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Theorem hoidmvle 42751
 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
hoidmvle.x (𝜑𝑋 ∈ Fin)
hoidmvle.a (𝜑𝐴:𝑋⟶ℝ)
hoidmvle.b (𝜑𝐵:𝑋⟶ℝ)
hoidmvle.c (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑋))
hoidmvle.d (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑋))
hoidmvle.s (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
Assertion
Ref Expression
hoidmvle (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
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 𝑖 𝑙 𝑜 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hoidmvle.s . 2 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
2 hoidmvle.d . . . 4 (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑋))
3 ovex 7181 . . . . . . 7 (ℝ ↑m 𝑋) ∈ V
4 nnex 11633 . . . . . . 7 ℕ ∈ V
53, 4pm3.2i 471 . . . . . 6 ((ℝ ↑m 𝑋) ∈ V ∧ ℕ ∈ V)
65a1i 11 . . . . 5 (𝜑 → ((ℝ ↑m 𝑋) ∈ V ∧ ℕ ∈ V))
7 elmapg 8409 . . . . 5 (((ℝ ↑m 𝑋) ∈ V ∧ ℕ ∈ V) → (𝐷 ∈ ((ℝ ↑m 𝑋) ↑m ℕ) ↔ 𝐷:ℕ⟶(ℝ ↑m 𝑋)))
86, 7syl 17 . . . 4 (𝜑 → (𝐷 ∈ ((ℝ ↑m 𝑋) ↑m ℕ) ↔ 𝐷:ℕ⟶(ℝ ↑m 𝑋)))
92, 8mpbird 258 . . 3 (𝜑𝐷 ∈ ((ℝ ↑m 𝑋) ↑m ℕ))
10 hoidmvle.c . . . . 5 (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑋))
11 elmapg 8409 . . . . . 6 (((ℝ ↑m 𝑋) ∈ V ∧ ℕ ∈ V) → (𝐶 ∈ ((ℝ ↑m 𝑋) ↑m ℕ) ↔ 𝐶:ℕ⟶(ℝ ↑m 𝑋)))
126, 11syl 17 . . . . 5 (𝜑 → (𝐶 ∈ ((ℝ ↑m 𝑋) ↑m ℕ) ↔ 𝐶:ℕ⟶(ℝ ↑m 𝑋)))
1310, 12mpbird 258 . . . 4 (𝜑𝐶 ∈ ((ℝ ↑m 𝑋) ↑m ℕ))
14 hoidmvle.b . . . . . 6 (𝜑𝐵:𝑋⟶ℝ)
15 reex 10617 . . . . . . . . 9 ℝ ∈ V
1615a1i 11 . . . . . . . 8 (𝜑 → ℝ ∈ V)
17 hoidmvle.x . . . . . . . 8 (𝜑𝑋 ∈ Fin)
1816, 17jca 512 . . . . . . 7 (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ Fin))
19 elmapg 8409 . . . . . . 7 ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
2018, 19syl 17 . . . . . 6 (𝜑 → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ))
2114, 20mpbird 258 . . . . 5 (𝜑𝐵 ∈ (ℝ ↑m 𝑋))
22 hoidmvle.a . . . . . . 7 (𝜑𝐴:𝑋⟶ℝ)
23 elmapg 8409 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → (𝐴 ∈ (ℝ ↑m 𝑋) ↔ 𝐴:𝑋⟶ℝ))
2418, 23syl 17 . . . . . . 7 (𝜑 → (𝐴 ∈ (ℝ ↑m 𝑋) ↔ 𝐴:𝑋⟶ℝ))
2522, 24mpbird 258 . . . . . 6 (𝜑𝐴 ∈ (ℝ ↑m 𝑋))
26 oveq2 7156 . . . . . . . . . 10 (𝑥 = ∅ → (ℝ ↑m 𝑥) = (ℝ ↑m ∅))
2726eleq2d 2903 . . . . . . . . 9 (𝑥 = ∅ → (𝑎 ∈ (ℝ ↑m 𝑥) ↔ 𝑎 ∈ (ℝ ↑m ∅)))
2826eleq2d 2903 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑏 ∈ (ℝ ↑m 𝑥) ↔ 𝑏 ∈ (ℝ ↑m ∅)))
2926oveq1d 7163 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((ℝ ↑m 𝑥) ↑m ℕ) = ((ℝ ↑m ∅) ↑m ℕ))
3029eleq2d 2903 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)))
3129eleq2d 2903 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)))
32 ixpeq1 8461 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)))
33 ixpeq1 8461 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
3433iuneq2d 4945 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
3532, 34sseq12d 4004 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
36 fveq2 6667 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → (𝐿𝑥) = (𝐿‘∅))
3736oveqd 7165 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → (𝑎(𝐿𝑥)𝑏) = (𝑎(𝐿‘∅)𝑏))
3836oveqd 7165 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ∅ → ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)) = ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))
3938mpteq2dv 5159 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))
4039fveq2d 6671 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))))
4137, 40breq12d 5076 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → ((𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) ↔ (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))))
4235, 41imbi12d 346 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → ((X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ (X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))))))
4331, 42imbi12d 346 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → (X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) → (X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))))))
4443ralbidv2 3200 . . . . . . . . . . . . 13 (𝑥 = ∅ → (∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))))))
4530, 44imbi12d 346 . . . . . . . . . . . 12 (𝑥 = ∅ → ((𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → ∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) → ∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))))))
4645ralbidv2 3200 . . . . . . . . . . 11 (𝑥 = ∅ → (∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))))))
4728, 46imbi12d 346 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑏 ∈ (ℝ ↑m 𝑥) → ∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑏 ∈ (ℝ ↑m ∅) → ∀𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))))))
4847ralbidv2 3200 . . . . . . . . 9 (𝑥 = ∅ → (∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑏 ∈ (ℝ ↑m ∅)∀𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))))))
4927, 48imbi12d 346 . . . . . . . 8 (𝑥 = ∅ → ((𝑎 ∈ (ℝ ↑m 𝑥) → ∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑎 ∈ (ℝ ↑m ∅) → ∀𝑏 ∈ (ℝ ↑m ∅)∀𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))))))
5049ralbidv2 3200 . . . . . . 7 (𝑥 = ∅ → (∀𝑎 ∈ (ℝ ↑m 𝑥)∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑎 ∈ (ℝ ↑m ∅)∀𝑏 ∈ (ℝ ↑m ∅)∀𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))))))
51 oveq2 7156 . . . . . . . . . 10 (𝑥 = 𝑦 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑦))
5251eleq2d 2903 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑎 ∈ (ℝ ↑m 𝑥) ↔ 𝑎 ∈ (ℝ ↑m 𝑦)))
5351eleq2d 2903 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑏 ∈ (ℝ ↑m 𝑥) ↔ 𝑏 ∈ (ℝ ↑m 𝑦)))
5451oveq1d 7163 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((ℝ ↑m 𝑥) ↑m ℕ) = ((ℝ ↑m 𝑦) ↑m ℕ))
5554eleq2d 2903 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)))
5654eleq2d 2903 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)))
57 ixpeq1 8461 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)))
58 ixpeq1 8461 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
5958iuneq2d 4945 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
6057, 59sseq12d 4004 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
61 fveq2 6667 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝐿𝑥) = (𝐿𝑦))
6261oveqd 7165 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (𝑎(𝐿𝑥)𝑏) = (𝑎(𝐿𝑦)𝑏))
6361oveqd 7165 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)) = ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))
6463mpteq2dv 5159 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))
6564fveq2d 6671 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))
6662, 65breq12d 5076 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) ↔ (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))))
6760, 66imbi12d 346 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ (X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))))
6856, 67imbi12d 346 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → (X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ) → (X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))))))
6968ralbidv2 3200 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))))
7055, 69imbi12d 346 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → ∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ) → ∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))))))
7170ralbidv2 3200 . . . . . . . . . . 11 (𝑥 = 𝑦 → (∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))))
7253, 71imbi12d 346 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑏 ∈ (ℝ ↑m 𝑥) → ∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑏 ∈ (ℝ ↑m 𝑦) → ∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))))))
7372ralbidv2 3200 . . . . . . . . 9 (𝑥 = 𝑦 → (∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))))
