿‘𝑥)𝑏) = (𝑎(𝐿‘∅)𝑏))
38	36	oveqd 7431	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ∅ → ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))
39	38	mpteq2dv 5243	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))
40	39	fveq2d 6894	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))
41	37, 40	breq12d 5154	. . . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . 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 3164	. . . . . . . . . . . . 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 343	. . . . . . . . . . . 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 3164	. . . . . . . . . . 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 343	. . . . . . . . . 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 3164	. . . . . . . . 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 343	. . . . . . . 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 3164	. . . . . . 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 7422	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑦))
52	51	eleq2d 2811	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑎 ∈ (ℝ ↑m 𝑥) ↔ 𝑎 ∈ (ℝ ↑m 𝑦)))
53	51	eleq2d 2811	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑏 ∈ (ℝ ↑m 𝑥) ↔ 𝑏 ∈ (ℝ ↑m 𝑦)))
54	51	oveq1d 7429	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((ℝ ↑m 𝑥) ↑m ℕ) = ((ℝ ↑m 𝑦) ↑m ℕ))
55	54	eleq2d 2811	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)))
56	54	eleq2d 2811	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)))
57		ixpeq1 8923	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)))
58		ixpeq1 8923	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
59	58	iuneq2d 5018	. . . . . . . . . . . . . . . . 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 4005	. . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝐿‘𝑥) = (𝐿‘𝑦))
62	61	oveqd 7431	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑎(𝐿‘𝑥)𝑏) = (𝑎(𝐿‘𝑦)𝑏))
63	61	oveqd 7431	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))
64	63	mpteq2dv 5243	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))
65	64	fveq2d 6894	. . . . . . . . . . . . . . . . 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 5154	. . . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . 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 3164	. . . . . . . . . . . . 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 343	. . . . . . . . . . . 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 3164	. . . . . . . . . . 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 343	. . . . . . . . . 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 3164	. . . . . . . . 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 343	. . . . . . . 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 3164	. . . . . . 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 7422	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (ℝ ↑m 𝑥) = (ℝ ↑m (𝑦 ∪ {𝑧})))
77	76	eleq2d 2811	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑎 ∈ (ℝ ↑m 𝑥) ↔ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))))
78	76	eleq2d 2811	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑏 ∈ (ℝ ↑m 𝑥) ↔ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))))
79	76	oveq1d 7429	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((ℝ ↑m 𝑥) ↑m ℕ) = ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ))
80	79	eleq2d 2811	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)))
81	79	eleq2d 2811	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)))
82		ixpeq1 8923	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)))
83		ixpeq1 8923	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
84	83	iuneq2d 5018	. . . . . . . . . . . . . . . . 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 4005	. . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐿‘𝑥) = (𝐿‘(𝑦 ∪ {𝑧})))
87	86	oveqd 7431	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑎(𝐿‘𝑥)𝑏) = (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏))
88	86	oveqd 7431	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))
89	88	mpteq2dv 5243	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))
90	89	fveq2d 6894	. . . . . . . . . . . . . . . . 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 5154	. . . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . 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 3164	. . . . . . . . . . . . 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 343	. . . . . . . . . . . 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 3164	. . . . . . . . . . 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 343	. . . . . . . . . 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 3164	. . . . . . . . 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 343	. . . . . . . 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 3164	. . . . . . 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 7422	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑋))
102	101	eleq2d 2811	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑎 ∈ (ℝ ↑m 𝑥) ↔ 𝑎 ∈ (ℝ ↑m 𝑋)))
103	101	eleq2d 2811	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑏 ∈ (ℝ ↑m 𝑥) ↔ 𝑏 ∈ (ℝ ↑m 𝑋)))
104	101	oveq1d 7429	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → ((ℝ ↑m 𝑥) ↑m ℕ) = ((ℝ ↑m 𝑋) ↑m ℕ))
105	104	eleq2d 2811	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)))
106	104	eleq2d 2811	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → (𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)))
107		ixpeq1 8923	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)))
108		ixpeq1 8923	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑋 → X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
109	108	iuneq2d 5018	. . . . . . . . . . . . . . . . 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 4005	. . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑋 → (𝐿‘𝑥) = (𝐿‘𝑋))
112	111	oveqd 7431	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → (𝑎(𝐿‘𝑥)𝑏) = (𝑎(𝐿‘𝑋)𝑏))
113	111	oveqd 7431	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑋 → ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))
