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Theorem hoidmvle 40121
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 6632 . . . . . . 7 (ℝ ↑𝑚 𝑋) ∈ V
4 nnex 10970 . . . . . . 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 7815 . . . . 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 247 . . 3 (𝜑𝐷 ∈ ((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ))
10 hoidmvle.c . . . . 5 (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑋))
11 elmapg 7815 . . . . . 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 247 . . . 4 (𝜑𝐶 ∈ ((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ))
14 hoidmvle.b . . . . . 6 (𝜑𝐵:𝑋⟶ℝ)
15 reex 9971 . . . . . . . . 9 ℝ ∈ V
1615a1i 11 . . . . . . . 8 (𝜑 → ℝ ∈ V)
17 hoidmvle.x . . . . . . . 8 (𝜑𝑋 ∈ Fin)
1816, 17jca 554 . . . . . . 7 (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ Fin))
19 elmapg 7815 . . . . . . 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 247 . . . . 5 (𝜑𝐵 ∈ (ℝ ↑𝑚 𝑋))
22 hoidmvle.a . . . . . . 7 (𝜑𝐴:𝑋⟶ℝ)
23 elmapg 7815 . . . . . . . 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 247 . . . . . 6 (𝜑𝐴 ∈ (ℝ ↑𝑚 𝑋))
26 oveq2 6612 . . . . . . . . . 10 (𝑥 = ∅ → (ℝ ↑𝑚 𝑥) = (ℝ ↑𝑚 ∅))
2726eleq2d 2684 . . . . . . . . 9 (𝑥 = ∅ → (𝑎 ∈ (ℝ ↑𝑚 𝑥) ↔ 𝑎 ∈ (ℝ ↑𝑚 ∅)))
2826eleq2d 2684 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑏 ∈ (ℝ ↑𝑚 𝑥) ↔ 𝑏 ∈ (ℝ ↑𝑚 ∅)))
2926oveq1d 6619 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ) = ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ))
3029eleq2d 2684 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑐 ∈ ((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ) ↔ 𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ)))
3129eleq2d 2684 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (𝑑 ∈ ((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ) ↔ 𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ)))
32 ixpeq1 7863 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)))
33 ixpeq1 7863 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
3433iuneq2d 4513 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
3532, 34sseq12d 3613 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘 ∈ ∅ ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
36 fveq2 6148 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → (𝐿𝑥) = (𝐿‘∅))
3736oveqd 6621 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → (𝑎(𝐿𝑥)𝑏) = (𝑎(𝐿‘∅)𝑏))
3836oveqd 6621 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ∅ → ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)) = ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))
3938mpteq2dv 4705 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))
4039fveq2d 6152 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))))
4137, 40breq12d 4626 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → ((𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) ↔ (𝑎(𝐿‘∅)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗))))))
4235, 41imbi12d 334 . . . . . . . . . . . . . . 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 334 . . . . . . . . . . . . . 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 2978 . . . . . . . . . . . . 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 334 . . . . . . . . . . . 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 2978 . . . . . . . . . . 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 334 . . . . . . . . . 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 2978 . . . . . . . . 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 334 . . . . . . . 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 2978 . . . . . . 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 6612 . . . . . . . . . 10 (𝑥 = 𝑦 → (ℝ ↑𝑚 𝑥) = (ℝ ↑𝑚 𝑦))
5251eleq2d 2684 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑎 ∈ (ℝ ↑𝑚 𝑥) ↔ 𝑎 ∈ (ℝ ↑𝑚 𝑦)))
5351eleq2d 2684 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑏 ∈ (ℝ ↑𝑚 𝑥) ↔ 𝑏 ∈ (ℝ ↑𝑚 𝑦)))
5451oveq1d 6619 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ) = ((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ))
5554eleq2d 2684 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑐 ∈ ((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ) ↔ 𝑐 ∈ ((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)))
5654eleq2d 2684 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑑 ∈ ((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ) ↔ 𝑑 ∈ ((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)))
57 ixpeq1 7863 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)))
58 ixpeq1 7863 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
5958iuneq2d 4513 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
6057, 59sseq12d 3613 . . . . . . . . . . . . . . . 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 6148 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝐿𝑥) = (𝐿𝑦))
6261oveqd 6621 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (𝑎(𝐿𝑥)𝑏) = (𝑎(𝐿𝑦)𝑏))
6361oveqd 6621 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)) = ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))
6463mpteq2dv 4705 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))
6564fveq2d 6152 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))))
6662, 65breq12d 4626 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) ↔ (𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))))
6760, 66imbi12d 334 . . . . . . . . . . . . . . 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 334 . . . . . . . . . . . . . 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 2978 . . . . . . . . . . . . 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 334 . . . . . . . . . . . 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 2978 . . . . . . . . . . 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 334 . . . . . . . . . 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 2978 . . . . . . . . 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 334 . . . . . . . 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 2978 . . . . . . 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 6612 . . . . . . . . . 10 (𝑥 = (𝑦 ∪ {𝑧}) → (ℝ ↑𝑚 𝑥) = (ℝ ↑𝑚 (𝑦 ∪ {𝑧})))