7452, 73imbi12d 346 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑎 ∈ (ℝ ↑m 𝑥) → ∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑎 ∈ (ℝ ↑m 𝑦) → ∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))))))
7574ralbidv2 3200 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑎 ∈ (ℝ ↑m 𝑥)∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑎 ∈ (ℝ ↑m 𝑦)∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))))
76 oveq2 7156 . . . . . . . . . 10 (𝑥 = (𝑦 ∪ {𝑧}) → (ℝ ↑m 𝑥) = (ℝ ↑m (𝑦 ∪ {𝑧})))
7776eleq2d 2903 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑎 ∈ (ℝ ↑m 𝑥) ↔ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))))
7876eleq2d 2903 . . . . . . . . . . 11 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑏 ∈ (ℝ ↑m 𝑥) ↔ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))))
7976oveq1d 7163 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 ∪ {𝑧}) → ((ℝ ↑m 𝑥) ↑m ℕ) = ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ))
8079eleq2d 2903 . . . . . . . . . . . . 13 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)))
8179eleq2d 2903 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)))
82 ixpeq1 8461 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∪ {𝑧}) → X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)))
83 ixpeq1 8461 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑦 ∪ {𝑧}) → X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
8483iuneq2d 4945 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
8582, 84sseq12d 4004 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 ∪ {𝑧}) → (X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
86 fveq2 6667 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐿𝑥) = (𝐿‘(𝑦 ∪ {𝑧})))
8786oveqd 7165 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑎(𝐿𝑥)𝑏) = (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏))
8886oveqd 7165 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)) = ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))
8988mpteq2dv 5159 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))
9089fveq2d 6671 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∪ {𝑧}) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))
9187, 90breq12d 5076 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) ↔ (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))))
9285, 91imbi12d 346 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 ∪ {𝑧}) → ((X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ (X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))))
9381, 92imbi12d 346 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → (X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) → (X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))))))
9493ralbidv2 3200 . . . . . . . . . . . . 13 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))))
9580, 94imbi12d 346 . . . . . . . . . . . 12 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → ∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) → ∀𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))))))
9695ralbidv2 3200 . . . . . . . . . . 11 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))))
9778, 96imbi12d 346 . . . . . . . . . 10 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑏 ∈ (ℝ ↑m 𝑥) → ∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) → ∀𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))))))
9897ralbidv2 3200 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))∀𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))))
9977, 98imbi12d 346 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑎 ∈ (ℝ ↑m 𝑥) → ∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))))))
10099ralbidv2 3200 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑎 ∈ (ℝ ↑m 𝑥)∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))))
101 oveq2 7156 . . . . . . . . . 10 (𝑥 = 𝑋 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑋))
102101eleq2d 2903 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑎 ∈ (ℝ ↑m 𝑥) ↔ 𝑎 ∈ (ℝ ↑m 𝑋)))
103101eleq2d 2903 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑏 ∈ (ℝ ↑m 𝑥) ↔ 𝑏 ∈ (ℝ ↑m 𝑋)))
104101oveq1d 7163 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → ((ℝ ↑m 𝑥) ↑m ℕ) = ((ℝ ↑m 𝑋) ↑m ℕ))
105104eleq2d 2903 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)))
106104eleq2d 2903 . . . . . . . . . . . . . . 15 (𝑥 = 𝑋 → (𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) ↔ 𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)))
107 ixpeq1 8461 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑋X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)))
108 ixpeq1 8461 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑋X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
109108iuneq2d 4945 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑋 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
110107, 109sseq12d 4004 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑋 → (X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
111 fveq2 6667 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑋 → (𝐿𝑥) = (𝐿𝑋))
112111oveqd 7165 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑋 → (𝑎(𝐿𝑥)𝑏) = (𝑎(𝐿𝑋)𝑏))
113111oveqd 7165 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑋 → ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)) = ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))
114113mpteq2dv 5159 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑋 → (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))
115114fveq2d 6671 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑋 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))
116112, 115breq12d 5076 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑋 → ((𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) ↔ (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))
117110, 116imbi12d 346 . . . . . . . . . . . . . . 15 (𝑥 = 𝑋 → ((X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ (X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))))
118106, 117imbi12d 346 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → ((𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → (X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ) → (X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))))
119118ralbidv2 3200 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))))
120105, 119imbi12d 346 . . . . . . . . . . . 12 (𝑥 = 𝑋 → ((𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ) → ∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ) → ∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))))
121120ralbidv2 3200 . . . . . . . . . . 11 (𝑥 = 𝑋 → (∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))))
122103, 121imbi12d 346 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝑏 ∈ (ℝ ↑m 𝑥) → ∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑏 ∈ (ℝ ↑m 𝑋) → ∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))))
123122ralbidv2 3200 . . . . . . . . 9 (𝑥 = 𝑋 → (∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))))
124102, 123imbi12d 346 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑎 ∈ (ℝ ↑m 𝑥) → ∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))))) ↔ (𝑎 ∈ (ℝ ↑m 𝑋) → ∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))))
125124ralbidv2 3200 . . . . . . 7 (𝑥 = 𝑋 → (∀𝑎 ∈ (ℝ ↑m 𝑥)∀𝑏 ∈ (ℝ ↑m 𝑥)∀𝑐 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑥) ↑m ℕ)(X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))))) ↔ ∀𝑎 ∈ (ℝ ↑m 𝑋)∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))))
126 hoidmvle.l . . . . . . . . . . . . . . . 16 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
127 fveq1 6666 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑒 → (𝑎𝑘) = (𝑒𝑘))
128127oveq1d 7163 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 𝑒 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑒𝑘)[,)(𝑏𝑘)))