114	113	mpteq2dv 5243	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑋 → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))
115	114	fveq2d 6894	. . . . . . . . . . . . . . . . 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 5154	. . . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . 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 3164	. . . . . . . . . . . . 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 343	. . . . . . . . . . . 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 3164	. . . . . . . . . . 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 343	. . . . . . . . . 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 3164	. . . . . . . . 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 343	. . . . . . . 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 3164	. . . . . . 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 6889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑒 → (𝑎‘𝑘) = (𝑒‘𝑘))
128	127	oveq1d 7429	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑒 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑒‘𝑘)[,)(𝑏‘𝑘)))
129	128	fveq2d 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑒 → (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘))))
130	129	prodeq2ad 45015	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑒 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘))))
131	130	ifeq2d 4542	. . . . . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝑓 → (𝑏‘𝑘) = (𝑓‘𝑘))
133	132	oveq2d 7430	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = 𝑓 → ((𝑒‘𝑘)[,)(𝑏‘𝑘)) = ((𝑒‘𝑘)[,)(𝑓‘𝑘)))
134	133	fveq2d 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑓 → (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑒‘𝑘)[,)(𝑓‘𝑘))))
135	134	prodeq2ad 45015	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑓 → ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑓‘𝑘))))
136	135	ifeq2d 4542	. . . . . . . . . . . . . . . . . 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 7510	. . . . . . . . . . . . . . . . 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 5246	. . . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . 15 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑒 ∈ (ℝ ↑m 𝑥), 𝑓 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑓‘𝑘))))))
140		elmapi 8864	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (ℝ ↑m ∅) → 𝑎:∅⟶ℝ)
141	140	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ (ℝ ↑m ∅) ∧ 𝑏 ∈ (ℝ ↑m ∅)) → 𝑎:∅⟶ℝ)
142		elmapi 8864	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ (ℝ ↑m ∅) → 𝑏:∅⟶ℝ)
143	142	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ (ℝ ↑m ∅) ∧ 𝑏 ∈ (ℝ ↑m ∅)) → 𝑏:∅⟶ℝ)
144	139, 141, 143	hoidmv0val 46006	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ (ℝ ↑m ∅) ∧ 𝑏 ∈ (ℝ ↑m ∅)) → (𝑎(𝐿‘∅)𝑏) = 0)
145	144	ad5ant23 758	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (ℝ ↑m ∅)) ∧ 𝑏 ∈ (ℝ ↑m ∅)) ∧ 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → (𝑎(𝐿‘∅)𝑏) = 0)
146		nfv 1909	. . . . . . . . . . . . . . 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 13443	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
149		0fin 9192	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ Fin
150	149	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → ∅ ∈ Fin)
151		ovexd 7449	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ))
154		simpr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
155	151, 152, 153, 154	fvmap 44607	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑐‘𝑗) ∈ (ℝ ↑m ∅))
156		elmapi 8864	. . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → (𝑐‘𝑗):∅⟶ℝ)
159		ovexd 7449	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ))
162		simpr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
163	159, 160, 161, 162	fvmap 44607	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑑‘𝑗) ∈ (ℝ ↑m ∅))
164		elmapi 8864	. . . . . . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → (𝑑‘𝑗):∅⟶ℝ)
167	126, 150, 158, 166	hoidmvcl 46005	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)) ∈ (0[,)+∞))
168	148, 167	sselid 3970	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)) ∈ (0[,]+∞))
169	146, 147, 168	sge0ge0mpt 45861	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))
170	169	adantll 712	. . . . . . . . . . . . 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 5163	. . . . . . . . . . . 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 3136	. . . . . . . . . 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 3136	. . . . . . . . 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 3136	. . . . . . . 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 3136	. . . . . . 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 481	. . . . . . . . 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 8928	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑒 → X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)))
179	178	sseq1d 4003	. . . . . . . . . . . . . . . . 17 ⊢ (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 7421	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑒 → (𝑎(𝐿‘𝑦)𝑏) = (𝑒(𝐿‘𝑦)𝑏))
181	180	breq1d 5151	. . . . . . . . . . . . . . . . 17 ⊢ (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 343	. . . . . . . . . . . . . . . 16 ⊢ (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 3168	. . . . . . . . . . . . . . 15 ⊢ (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 3168	. . . . . . . . . . . . . 14 ⊢ (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 3168	. . . . . . . . . . . . 13 ⊢ (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 8928	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑓 → X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)))
187	186	sseq1d 4003	. . . . . . . . . . . . . . . . . . 19 ⊢ (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 7422	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑓 → (𝑒(𝐿‘𝑦)𝑏) = (𝑒(𝐿‘𝑦)𝑓))