7776eleq2d 2684 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑎 ∈ (ℝ ↑𝑚 𝑥) ↔ 𝑎 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧}))))
7876eleq2d 2684 . . . . . . . . . . 11 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑏 ∈ (ℝ ↑𝑚 𝑥) ↔ 𝑏 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧}))))
7976oveq1d 6619 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 ∪ {𝑧}) → ((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ) = ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
8079eleq2d 2684 . . . . . . . . . . . . 13 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑐 ∈ ((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ) ↔ 𝑐 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)))
8179eleq2d 2684 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑑 ∈ ((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ) ↔ 𝑑 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)))
82 ixpeq1 7863 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∪ {𝑧}) → X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)))
83 ixpeq1 7863 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑦 ∪ {𝑧}) → X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
8483iuneq2d 4513 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
8582, 84sseq12d 3613 . . . . . . . . . . . . . . . 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 6148 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐿𝑥) = (𝐿‘(𝑦 ∪ {𝑧})))
8786oveqd 6621 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑎(𝐿𝑥)𝑏) = (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏))
8886oveqd 6621 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)) = ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))
8988mpteq2dv 4705 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))
9089fveq2d 6152 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∪ {𝑧}) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))))
9187, 90breq12d 4626 . . . . . . . . . . . . . . . 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 334 . . . . . . . . . . . . . . 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 334 . . . . . . . . . . . . . 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 2978 . . . . . . . . . . . . 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 334 . . . . . . . . . . . 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 2978 . . . . . . . . . . 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 334 . . . . . . . . . 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 2978 . . . . . . . . 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 334 . . . . . . . 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 2978 . . . . . . 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 6612 . . . . . . . . . 10 (𝑥 = 𝑋 → (ℝ ↑𝑚 𝑥) = (ℝ ↑𝑚 𝑋))
102101eleq2d 2684 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑎 ∈ (ℝ ↑𝑚 𝑥) ↔ 𝑎 ∈ (ℝ ↑𝑚 𝑋)))
103101eleq2d 2684 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑏 ∈ (ℝ ↑𝑚 𝑥) ↔ 𝑏 ∈ (ℝ ↑𝑚 𝑋)))
104101oveq1d 6619 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → ((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ) = ((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ))
105104eleq2d 2684 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝑐 ∈ ((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ) ↔ 𝑐 ∈ ((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ)))
106104eleq2d 2684 . . . . . . . . . . . . . . 15 (𝑥 = 𝑋 → (𝑑 ∈ ((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ) ↔ 𝑑 ∈ ((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ)))
107 ixpeq1 7863 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑋X𝑘𝑥 ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)))
108 ixpeq1 7863 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑋X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
109108iuneq2d 4513 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑋 𝑗 ∈ ℕ X𝑘𝑥 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
110107, 109sseq12d 3613 . . . . . . . . . . . . . . . 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 6148 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑋 → (𝐿𝑥) = (𝐿𝑋))
112111oveqd 6621 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑋 → (𝑎(𝐿𝑥)𝑏) = (𝑎(𝐿𝑋)𝑏))
113111oveqd 6621 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑋 → ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)) = ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))
114113mpteq2dv 4705 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑋 → (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))
115114fveq2d 6152 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑋 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))))
116112, 115breq12d 4626 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑋 → ((𝑎(𝐿𝑥)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑥)(𝑑𝑗)))) ↔ (𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))
117110, 116imbi12d 334 . . . . . . . . . . . . . . 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 334 . . . . . . . . . . . . . 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 2978 . . . . . . . . . . . . 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 334 . . . . . . . . . . . 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 2978 . . . . . . . . . . 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 334 . . . . . . . . . 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 2978 . . . . . . . . 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 334 . . . . . . . 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 2978 . . . . . . 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 6147 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑒 → (𝑎𝑘) = (𝑒𝑘))