129128fveq2d 6671 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑒 → (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = (vol‘((𝑒𝑘)[,)(𝑏𝑘))))
130129prodeq2ad 41741 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑒 → ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑏𝑘))))
131130ifeq2d 4489 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑒 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑏𝑘)))))
132 fveq1 6666 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝑓 → (𝑏𝑘) = (𝑓𝑘))
133132oveq2d 7164 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = 𝑓 → ((𝑒𝑘)[,)(𝑏𝑘)) = ((𝑒𝑘)[,)(𝑓𝑘)))
134133fveq2d 6671 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑓 → (vol‘((𝑒𝑘)[,)(𝑏𝑘))) = (vol‘((𝑒𝑘)[,)(𝑓𝑘))))
135134prodeq2ad 41741 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑓 → ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑓𝑘))))
136135ifeq2d 4489 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑓 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑓𝑘)))))
137131, 136cbvmpov 7239 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))) = (𝑒 ∈ (ℝ ↑m 𝑥), 𝑓 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑓𝑘)))))
138137mpteq2i 5155 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘)))))) = (𝑥 ∈ Fin ↦ (𝑒 ∈ (ℝ ↑m 𝑥), 𝑓 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑓𝑘))))))
139126, 138eqtri 2849 . . . . . . . . . . . . . . 15 𝐿 = (𝑥 ∈ Fin ↦ (𝑒 ∈ (ℝ ↑m 𝑥), 𝑓 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑓𝑘))))))
140 elmapi 8418 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (ℝ ↑m ∅) → 𝑎:∅⟶ℝ)
141140adantr 481 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (ℝ ↑m ∅) ∧ 𝑏 ∈ (ℝ ↑m ∅)) → 𝑎:∅⟶ℝ)
142 elmapi 8418 . . . . . . . . . . . . . . . 16 (𝑏 ∈ (ℝ ↑m ∅) → 𝑏:∅⟶ℝ)
143142adantl 482 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (ℝ ↑m ∅) ∧ 𝑏 ∈ (ℝ ↑m ∅)) → 𝑏:∅⟶ℝ)
144139, 141, 143hoidmv0val 42734 . . . . . . . . . . . . . 14 ((𝑎 ∈ (ℝ ↑m ∅) ∧ 𝑏 ∈ (ℝ ↑m ∅)) → (𝑎(𝐿‘∅)𝑏) = 0)
145144ad5ant23 756 . . . . . . . . . . . . 13 (((((𝜑𝑎 ∈ (ℝ ↑m ∅)) ∧ 𝑏 ∈ (ℝ ↑m ∅)) ∧ 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → (𝑎(𝐿‘∅)𝑏) = 0)
146 nfv 1908 . . . . . . . . . . . . . . 15 𝑗(𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ))
1474a1i 11 . . . . . . . . . . . . . . 15 ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → ℕ ∈ V)
148 icossicc 12814 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ (0[,]+∞)
149 0fin 8735 . . . . . . . . . . . . . . . . . 18 ∅ ∈ Fin
150149a1i 11 . . . . . . . . . . . . . . . . 17 (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → ∅ ∈ Fin)
151 ovexd 7183 . . . . . . . . . . . . . . . . . . . 20 ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (ℝ ↑m ∅) ∈ V)
1524a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → ℕ ∈ V)
153 simpl 483 . . . . . . . . . . . . . . . . . . . 20 ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ))
154 simpr 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
155151, 152, 153, 154fvmap 41328 . . . . . . . . . . . . . . . . . . 19 ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑐𝑗) ∈ (ℝ ↑m ∅))
156 elmapi 8418 . . . . . . . . . . . . . . . . . . 19 ((𝑐𝑗) ∈ (ℝ ↑m ∅) → (𝑐𝑗):∅⟶ℝ)
157155, 156syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑐𝑗):∅⟶ℝ)
158157adantlr 711 . . . . . . . . . . . . . . . . 17 (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → (𝑐𝑗):∅⟶ℝ)
159 ovexd 7183 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (ℝ ↑m ∅) ∈ V)
1604a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → ℕ ∈ V)
161 simpl 483 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ))
162 simpr 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
163159, 160, 161, 162fvmap 41328 . . . . . . . . . . . . . . . . . . 19 ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑑𝑗) ∈ (ℝ ↑m ∅))
164 elmapi 8418 . . . . . . . . . . . . . . . . . . 19 ((𝑑𝑗) ∈ (ℝ ↑m ∅) → (𝑑𝑗):∅⟶ℝ)
165163, 164syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑑𝑗):∅⟶ℝ)
166165adantll 710 . . . . . . . . . . . . . . . . 17 (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → (𝑑𝑗):∅⟶ℝ)
167126, 150, 158, 166hoidmvcl 42733 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)) ∈ (0[,)+∞))
168148, 167sseldi 3969 . . . . . . . . . . . . . . 15 (((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑗 ∈ ℕ) → ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)) ∈ (0[,]+∞))
169146, 147, 168sge0ge0mpt 42589 . . . . . . . . . . . . . 14 ((𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))))
170169adantll 710 . . . . . . . . . . . . 13 (((((𝜑𝑎 ∈ (ℝ ↑m ∅)) ∧ 𝑏 ∈ (ℝ ↑m ∅)) ∧ 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))))
171145, 170eqbrtrd 5085 . . . . . . . . . . . 12 (((((𝜑𝑎 ∈ (ℝ ↑m ∅)) ∧ 𝑏 ∈ (ℝ ↑m ∅)) ∧ 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))))
172171a1d 25 . . . . . . . . . . 11 (((((𝜑𝑎 ∈ (ℝ ↑m ∅)) ∧ 𝑏 ∈ (ℝ ↑m ∅)) ∧ 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → (X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))))
173172ralrimiva 3187 . . . . . . . . . 10 ((((𝜑𝑎 ∈ (ℝ ↑m ∅)) ∧ 𝑏 ∈ (ℝ ↑m ∅)) ∧ 𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)) → ∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))))
174173ralrimiva 3187 . . . . . . . . 9 (((𝜑𝑎 ∈ (ℝ ↑m ∅)) ∧ 𝑏 ∈ (ℝ ↑m ∅)) → ∀𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))))
175174ralrimiva 3187 . . . . . . . 8 ((𝜑𝑎 ∈ (ℝ ↑m ∅)) → ∀𝑏 ∈ (ℝ ↑m ∅)∀𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))))
176175ralrimiva 3187 . . . . . . 7 (𝜑 → ∀𝑎 ∈ (ℝ ↑m ∅)∀𝑏 ∈ (ℝ ↑m ∅)∀𝑐 ∈ ((ℝ ↑m ∅) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m ∅) ↑m ℕ)(X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))))
177 simpl 483 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑎 ∈ (ℝ ↑m 𝑦)∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))) → (𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))))
178128ixpeq2dv 8466 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑒X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)))
179178sseq1d 4002 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑒 → (X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
180 oveq1 7155 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑒 → (𝑎(𝐿𝑦)𝑏) = (𝑒(𝐿𝑦)𝑏))
181180breq1d 5073 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑒 → ((𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))) ↔ (𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))))
182179, 181imbi12d 346 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑒 → ((X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ (X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))))
183182ralbidv 3202 . . . . . . . . . . . . . . 15 (𝑎 = 𝑒 → (∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))))
184183ralbidv 3202 . . . . . . . . . . . . . 14 (𝑎 = 𝑒 → (∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))))
185184ralbidv 3202 . . . . . . . . . . . . 13 (𝑎 = 𝑒 → (∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))))
186133ixpeq2dv 8466 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑓X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) = X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)))
187186sseq1d 4002 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑓 → (X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
188 oveq2 7156 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑓 → (𝑒(𝐿𝑦)𝑏) = (𝑒(𝐿𝑦)𝑓))
189188breq1d 5073 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑓 → ((𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))) ↔ (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))))
190187, 189imbi12d 346 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑓 → ((X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ (X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))))