189	188	breq1d 5151	. . . . . . . . . . . . . . . . . . 19 ⊢ (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 343	. . . . . . . . . . . . . . . . . 18 ⊢ (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 3168	. . . . . . . . . . . . . . . . 17 ⊢ (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 3168	. . . . . . . . . . . . . . . 16 ⊢ (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 6889	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑐 = 𝑔 → (𝑐‘𝑗) = (𝑔‘𝑗))
194	193	fveq1d 6892	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 = 𝑔 → ((𝑐‘𝑗)‘𝑘) = ((𝑔‘𝑗)‘𝑘))
195	194	oveq1d 7429	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = 𝑔 → (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
196	195	ixpeq2dv 8928	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = 𝑔 → X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
197	196	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 = 𝑔 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
198	197	iuneq2dv 5013	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (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 4004	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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 7429	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = 𝑔 → ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)) = ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))
201	200	mpteq2dv 5243	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = 𝑔 → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))
202	201	fveq2d 6894	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = 𝑔 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))
203	202	breq2d 5153	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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 343	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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 3168	. . . . . . . . . . . . . . . . . . 19 ⊢ (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 6889	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑑 = ℎ → (𝑑‘𝑗) = (ℎ‘𝑗))
207	206	fveq1d 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑑 = ℎ → ((𝑑‘𝑗)‘𝑘) = ((ℎ‘𝑗)‘𝑘))
208	207	oveq2d 7430	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑑 = ℎ → (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
209	208	ixpeq2dv 8928	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑑 = ℎ → X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
210	209	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑑 = ℎ ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
211	210	iuneq2dv 5013	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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 4004	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (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 7430	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑑 = ℎ → ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)) = ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))
214	213	mpteq2dv 5243	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑑 = ℎ → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))))
215	214	fveq2d 6894	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 = ℎ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))))
216	215	breq2d 5153	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (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 343	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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 3225	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀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	. . . . . . . . . . . . . . . . . . 19 ⊢ (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 278	. . . . . . . . . . . . . . . . . 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))))) ↔ ∀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 3225	. . . . . . . . . . . . . . . . 17 ⊢ (∀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	. . . . . . . . . . . . . . . 16 ⊢ (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 278	. . . . . . . . . . . . . . 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))))) ↔ ∀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 3225	. . . . . . . . . . . . . 14 ⊢ (∀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	. . . . . . . . . . . . 13 ⊢ (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 278	. . . . . . . . . . . 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))))) ↔ ∀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 3225	. . . . . . . . . . 11 ⊢ (∀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	biimpi 215	. . . . . . . . . 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))))))
229	228	adantl 480	. . . . . . . . 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))))))
230		simplll 773	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝜑)
231		eldifi 4117	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ (𝑋 ∖ 𝑦) → 𝑧 ∈ 𝑋)
232	231	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦)) → 𝑧 ∈ 𝑋)
233	232	adantrl 714	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) → 𝑧 ∈ 𝑋)
234	233	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝑧 ∈ 𝑋)
235		simpl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})))
236		uneq1 4147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = (∅ ∪ {𝑧}))
237		0un 4386	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∅ ∪ {𝑧}) = {𝑧}
238	237	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → (∅ ∪ {𝑧}) = {𝑧})
239	236, 238	eqtr2d 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = ∅ → {𝑧} = (𝑦 ∪ {𝑧}))
240	239	eqcomd 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = {𝑧})
241	240	oveq2d 7430	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = ∅ → (ℝ ↑m (𝑦 ∪ {𝑧})) = (ℝ ↑m {𝑧}))
242	241	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → (ℝ ↑m (𝑦 ∪ {𝑧})) = (ℝ ↑m {𝑧}))
243	235, 242	eleqtrd 2827	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑m {𝑧}))
244	243	adantll 712	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑m {𝑧}))
245	230, 234, 244	jca31 513	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → ((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})))
246	245	adantllr 717	. . . . . . . . . . . . . . . . . . . . . 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})))