191190ralbidv 3202 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑓 → (∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))))
192191ralbidv 3202 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑓 → (∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))))
193 fveq1 6666 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑐 = 𝑔 → (𝑐𝑗) = (𝑔𝑗))
194193fveq1d 6669 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = 𝑔 → ((𝑐𝑗)‘𝑘) = ((𝑔𝑗)‘𝑘))
195194oveq1d 7163 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = 𝑔 → (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
196195ixpeq2dv 8466 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = 𝑔X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
197196adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐 = 𝑔𝑗 ∈ ℕ) → X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
198197iuneq2dv 4940 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = 𝑔 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
199198sseq2d 4003 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑔 → (X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
200193oveq1d 7163 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = 𝑔 → ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)) = ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗)))
201200mpteq2dv 5159 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = 𝑔 → (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗))))
202201fveq2d 6671 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = 𝑔 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗)))))
203202breq2d 5075 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑔 → ((𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))) ↔ (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗))))))
204199, 203imbi12d 346 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = 𝑔 → ((X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ (X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗)))))))
205204ralbidv 3202 . . . . . . . . . . . . . . . . . . 19 (𝑐 = 𝑔 → (∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗)))))))
206 fveq1 6666 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑑 = → (𝑑𝑗) = (𝑗))
207206fveq1d 6669 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑 = → ((𝑑𝑗)‘𝑘) = ((𝑗)‘𝑘))
208207oveq2d 7164 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = → (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)))
209208ixpeq2dv 8466 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)))
210209adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑑 = 𝑗 ∈ ℕ) → X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)))
211210iuneq2dv 4940 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)))
212211sseq2d 4003 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 = → (X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘))))
213206oveq2d 7164 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = → ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗)) = ((𝑔𝑗)(𝐿𝑦)(𝑗)))
214213mpteq2dv 5159 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = → (𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))
215214fveq2d 6671 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))
216215breq2d 5075 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 = → ((𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗)))) ↔ (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))))
217212, 216imbi12d 346 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = → ((X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ (X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))))
218217cbvralv 3458 . . . . . . . . . . . . . . . . . . . 20 (∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))))
219218a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑐 = 𝑔 → (∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))))
220205, 219bitrd 280 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝑔 → (∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))))
221220cbvralv 3458 . . . . . . . . . . . . . . . . 17 (∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))))
222221a1i 11 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑓 → (∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))))
223192, 222bitrd 280 . . . . . . . . . . . . . . 15 (𝑏 = 𝑓 → (∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))))
224223cbvralv 3458 . . . . . . . . . . . . . 14 (∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))))
225224a1i 11 . . . . . . . . . . . . 13 (𝑎 = 𝑒 → (∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))))
226185, 225bitrd 280 . . . . . . . . . . . 12 (𝑎 = 𝑒 → (∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))))
227226cbvralv 3458 . . . . . . . . . . 11 (∀𝑎 ∈ (ℝ ↑m 𝑦)∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) ↔ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))))
228227biimpi 217 . . . . . . . . . 10 (∀𝑎 ∈ (ℝ ↑m 𝑦)∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) → ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))))
229228adantl 482 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑎 ∈ (ℝ ↑m 𝑦)∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))) → ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))))
230 simplll 771 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝜑)
231 eldifi 4107 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ (𝑋𝑦) → 𝑧𝑋)
232231adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑧 ∈ (𝑋𝑦)) → 𝑧𝑋)
233232adantrl 712 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) → 𝑧𝑋)
234233ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝑧𝑋)
235 simpl 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})))
236 uneq1 4136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = (∅ ∪ {𝑧}))
237 0un 4350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∅ ∪ {𝑧}) = {𝑧}
238237a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → (∅ ∪ {𝑧}) = {𝑧})
239236, 238eqtr2d 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ∅ → {𝑧} = (𝑦 ∪ {𝑧}))
240239eqcomd 2832 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = {𝑧})
241240oveq2d 7164 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = ∅ → (ℝ ↑m (𝑦 ∪ {𝑧})) = (ℝ ↑m {𝑧}))
242241adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → (ℝ ↑m (𝑦 ∪ {𝑧})) = (ℝ ↑m {𝑧}))
243235, 242eleqtrd 2920 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑m {𝑧}))
244243adantll 710 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑m {𝑧}))
245230, 234, 244jca31 515 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → ((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})))
246245adantllr 715 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → ((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})))
247246adantlr 711 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → ((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})))
248247adantlr 711 . . . . . . . . . . . . . . . . . . . 20 (((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)) ∧ 𝑦 = ∅) → ((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})))
249 simpl 483 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})))
250241adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → (ℝ ↑m (𝑦 ∪ {𝑧})) = (ℝ ↑m {𝑧}))
251249, 250eleqtrd 2920 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑m {𝑧}))
252251adantlr 711 . . . . . . . . . . . . . . . . . . . . 21 (((𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑m {𝑧}))
253252adantlll 714 . . . . . . . . . . . . . . . . . . . 20 (((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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 483 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ))
255241oveq1d 7163 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = ∅ → ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) = ((ℝ ↑m {𝑧}) ↑m ℕ))
256255adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) = ((ℝ ↑m {𝑧}) ↑m ℕ))
257254, 256eleqtrd 2920 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ))
258257adantll 710 . . . . . . . . . . . . . . . . . . . 20 (((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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))
259248, 253, 258jca31 515 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)) ∧ 𝑦 = ∅) → ((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)))
260259adantlr 711 . . . . . . . . . . . . . . . . . 18 ((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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 = ∅) → ((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)))
261260adantlr 711 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ 𝑦 = ∅) → ((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)))
262 simpl 483 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ))
263255adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) = ((ℝ ↑m {𝑧}) ↑m ℕ))
264262, 263eleqtrd 2920 . . . . . . . . . . . . . . . . . . 19 ((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ))
265264adantlr 711 . . . . . . . . . . . . . . . . . 18 (((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ))
266265adantlll 714 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ))
267 simpl 483 . . . . . . . . . . . . . . . . . . 19 ((X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ∧ 𝑦 = ∅) → X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
268239ixpeq1d 8462 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = ∅ → X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)))
269268adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ∧ 𝑦 = ∅) → X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)))
270239ixpeq1d 8462 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = ∅ → X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)))
271270adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 = ∅ ∧ 𝑖 ∈ ℕ) → X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)))
272271iuneq2dv 4940 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = ∅ → 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)))
273 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 = 𝑗 → (𝑐𝑖) = (𝑐𝑗))
274273fveq1d 6669 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 = 𝑗 → ((𝑐𝑖)‘𝑘) = ((𝑐𝑗)‘𝑘))
275 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 = 𝑗 → (𝑑𝑖) = (𝑑𝑗))
276275fveq1d 6669 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 = 𝑗 → ((𝑑𝑖)‘𝑘) = ((𝑑𝑗)‘𝑘))
277274, 276oveq12d 7166 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑗 → (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
278277ixpeq2dv 8466 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑗X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
279278cbviunv 4962 . . . . . . . . . . . . . . . . . . . . . . 23 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))
280279a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = ∅ → 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
281272, 280eqtrd 2861 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = ∅ → 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
282281adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ∧ 𝑦 = ∅) → 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
283269, 282sseq12d 4004 . . . . . . . . . . . . . . . . . . 19 ((X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ∧ 𝑦 = ∅) → (X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) ↔ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
284267, 283mpbird 258 . . . . . . . . . . . . . . . . . 18 ((X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ∧ 𝑦 = ∅) → X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)))
285284adantll 710 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ 𝑦 = ∅) → X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)))
286261, 266, 285jca31 515 . . . . . . . . . . . . . . . 16 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ 𝑦 = ∅) → ((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))))
287 simpr 485 . . . . . . . . . . . . . . . 16 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ 𝑦 = ∅) → 𝑦 = ∅)
288 fveq1 6666 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑢 → (𝑎𝑘) = (𝑢𝑘))
289288oveq1d 7163 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑢 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑢𝑘)[,)(𝑏𝑘)))
290289fveq2d 6671 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑢 → (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = (vol‘((𝑢𝑘)[,)(𝑏𝑘))))
291290prodeq2ad 41741 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝑢 → ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑢𝑘)[,)(𝑏𝑘))))
292291ifeq2d 4489 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑢 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑢𝑘)[,)(𝑏𝑘)))))
293 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑙 → (𝑢𝑘) = (𝑢𝑙))
294 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑙 → (𝑏𝑘) = (𝑏𝑙))
295293, 294oveq12d 7166 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 = 𝑙 → ((𝑢𝑘)[,)(𝑏𝑘)) = ((𝑢𝑙)[,)(𝑏𝑙)))
296295fveq2d 6671 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 𝑙 → (vol‘((𝑢𝑘)[,)(𝑏𝑘))) = (vol‘((𝑢𝑙)[,)(𝑏𝑙))))
297296cbvprodv 15260 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘𝑥 (vol‘((𝑢𝑘)[,)(𝑏𝑘))) = ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑏𝑙)))
298297a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = 𝑣 → ∏𝑘𝑥 (vol‘((𝑢𝑘)[,)(𝑏𝑘))) = ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑏𝑙))))
299 fveq1 6666 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑏 = 𝑣 → (𝑏𝑙) = (𝑣𝑙))
300299oveq2d 7164 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 = 𝑣 → ((𝑢𝑙)[,)(𝑏𝑙)) = ((𝑢𝑙)[,)(𝑣𝑙)))
301300fveq2d 6671 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 = 𝑣 → (vol‘((𝑢𝑙)[,)(𝑏𝑙))) = (vol‘((𝑢𝑙)[,)(𝑣𝑙))))
302301prodeq2ad 41741 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = 𝑣 → ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑏𝑙))) = ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑣𝑙))))
303298, 302eqtrd 2861 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = 𝑣 → ∏𝑘𝑥 (vol‘((𝑢𝑘)[,)(𝑏𝑘))) = ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑣𝑙))))
304303ifeq2d 4489 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝑣 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑢𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑣𝑙)))))
305292, 304cbvmpov 7239 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))) = (𝑢 ∈ (ℝ ↑m 𝑥), 𝑣 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑣𝑙)))))
306305mpteq2i 5155 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘)))))) = (𝑥 ∈ Fin ↦ (𝑢 ∈ (ℝ ↑m 𝑥), 𝑣 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑣𝑙))))))
307126, 306eqtri 2849 . . . . . . . . . . . . . . . . . . 19 𝐿 = (𝑥 ∈ Fin ↦ (𝑢 ∈ (ℝ ↑m 𝑥), 𝑣 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑣𝑙))))))
308 simp-6r 784 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))) → 𝑧𝑋)
309 eqid 2826 . . . . . . . . . . . . . . . . . . 19 {𝑧} = {𝑧}
310 elmapi 8418 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ (ℝ ↑m {𝑧}) → 𝑎:{𝑧}⟶ℝ)
311310ad2antlr 723 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) → 𝑎:{𝑧}⟶ℝ)
312311ad3antrrr 726 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))) → 𝑎:{𝑧}⟶ℝ)
313 elmapi 8418 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 ∈ (ℝ ↑m {𝑧}) → 𝑏:{𝑧}⟶ℝ)
314313adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) → 𝑏:{𝑧}⟶ℝ)
315314ad3antrrr 726 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))) → 𝑏:{𝑧}⟶ℝ)
316 elmapi 8418 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ) → 𝑐:ℕ⟶(ℝ ↑m {𝑧}))
317316adantl 482 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) → 𝑐:ℕ⟶(ℝ ↑m {𝑧}))