247	246	adantlr 713	. . . . . . . . . . . . . . . . . . . . 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})))
248	247	adantlr 713	. . . . . . . . . . . . . . . . . . . 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})))
249		simpl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})))
250	241	adantl 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → (ℝ ↑m (𝑦 ∪ {𝑧})) = (ℝ ↑m {𝑧}))
251	249, 250	eleqtrd 2827	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑m {𝑧}))
252	251	adantlr 713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑m {𝑧}))
253	252	adantlll 716	. . . . . . . . . . . . . . . . . . . 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}))
254		simpl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ))
255	241	oveq1d 7429	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = ∅ → ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) = ((ℝ ↑m {𝑧}) ↑m ℕ))
256	255	adantl 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) = ((ℝ ↑m {𝑧}) ↑m ℕ))
257	254, 256	eleqtrd 2827	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ))
258	257	adantll 712	. . . . . . . . . . . . . . . . . . . 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))
259	248, 253, 258	jca31 513	. . . . . . . . . . . . . . . . . . 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)))
260	259	adantlr 713	. . . . . . . . . . . . . . . . . 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)))
261	260	adantlr 713	. . . . . . . . . . . . . . . . 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)))
262		simpl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ))
263	255	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) = ((ℝ ↑m {𝑧}) ↑m ℕ))
264	262, 263	eleqtrd 2827	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ))
265	264	adantlr 713	. . . . . . . . . . . . . . . . . 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))
266	265	adantlll 716	. . . . . . . . . . . . . . . . 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))
267		simpl 481	. . . . . . . . . . . . . . . . . . 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)))
268	239	ixpeq1d 8924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = ∅ → X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)))
269	268	adantl 480	. . . . . . . . . . . . . . . . . . . 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)))
270	239	ixpeq1d 8924	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = ∅ → X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)))
271	270	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 = ∅ ∧ 𝑖 ∈ ℕ) → X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)))
272	271	iuneq2dv 5013	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = ∅ → ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪ 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)))
273		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝑗 → (𝑐‘𝑖) = (𝑐‘𝑗))
274	273	fveq1d 6892	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖)‘𝑘) = ((𝑐‘𝑗)‘𝑘))
275		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝑗 → (𝑑‘𝑖) = (𝑑‘𝑗))
276	275	fveq1d 6892	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = 𝑗 → ((𝑑‘𝑖)‘𝑘) = ((𝑑‘𝑗)‘𝑘))
277	274, 276	oveq12d 7432	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑗 → (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
278	277	ixpeq2dv 8928	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 𝑗 → X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
279	278	cbviunv 5036	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∪ 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))
280	279	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)))
281	272, 280	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = ∅ → ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
282	281	adantl 480	. . . . . . . . . . . . . . . . . . . 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)))
283	269, 282	sseq12d 4005	. . . . . . . . . . . . . . . . . . 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))))
284	267, 283	mpbird 256	. . . . . . . . . . . . . . . . . 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)))
285	284	adantll 712	. . . . . . . . . . . . . . . . 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)))
286	261, 266, 285	jca31 513	. . . . . . . . . . . . . . . 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))))
287		simpr 483	. . . . . . . . . . . . . . . 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 = ∅)
288		fveq1 6889	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 𝑢 → (𝑎‘𝑘) = (𝑢‘𝑘))
289	288	oveq1d 7429	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑢 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑢‘𝑘)[,)(𝑏‘𝑘)))
290	289	fveq2d 6894	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑢 → (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))))
291	290	prodeq2ad 45015	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑢 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))))
292	291	ifeq2d 4542	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑢 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘)))))
293		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑙 → (𝑢‘𝑘) = (𝑢‘𝑙))
294		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑙 → (𝑏‘𝑘) = (𝑏‘𝑙))
295	293, 294	oveq12d 7432	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑙 → ((𝑢‘𝑘)[,)(𝑏‘𝑘)) = ((𝑢‘𝑙)[,)(𝑏‘𝑙)))
296	295	fveq2d 6894	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 = 𝑙 → (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙))))
297	296	cbvprodv 15890	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙)))
298	297	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = 𝑣 → ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙))))
299		fveq1 6889	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑏 = 𝑣 → (𝑏‘𝑙) = (𝑣‘𝑙))
300	299	oveq2d 7430	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 = 𝑣 → ((𝑢‘𝑙)[,)(𝑏‘𝑙)) = ((𝑢‘𝑙)[,)(𝑣‘𝑙)))
301	300	fveq2d 6894	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = 𝑣 → (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙))) = (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙))))