318317ad2antrr 722 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))) → 𝑐:ℕ⟶(ℝ ↑m {𝑧}))
319 elmapi 8418 . . . . . . . . . . . . . . . . . . . 20 (𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ) → 𝑑:ℕ⟶(ℝ ↑m {𝑧}))
320319ad2antlr 723 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))) → 𝑑:ℕ⟶(ℝ ↑m {𝑧}))
321 id 22 . . . . . . . . . . . . . . . . . . . . 21 (X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) → X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)))
322 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 𝑙 → (𝑎𝑘) = (𝑎𝑙))
323322, 294oveq12d 7166 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 𝑙 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑎𝑙)[,)(𝑏𝑙)))
324 eqcom 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 = 𝑙𝑙 = 𝑘)
325324imbi1i 351 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 = 𝑙 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑎𝑙)[,)(𝑏𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑎𝑙)[,)(𝑏𝑙))))
326 eqcom 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑎𝑙)[,)(𝑏𝑙)) ↔ ((𝑎𝑙)[,)(𝑏𝑙)) = ((𝑎𝑘)[,)(𝑏𝑘)))
327326imbi2i 337 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑙 = 𝑘 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑎𝑙)[,)(𝑏𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎𝑙)[,)(𝑏𝑙)) = ((𝑎𝑘)[,)(𝑏𝑘))))
328325, 327bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 = 𝑙 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑎𝑙)[,)(𝑏𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎𝑙)[,)(𝑏𝑙)) = ((𝑎𝑘)[,)(𝑏𝑘))))
329323, 328mpbi 231 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = 𝑘 → ((𝑎𝑙)[,)(𝑏𝑙)) = ((𝑎𝑘)[,)(𝑏𝑘)))
330329cbvixpv 8468 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑙 ∈ {𝑧} ((𝑎𝑙)[,)(𝑏𝑙)) = X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘))
331330a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) → X𝑙 ∈ {𝑧} ((𝑎𝑙)[,)(𝑏𝑙)) = X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)))
332277ixpeq2dv 8466 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑗X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = X𝑘 ∈ {𝑧} (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
333332cbviunv 4962 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))
334 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑙 → ((𝑐𝑗)‘𝑘) = ((𝑐𝑗)‘𝑙))
335 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑙 → ((𝑑𝑗)‘𝑘) = ((𝑑𝑗)‘𝑙))
336334, 335oveq12d 7166 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 = 𝑙 → (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)))
337336cbvixpv 8468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 X𝑘 ∈ {𝑧} (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑙 ∈ {𝑧} (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙))
338337a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ℕ → X𝑘 ∈ {𝑧} (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑙 ∈ {𝑧} (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)))
339338iuneq2i 4937 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑗 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙))
340333, 339eqtr2i 2850 . . . . . . . . . . . . . . . . . . . . . . 23 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)) = 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))
341340a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) → 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)) = 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)))
342331, 341sseq12d 4004 . . . . . . . . . . . . . . . . . . . . 21 (X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) → (X𝑙 ∈ {𝑧} ((𝑎𝑙)[,)(𝑏𝑙)) ⊆ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)) ↔ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))))
343321, 342mpbird 258 . . . . . . . . . . . . . . . . . . . 20 (X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) → X𝑙 ∈ {𝑧} ((𝑎𝑙)[,)(𝑏𝑙)) ⊆ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)))
344343adantl 482 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))) → X𝑙 ∈ {𝑧} ((𝑎𝑙)[,)(𝑏𝑙)) ⊆ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)))
345307, 308, 309, 312, 315, 318, 320, 344hoidmv1le 42745 . . . . . . . . . . . . . . . . . 18 (((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))) → (𝑎(𝐿‘{𝑧})𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘{𝑧})(𝑑𝑗)))))
346345adantr 481 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))) ∧ 𝑦 = ∅) → (𝑎(𝐿‘{𝑧})𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘{𝑧})(𝑑𝑗)))))
347236, 238eqtrd 2861 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = {𝑧})
348347fveq2d 6671 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ∅ → (𝐿‘(𝑦 ∪ {𝑧})) = (𝐿‘{𝑧}))
349348oveqd 7165 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) = (𝑎(𝐿‘{𝑧})𝑏))
350349adantl 482 . . . . . . . . . . . . . . . . . 18 ((((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))) ∧ 𝑦 = ∅) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) = (𝑎(𝐿‘{𝑧})𝑏))
351348oveqd 7165 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = ∅ → ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)) = ((𝑐𝑗)(𝐿‘{𝑧})(𝑑𝑗)))
352351mpteq2dv 5159 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ∅ → (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘{𝑧})(𝑑𝑗))))
353352fveq2d 6671 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘{𝑧})(𝑑𝑗)))))
354353adantl 482 . . . . . . . . . . . . . . . . . 18 ((((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))) ∧ 𝑦 = ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘{𝑧})(𝑑𝑗)))))
355350, 354breq12d 5076 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))) ∧ 𝑦 = ∅) → ((𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))) ↔ (𝑎(𝐿‘{𝑧})𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘{𝑧})(𝑑𝑗))))))
356346, 355mpbird 258 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑m {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ 𝑑 ∈ ((ℝ ↑m {𝑧}) ↑m ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))) ∧ 𝑦 = ∅) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))
357286, 287, 356syl2anc 584 . . . . . . . . . . . . . . 15 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ 𝑦 = ∅) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))
35817ad8antr 736 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑋 ∈ Fin)
359 simplrl 773 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))) → 𝑦𝑋)
360359ad5ant12 752 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)) → 𝑦𝑋)
361360ad5ant12 752 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑦𝑋)
362 simplrr 774 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))) → 𝑧 ∈ (𝑋𝑦))
363362ad5ant12 752 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)) → 𝑧 ∈ (𝑋𝑦))
364363ad5ant12 752 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑧 ∈ (𝑋𝑦))
365 eqid 2826 . . . . . . . . . . . . . . . . 17 (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧})
366 elmapi 8418 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ)
367366adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ)
368367ad4ant23 749 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ)
369368ad3antrrr 726 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ)
370 elmapi 8418 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ)
371370adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧})) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ)
372371ad4ant23 749 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ)
373372ad3antrrr 726 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ)
374 elmapi 8418 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) → 𝑐:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
375374adantl 482 . . . . . . . . . . . . . . . . . 18 ((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))) ∧ 𝑎 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑m (𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ)) → 𝑐:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