302	301	prodeq2ad 45015	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = 𝑣 → ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙))) = ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙))))
303	298, 302	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = 𝑣 → ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙))))
304	303	ifeq2d 4542	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝑣 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙)))))
305	292, 304	cbvmpov 7510	. . . . . . . . . . . . . . . . . . . . 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)))))
306	305	mpteq2i 5246	. . . . . . . . . . . . . . . . . . . 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))))))
307	126, 306	eqtri 2753	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑢 ∈ (ℝ ↑m 𝑥), 𝑣 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙))))))
308		simp-6r 786	. . . . . . . . . . . . . . . . . . 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)
309		eqid 2725	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑧} = {𝑧}
310		elmapi 8864	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ (ℝ ↑m {𝑧}) → 𝑎:{𝑧}⟶ℝ)
311	310	ad2antlr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) → 𝑎:{𝑧}⟶ℝ)
312	311	ad3antrrr 728	. . . . . . . . . . . . . . . . . . 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)
313		elmapi 8864	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 ∈ (ℝ ↑m {𝑧}) → 𝑏:{𝑧}⟶ℝ)
314	313	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) → 𝑏:{𝑧}⟶ℝ)
315	314	ad3antrrr 728	. . . . . . . . . . . . . . . . . . 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)
316		elmapi 8864	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ) → 𝑐:ℕ⟶(ℝ ↑m {𝑧}))
317	316	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) → 𝑐:ℕ⟶(ℝ ↑m {𝑧}))
318	317	ad2antrr 724	. . . . . . . . . . . . . . . . . . 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}))
319		elmapi 8864	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ) → 𝑑:ℕ⟶(ℝ ↑m {𝑧}))
320	319	ad2antlr 725	. . . . . . . . . . . . . . . . . . 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}))
321		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)))
322		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 = 𝑙 → (𝑎‘𝑘) = (𝑎‘𝑙))
323	322, 294	oveq12d 7432	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 𝑙 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙)))
324		eqcom 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑙 ↔ 𝑙 = 𝑘)
325	324	imbi1i 348	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 = 𝑙 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙))))
326		eqcom 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙)) ↔ ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘)))
327	326	imbi2i 335	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑙 = 𝑘 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘))))
328	325, 327	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 = 𝑙 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘))))
329	323, 328	mpbi 229	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 = 𝑘 → ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘)))
330	329	cbvixpv 8930	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ X𝑙 ∈ {𝑧} ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘))
331	330	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)))
332	277	ixpeq2dv 8928	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑗 → X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = X𝑘 ∈ {𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
333	332	cbviunv 5036	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))
334		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑙 → ((𝑐‘𝑗)‘𝑘) = ((𝑐‘𝑗)‘𝑙))
335		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑙 → ((𝑑‘𝑗)‘𝑘) = ((𝑑‘𝑗)‘𝑙))
336	334, 335	oveq12d 7432	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑙 → (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)))
337	336	cbvixpv 8930	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ X𝑘 ∈ {𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙))
338	337	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ ℕ → X𝑘 ∈ {𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)))
339	338	iuneq2i 5010	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙))
340	333, 339	eqtr2i 2754	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∪ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) = ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))
341	340	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)))
342	331, 341	sseq12d 4005	. . . . . . . . . . . . . . . . . . . . 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))))
343	321, 342	mpbird 256	. . . . . . . . . . . . . . . . . . . 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)))
344	343	adantl 480	. . . . . . . . . . . . . . . . . . 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)))
345	307, 308, 309, 312, 315, 318, 320, 344	hoidmv1le 46017	. . . . . . . . . . . . . . . . . 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)))))
346	345	adantr 479	. . . . . . . . . . . . . . . . 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)))))
347	236, 238	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = {𝑧})
348	347	fveq2d 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ∅ → (𝐿‘(𝑦 ∪ {𝑧})) = (𝐿‘{𝑧}))
349	348	oveqd 7431	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ∅ → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) = (𝑎(𝐿‘{𝑧})𝑏))
350	349	adantl 480	. . . . . . . . . . . . . . . . . 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))
351	348	oveqd 7431	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = ∅ → ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗)))
352	351	mpteq2dv 5243	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ∅ → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗))))
353	352	fveq2d 6894	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ∅ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗)))))
354	353	adantl 480	. . . . . . . . . . . . . . . . . 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)))))
355	350, 354	breq12d 5154	. . . . . . . . . . . . . . . . 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))))))
356	346, 355	mpbird 256	. . . . . . . . . . . . . . . 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)))))
357	286, 287, 356	syl2anc 582	. . . . . . . . . . . . . . 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)))))
358	17	ad8antr 738	. . . . . . . . . . . . . . . . 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)
359		simplrl 775	. . . . . . . . . . . . . . . . . . 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)
360	359	ad5ant12 754	. . . . . . . . . . . . . . . . . 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)
361	360	ad5ant12 754	. . . . . . . . . . . . . . . . 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)
362		simplrr 776	. . . . . . . . . . . . . . . . . . 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))
363	362	ad5ant12 754	. . . . . . . . . . . . . . . . . 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))
364	363	ad5ant12 754	. . . . . . . . . . . . . . . . 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))
365		eqid 2725	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧})
366		elmapi 8864	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ)
367	366	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ)
368	367	ad4ant23 751	. . . . . . . . . . . . . . . . . 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)
369	368	ad3antrrr 728	. . . . . . . . . . . . . . . . 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)
370		elmapi 8864	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ)
371	370	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ)
372	371	ad4ant23 751	. . . . . . . . . . . . . . . . . 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)
373	372	ad3antrrr 728	. . . . . . . . . . . . . . . . 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)
374		elmapi 8864	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) → 𝑐:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
375	374	adantl 480	. . . . . . . . . . . . . . . . . 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})))
376	375	ad3antrrr 728	. . . . . . . . . . . . . . . . 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})))
377		elmapi 8864	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) → 𝑑:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
378	377	ad2antrr 724	. . . . . . . . . . . . . . . . . 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})))
379	378	adantlll 716	. . . . . . . . . . . . . . . . 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})))
380		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 = 𝑙 → (𝑒‘𝑘) = (𝑒‘𝑙))
381		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 = 𝑙 → (𝑓‘𝑘) = (𝑓‘𝑙))
382	380, 381	oveq12d 7432	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑙 → ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = ((𝑒‘𝑙)[,)(𝑓‘𝑙)))
383	382	cbvixpv 8930	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙))
384	383	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ = 𝑜 → X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)))
385		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 = 𝑖 → (𝑔‘𝑗) = (𝑔‘𝑖))
386	385	fveq1d 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 = 𝑖 → ((𝑔‘𝑗)‘𝑘) = ((𝑔‘𝑖)‘𝑘))
387		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 = 𝑖 → (ℎ‘𝑗) = (ℎ‘𝑖))
388	387	fveq1d 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 = 𝑖 → ((ℎ‘𝑗)‘𝑘) = ((ℎ‘𝑖)‘𝑘))
389	386, 388	oveq12d 7432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 = 𝑖 → (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)))
390	389	ixpeq2dv 8928	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = 𝑖 → X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)))
391	390	cbviunv 5036	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪ 𝑖 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘))
392	391	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℎ = 𝑜 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪ 𝑖 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)))
393		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑘 = 𝑙 → ((𝑔‘𝑖)‘𝑘) = ((𝑔‘𝑖)‘𝑙))
394		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑘 = 𝑙 → ((ℎ‘𝑖)‘𝑘) = ((ℎ‘𝑖)‘𝑙))
395	393, 394	oveq12d 7432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑙 → (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙)))
396	395	cbvixpv 8930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙))
397	396	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (ℎ = 𝑜 → X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙)))
398		fveq1 6889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (ℎ = 𝑜 → (ℎ‘𝑖) = (𝑜‘𝑖))
399	398	fveq1d 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (ℎ = 𝑜 → ((ℎ‘𝑖)‘𝑙) = ((𝑜‘𝑖)‘𝑙))
400	399	oveq2d 7430	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (ℎ = 𝑜 → (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙)) = (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))
401	400	ixpeq2dv 8928	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (ℎ = 𝑜 → X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))
402	397, 401	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = 𝑜 → X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))
403	402	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((ℎ = 𝑜 ∧ 𝑖 ∈ ℕ) → X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))
404	403	iuneq2dv 5013	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℎ = 𝑜 → ∪ 𝑖 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))
405	392, 404	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ = 𝑜 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))
406	384, 405	sseq12d 4005	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (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))))
407	385, 387	oveq12d 7432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 = 𝑖 → ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)) = ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖)))
408	407	cbvmptv 5254	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖)))