376375ad3antrrr 726 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑐:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
377 elmapi 8418 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) → 𝑑:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
378377ad2antrr 722 . . . . . . . . . . . . . . . . . 18 (((𝑑 ∈ ((ℝ ↑m (𝑦 ∪ {𝑧})) ↑m ℕ) ∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑑:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
379378adantlll 714 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑑:ℕ⟶(ℝ ↑m (𝑦 ∪ {𝑧})))
380 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑙 → (𝑒𝑘) = (𝑒𝑙))
381 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑙 → (𝑓𝑘) = (𝑓𝑙))
382380, 381oveq12d 7166 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑙 → ((𝑒𝑘)[,)(𝑓𝑘)) = ((𝑒𝑙)[,)(𝑓𝑙)))
383382cbvixpv 8468 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) = X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙))
384383a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = 𝑜X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) = X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙)))
385 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 = 𝑖 → (𝑔𝑗) = (𝑔𝑖))
386385fveq1d 6669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑖 → ((𝑔𝑗)‘𝑘) = ((𝑔𝑖)‘𝑘))
387 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 = 𝑖 → (𝑗) = (𝑖))
388387fveq1d 6669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑖 → ((𝑗)‘𝑘) = ((𝑖)‘𝑘))
389386, 388oveq12d 7166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = 𝑖 → (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) = (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)))
390389ixpeq2dv 8466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑖X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) = X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)))
391390cbviunv 4962 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) = 𝑖 ∈ ℕ X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘))
392391a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = 𝑜 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) = 𝑖 ∈ ℕ X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)))
393 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑘 = 𝑙 → ((𝑔𝑖)‘𝑘) = ((𝑔𝑖)‘𝑙))
394 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑘 = 𝑙 → ((𝑖)‘𝑘) = ((𝑖)‘𝑙))
395393, 394oveq12d 7166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 = 𝑙 → (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)) = (((𝑔𝑖)‘𝑙)[,)((𝑖)‘𝑙)))
396395cbvixpv 8468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)) = X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑖)‘𝑙))
397396a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ( = 𝑜X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)) = X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑖)‘𝑙)))
398 fveq1 6666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ( = 𝑜 → (𝑖) = (𝑜𝑖))
399398fveq1d 6669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ( = 𝑜 → ((𝑖)‘𝑙) = ((𝑜𝑖)‘𝑙))
400399oveq2d 7164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ( = 𝑜 → (((𝑔𝑖)‘𝑙)[,)((𝑖)‘𝑙)) = (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)))
401400ixpeq2dv 8466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ( = 𝑜X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑖)‘𝑙)) = X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)))
402397, 401eqtrd 2861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ( = 𝑜X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)) = X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)))
403402adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (( = 𝑜𝑖 ∈ ℕ) → X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)) = X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)))
404403iuneq2dv 4940 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = 𝑜 𝑖 ∈ ℕ X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)) = 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)))
405392, 404eqtrd 2861 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = 𝑜 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) = 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)))
406384, 405sseq12d 4004 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( = 𝑜 → (X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) ↔ X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙))))
407385, 387oveq12d 7166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = 𝑖 → ((𝑔𝑗)(𝐿𝑦)(𝑗)) = ((𝑔𝑖)(𝐿𝑦)(𝑖)))
408407cbvmptv 5166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑖)))
409408a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( = 𝑜 → (𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑖))))
410398oveq2d 7164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ( = 𝑜 → ((𝑔𝑖)(𝐿𝑦)(𝑖)) = ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖)))
411410mpteq2dv 5159 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( = 𝑜 → (𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑖))) = (𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))
412409, 411eqtrd 2861 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = 𝑜 → (𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))
413412fveq2d 6671 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = 𝑜 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))) = (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖)))))
414413breq2d 5075 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( = 𝑜 → ((𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))) ↔ (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))))
415406, 414imbi12d 346 . . . . . . . . . . . . . . . . . . . . . . . 24 ( = 𝑜 → ((X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))) ↔ (X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖)))))))
416415cbvralv 3458 . . . . . . . . . . . . . . . . . . . . . . 23 (∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))) ↔ ∀𝑜 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))))
417416ralbii 3170 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))) ↔ ∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑜 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))))
418417ralbii 3170 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))) ↔ ∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑜 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))))
419418ralbii 3170 . . . . . . . . . . . . . . . . . . . 20 (∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))) ↔ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑜 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))))
420419biimpi 217 . . . . . . . . . . . . . . . . . . 19 (∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))) → ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑜 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))))
421420adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))) → ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑜 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))))
422421ad6antr 732 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑜 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))))
423323cbvixpv 8468 . . . . . . . . . . . . . . . . . . . 20 X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎𝑙)[,)(𝑏𝑙))
424336cbvixpv 8468 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙))
425424a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑖X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)))
426 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑖 → (𝑐𝑗) = (𝑐𝑖))
427426fveq1d 6669 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 𝑖 → ((𝑐𝑗)‘𝑙) = ((𝑐𝑖)‘𝑙))