409	408	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ℎ = 𝑜 → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖))))
410	398	oveq2d 7430	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = 𝑜 → ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖)) = ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))
411	410	mpteq2dv 5243	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ℎ = 𝑜 → (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖))) = (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))
412	409, 411	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℎ = 𝑜 → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))
413	412	fveq2d 6894	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ = 𝑜 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))) = (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))))
414	413	breq2d 5153	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ = 𝑜 → ((𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))) ↔ (𝑒(𝐿‘𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))))
415	406, 414	imbi12d 343	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (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)))))))
416	415	cbvralvw 3225	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀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))))))
417	416	ralbii 3083	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀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))))))
418	417	ralbii 3083	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀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))))))
419	418	ralbii 3083	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀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	biimpi 215	. . . . . . . . . . . . . . . . . . 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))))))
421	420	adantl 480	. . . . . . . . . . . . . . . . . 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))))))
422	421	ad6antr 734	. . . . . . . . . . . . . . . . 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))))))
423	323	cbvixpv 8930	. . . . . . . . . . . . . . . . . . . 20 ⊢ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑙)[,)(𝑏‘𝑙))
424	336	cbvixpv 8930	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙))
425	424	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑖 → X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)))
426		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → (𝑐‘𝑗) = (𝑐‘𝑖))
427	426	fveq1d 6892	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑖 → ((𝑐‘𝑗)‘𝑙) = ((𝑐‘𝑖)‘𝑙))
428		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → (𝑑‘𝑗) = (𝑑‘𝑖))
429	428	fveq1d 6892	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑖 → ((𝑑‘𝑗)‘𝑙) = ((𝑑‘𝑖)‘𝑙))
430	427, 429	oveq12d 7432	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑖 → (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) = (((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙)))
431	430	ixpeq2dv 8928	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑖 → X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙)))
432	425, 431	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑖 → X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙)))
433	432	cbviunv 5036	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙))
434	423, 433	sseq12i 4002	. . . . . . . . . . . . . . . . . . 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)))
435	434	biimpi 215	. . . . . . . . . . . . . . . . . 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)))
436	435	ad2antlr 725	. . . . . . . . . . . . . . . . 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)))
437		neqne 2938	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑦 = ∅ → 𝑦 ≠ ∅)
438	437	adantl 480	. . . . . . . . . . . . . . . . 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 ≠ ∅)
439	307, 358, 361, 364, 365, 369, 373, 376, 379, 422, 436, 438	hoidmvlelem5 46022	. . . . . . . . . . . . . . . 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)))))
440	273, 275	oveq12d 7432	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑖)) = ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))
441	440	cbvmptv 5254	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℕ ↦ ((𝑐‘𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑖))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))
442	441	fveq2i 6893	. . . . . . . . . . . . . . . . 17 ⊢ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑐‘𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑖)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))
443	442	breq2i 5149	. . . . . . . . . . . . . . . 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)))))
444	439, 443	sylib 217	. . . . . . . . . . . . . . 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)))))
445	357, 444	pm2.61dan 811	. . . . . . . . . . . . . 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)))))
446	445	ex 411	. . . . . . . . . . . . 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))))))
447	446	ralrimiva 3136	. . . . . . . . . . . 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))))))
448	447	ralrimiva 3136	. . . . . . . . . . 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))))))
449	448	ralrimiva 3136	. . . . . . . . . 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))))))
450	449	ralrimiva 3136	. . . . . . . . 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))))))
451	177, 229, 450	syl2anc 582	. . . . . . . 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))))))
452	451	ex 411	. . . . . . 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)))))))
453	50, 75, 100, 125, 176, 452, 17	findcard2d 9187	. . . . . 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))))))
454		fveq1 6889	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘))
455	454	oveq1d 7429	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝐴‘𝑘)[,)(𝑏‘𝑘)))
456	455	ixpeq2dv 8928	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)))
457	456	sseq1d 4003	. . . . . . . . . . 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))))
458		oveq1 7421	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (𝑎(𝐿‘𝑋)𝑏) = (𝐴(𝐿‘𝑋)𝑏))
459	458	breq1d 5151	. . . . . . . . . . 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))))))
460	457, 459	imbi12d 343	. . . . . . . . . 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)))))))
461	460	ralbidv 3168	. . . . . . . . 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)))))))
462	461	ralbidv 3168	. . . . . . . 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)))))))
463	462	ralbidv 3168	. . . . . . 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)))))))
464	463	rspcva 3599	. . . . . 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))))))
465	25, 453, 464	syl2anc 582	. . . . 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))))))
466		fveq1 6889	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → (𝑏‘𝑘) = (𝐵‘𝑘))
467	466	oveq2d 7430	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → ((𝐴‘𝑘)[,)(𝑏‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
468	467	ixpeq2dv 8928	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
469	468	sseq1d 4003	. . . . . . . . 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))))
470		oveq2 7422	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝐴(𝐿‘𝑋)𝑏) = (𝐴(𝐿‘𝑋)𝐵))
471	470	breq1d 5151	. . . . . . . . 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))))))
472	469, 471	imbi12d 343	. . . . . . . 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)))))))
473	472	ralbidv 3168	. . . . . . 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)))))))
474	473	ralbidv 3168	. . . . . 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)))))))
475	474	rspcva 3599	. . . . 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))))))
476	21, 465, 475	syl2anc 582	. . . 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))))))
477		fveq1 6889	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑗) = (𝐶‘𝑗))
478	477	fveq1d 6892	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → ((𝑐‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
479	478	oveq1d 7429	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
480	479	ixpeq2dv 8928	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
481	480	adantr 479	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
482	481	iuneq2dv 5013	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))
483	482	sseq2d 4004	. . . . . . 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))))
484	477	oveq1d 7429	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)) = ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))
485	484	mpteq2dv 5243	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))
486	485	fveq2d 6894	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))
487	486	breq2d 5153	. . . . . . 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))))))
488	483, 487	imbi12d 343	. . . . . 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)))))))
489	488	ralbidv 3168	. . . . 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)))))))
490	489	rspcva 3599	. . . 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))))))
491	13, 476, 490	syl2anc 582	. . 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))))))
492		fveq1 6889	. . . . . . . . . . 11 ⊢ (𝑑 = 𝐷 → (𝑑‘𝑗) = (𝐷‘𝑗))
493	492	fveq1d 6892	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝑑‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
494	493	oveq2d 7430	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
495	494	ixpeq2dv 8928	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
496	495	adantr 479	. . . . . . 7 ⊢ ((𝑑 = 𝐷 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
497	496	iuneq2dv 5013	. . . . . 6 ⊢ (𝑑 = 𝐷 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
498	497	sseq2d 4004	. . . . 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))))
499	492	oveq2d 7430	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)) = ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))
500	499	mpteq2dv 5243	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))))
501	500	fveq2d 6894	. . . . . 6 ⊢ (𝑑 = 𝐷 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
502	501	breq2d 5153	. . . . 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))))))
503	498, 502	imbi12d 343	. . . 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)))))))
504	503	rspcva 3599	. . 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))))))
505	9, 491, 504	syl2anc 582	. 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))))))
506	1, 505	mpd 15	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  ∀wral 3051  Vcvv 3463   ∖ cdif 3936   ∪ cun 3937   ⊆ wss 3939  ∅c0 4316  ifcif 4522  {csn 4622  ∪ ciun 4989   class class class wbr 5141   ↦ cmpt 5224  ⟶wf 6537  ‘cfv 6541  (class class class)co 7414   ∈ cmpo 7416   ↑_m cmap 8841  Xcixp 8912  Fincfn 8960  ℝcr 11135  0cc0 11136  +∞cpnf 11273   ≤ cle 11277  ℕcn 12240  [,)cico 13356  [,]cicc 13357  ∏cprod 15879  volcvol 25408  Σ^{^}csumge0 45785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736  ax-inf2 9662  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213  ax-pre-sup 11214

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4943  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7680  df-om 7867  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-2o 8484  df-er 8721  df-map 8843  df-pm 8844  df-ixp 8913  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-fi 9432  df-sup 9463  df-inf 9464  df-oi 9531  df-dju 9922  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-div 11900  df-nn 12241  df-2 12303  df-3 12304  df-n0 12501  df-z 12587  df-uz 12851  df-q 12961  df-rp 13005  df-xneg 13122  df-xadd 13123  df-xmul 13124  df-ioo 13358  df-ico 13360  df-icc 13361  df-fz 13515  df-fzo 13658  df-fl 13787  df-seq 13997  df-exp 14057  df-hash 14320  df-cj 15076  df-re 15077  df-im 15078  df-sqrt 15212  df-abs 15213  df-clim 15462  df-rlim 15463  df-sum 15663  df-prod 15880  df-rest 17401  df-topgen 17422  df-psmet 21273  df-xmet 21274  df-met 21275  df-bl 21276  df-mopn 21277  df-top 22812  df-topon 22829  df-bases 22865  df-cmp 23307  df-ovol 25409  df-vol 25410  df-sumge0 45786

This theorem is referenced by:  ovnhoilem2  46025