428 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑖 → (𝑑𝑗) = (𝑑𝑖))
429428fveq1d 6669 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 𝑖 → ((𝑑𝑗)‘𝑙) = ((𝑑𝑖)‘𝑙))
430427, 429oveq12d 7166 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑖 → (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)) = (((𝑐𝑖)‘𝑙)[,)((𝑑𝑖)‘𝑙)))
431430ixpeq2dv 8466 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑖X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑙)[,)((𝑑𝑖)‘𝑙)))
432425, 431eqtrd 2861 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑖X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑙)[,)((𝑑𝑖)‘𝑙)))
433432cbviunv 4962 . . . . . . . . . . . . . . . . . . . 20 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑙)[,)((𝑑𝑖)‘𝑙))
434423, 433sseq12i 4001 . . . . . . . . . . . . . . . . . . 19 (X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎𝑙)[,)(𝑏𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑙)[,)((𝑑𝑖)‘𝑙)))
435434biimpi 217 . . . . . . . . . . . . . . . . . 18 (X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎𝑙)[,)(𝑏𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑙)[,)((𝑑𝑖)‘𝑙)))
436435ad2antlr 723 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎𝑙)[,)(𝑏𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑙)[,)((𝑑𝑖)‘𝑙)))
437 neqne 3029 . . . . . . . . . . . . . . . . . 18 𝑦 = ∅ → 𝑦 ≠ ∅)
438437adantl 482 . . . . . . . . . . . . . . . . 17 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑦 ≠ ∅)
439307, 358, 361, 364, 365, 369, 373, 376, 379, 422, 436, 438hoidmvlelem5 42750 . . . . . . . . . . . . . . . 16 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑐𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑖)))))
440273, 275oveq12d 7166 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑗 → ((𝑐𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑖)) = ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))
441440cbvmptv 5166 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ ℕ ↦ ((𝑐𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑖))) = (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))
442441fveq2i 6670 . . . . . . . . . . . . . . . . 17 ^‘(𝑖 ∈ ℕ ↦ ((𝑐𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑖)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))
443442breq2i 5071 . . . . . . . . . . . . . . . 16 ((𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑐𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑖)))) ↔ (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))
444439, 443sylib 219 . . . . . . . . . . . . . . 15 (((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))
445357, 444pm2.61dan 809 . . . . . . . . . . . . . 14 ((((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))
446445ex 413 . . . . . . . . . . . . 13 (((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))))
447446ralrimiva 3187 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))))
448447ralrimiva 3187 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))))
449448ralrimiva 3187 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))))
450449ralrimiva 3187 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑒 ∈ (ℝ ↑m 𝑦)∀𝑓 ∈ (ℝ ↑m 𝑦)∀𝑔 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))))
451177, 229, 450syl2anc 584 . . . . . . . 8 (((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ ∀𝑎 ∈ (ℝ ↑m 𝑦)∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))) → ∀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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))))
452451ex 413 . . . . . . 7 ((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) → (∀𝑎 ∈ (ℝ ↑m 𝑦)∀𝑏 ∈ (ℝ ↑m 𝑦)∀𝑐 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑦) ↑m ℕ)(X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))) → ∀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})((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))))
45350, 75, 100, 125, 176, 452, 17findcard2d 8749 . . . . . 6 (𝜑 → ∀𝑎 ∈ (ℝ ↑m 𝑋)∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))
454 fveq1 6666 . . . . . . . . . . . . . 14 (𝑎 = 𝐴 → (𝑎𝑘) = (𝐴𝑘))
455454oveq1d 7163 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝐴𝑘)[,)(𝑏𝑘)))
456455ixpeq2dv 8466 . . . . . . . . . . . 12 (𝑎 = 𝐴X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)))
457456sseq1d 4002 . . . . . . . . . . 11 (𝑎 = 𝐴 → (X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
458 oveq1 7155 . . . . . . . . . . . 12 (𝑎 = 𝐴 → (𝑎(𝐿𝑋)𝑏) = (𝐴(𝐿𝑋)𝑏))
459458breq1d 5073 . . . . . . . . . . 11 (𝑎 = 𝐴 → ((𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))) ↔ (𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))
460457, 459imbi12d 346 . . . . . . . . . 10 (𝑎 = 𝐴 → ((X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))) ↔ (X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))))
461460ralbidv 3202 . . . . . . . . 9 (𝑎 = 𝐴 → (∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))))
462461ralbidv 3202 . . . . . . . 8 (𝑎 = 𝐴 → (∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))))
463462ralbidv 3202 . . . . . . 7 (𝑎 = 𝐴 → (∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))) ↔ ∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))))
464463rspcva 3625 . . . . . 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)(X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))) → ∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))
46525, 453, 464syl2anc 584 . . . . 5 (𝜑 → ∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))
466 fveq1 6666 . . . . . . . . . . . 12 (𝑏 = 𝐵 → (𝑏𝑘) = (𝐵𝑘))
467466oveq2d 7164 . . . . . . . . . . 11 (𝑏 = 𝐵 → ((𝐴𝑘)[,)(𝑏𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
468467ixpeq2dv 8466 . . . . . . . . . 10 (𝑏 = 𝐵X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
469468sseq1d 4002 . . . . . . . . 9 (𝑏 = 𝐵 → (X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
470 oveq2 7156 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴(𝐿𝑋)𝑏) = (𝐴(𝐿𝑋)𝐵))
471470breq1d 5073 . . . . . . . . 9 (𝑏 = 𝐵 → ((𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))) ↔ (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))
472469, 471imbi12d 346 . . . . . . . 8 (𝑏 = 𝐵 → ((X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))) ↔ (X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))))
473472ralbidv 3202 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))))
474473ralbidv 3202 . . . . . 6 (𝑏 = 𝐵 → (∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))))
475474rspcva 3625 . . . . 5 ((𝐵 ∈ (ℝ ↑m 𝑋) ∧ ∀𝑏 ∈ (ℝ ↑m 𝑋)∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))) → ∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))
47621, 465, 475syl2anc 584 . . . 4 (𝜑 → ∀𝑐 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)∀𝑑 ∈ ((ℝ ↑m 𝑋) ↑m ℕ)(X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) → (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))
477 fveq1 6666 . . . . . . . . . . . . 13 (𝑐 = 𝐶 → (𝑐𝑗) = (𝐶𝑗))
478477fveq1d 6669 . . . . . . . . . . . 12 (𝑐 = 𝐶 → ((𝑐𝑗)‘𝑘) = ((𝐶𝑗)‘𝑘))
479478oveq1d 7163 . . . . . . . . . . 11 (𝑐 = 𝐶 → (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = (((𝐶𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
480479ixpeq2dv 8466 . . . . . . . . . 10 (𝑐 = 𝐶X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
481480adantr 481 . . . . . . . . 9 ((𝑐 = 𝐶𝑗 ∈ ℕ) → X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
482481iuneq2dv 4940 . . . . . . . 8 (𝑐 = 𝐶 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
483482sseq2d 4003 . . . . . . 7 (𝑐 = 𝐶 → (X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
484477oveq1d 7163 . . . . . . . . . 10 (𝑐 = 𝐶 → ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)) = ((𝐶𝑗)(𝐿𝑋)(𝑑𝑗)))
485484mpteq2dv 5159 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑗 ∈ ℕ ↦ ((𝑐𝑗