128127oveq1d 6619 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 𝑒 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑒𝑘)[,)(𝑏𝑘)))
129128fveq2d 6152 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑒 → (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = (vol‘((𝑒𝑘)[,)(𝑏𝑘))))
130129prodeq2ad 39228 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑒 → ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑏𝑘))))
131130ifeq2d 4077 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑒 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑏𝑘)))))
132 fveq1 6147 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝑓 → (𝑏𝑘) = (𝑓𝑘))
133132oveq2d 6620 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = 𝑓 → ((𝑒𝑘)[,)(𝑏𝑘)) = ((𝑒𝑘)[,)(𝑓𝑘)))
134133fveq2d 6152 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑓 → (vol‘((𝑒𝑘)[,)(𝑏𝑘))) = (vol‘((𝑒𝑘)[,)(𝑓𝑘))))
135134prodeq2ad 39228 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑓 → ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑓𝑘))))
136135ifeq2d 4077 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑓 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑓𝑘)))))
137131, 136cbvmpt2v 6688 . . . . . . . . . . . . . . . . 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 4701 . . . . . . . . . . . . . . . 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 2643 . . . . . . . . . . . . . . 15 𝐿 = (𝑥 ∈ Fin ↦ (𝑒 ∈ (ℝ ↑𝑚 𝑥), 𝑓 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑒𝑘)[,)(𝑓𝑘))))))
140 elmapi 7823 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (ℝ ↑𝑚 ∅) → 𝑎:∅⟶ℝ)
141140adantr 481 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (ℝ ↑𝑚 ∅) ∧ 𝑏 ∈ (ℝ ↑𝑚 ∅)) → 𝑎:∅⟶ℝ)
142 elmapi 7823 . . . . . . . . . . . . . . . 16 (𝑏 ∈ (ℝ ↑𝑚 ∅) → 𝑏:∅⟶ℝ)
143142adantl 482 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (ℝ ↑𝑚 ∅) ∧ 𝑏 ∈ (ℝ ↑𝑚 ∅)) → 𝑏:∅⟶ℝ)
144139, 141, 143hoidmv0val 40104 . . . . . . . . . . . . . 14 ((𝑎 ∈ (ℝ ↑𝑚 ∅) ∧ 𝑏 ∈ (ℝ ↑𝑚 ∅)) → (𝑎(𝐿‘∅)𝑏) = 0)
145144ad5ant23 1301 . . . . . . . . . . . . 13 (((((𝜑𝑎 ∈ (ℝ ↑𝑚 ∅)) ∧ 𝑏 ∈ (ℝ ↑𝑚 ∅)) ∧ 𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ)) → (𝑎(𝐿‘∅)𝑏) = 0)
146 nfv 1840 . . . . . . . . . . . . . . 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 12202 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ (0[,]+∞)
149 0fin 8132 . . . . . . . . . . . . . . . . . 18 ∅ ∈ Fin
150149a1i 11 . . . . . . . . . . . . . . . . 17 (((𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ)) ∧ 𝑗 ∈ ℕ) → ∅ ∈ Fin)
151 ovex 6632 . . . . . . . . . . . . . . . . . . . . 21 (ℝ ↑𝑚 ∅) ∈ V
152151a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (ℝ ↑𝑚 ∅) ∈ V)
1534a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → ℕ ∈ V)
154 simpl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → 𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ))
155 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
156152, 153, 154, 155fvmap 38861 . . . . . . . . . . . . . . . . . . 19 ((𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (𝑐𝑗) ∈ (ℝ ↑𝑚 ∅))
157 elmapi 7823 . . . . . . . . . . . . . . . . . . 19 ((𝑐𝑗) ∈ (ℝ ↑𝑚 ∅) → (𝑐𝑗):∅⟶ℝ)
158156, 157syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (𝑐𝑗):∅⟶ℝ)
159158adantlr 750 . . . . . . . . . . . . . . . . 17 (((𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ)) ∧ 𝑗 ∈ ℕ) → (𝑐𝑗):∅⟶ℝ)
160151a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (ℝ ↑𝑚 ∅) ∈ V)
1614a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → ℕ ∈ V)
162 simpl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → 𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ))
163 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
164160, 161, 162, 163fvmap 38861 . . . . . . . . . . . . . . . . . . 19 ((𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (𝑑𝑗) ∈ (ℝ ↑𝑚 ∅))
165 elmapi 7823 . . . . . . . . . . . . . . . . . . 19 ((𝑑𝑗) ∈ (ℝ ↑𝑚 ∅) → (𝑑𝑗):∅⟶ℝ)
166164, 165syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (𝑑𝑗):∅⟶ℝ)
167166adantll 749 . . . . . . . . . . . . . . . . 17 (((𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ)) ∧ 𝑗 ∈ ℕ) → (𝑑𝑗):∅⟶ℝ)
168126, 150, 159, 167hoidmvcl 40103 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ)) ∧ 𝑗 ∈ ℕ) → ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)) ∈ (0[,)+∞))
169148, 168sseldi 3581 . . . . . . . . . . . . . . 15 (((𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ)) ∧ 𝑗 ∈ ℕ) → ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)) ∈ (0[,]+∞))
170146, 147, 169sge0ge0mpt 39962 . . . . . . . . . . . . . 14 ((𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ) ∧ 𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ)) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))))
171170adantll 749 . . . . . . . . . . . . 13 (((((𝜑𝑎 ∈ (ℝ ↑𝑚 ∅)) ∧ 𝑏 ∈ (ℝ ↑𝑚 ∅)) ∧ 𝑐 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚 ∅) ↑𝑚 ℕ)) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘∅)(𝑑𝑗)))))
172145, 171eqbrtrd 4635 . . . . . . . . . . . 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)))))
173172a1d 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))))))
174173ralrimiva 2960 . . . . . . . . . 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))))))
175174ralrimiva 2960 . . . . . . . . 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))))))
176175ralrimiva 2960 . . . . . . . 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))))))
177176ralrimiva 2960 . . . . . . 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))))))
178 simpl 473 . . . . . . . . 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))))
179128ixpeq2dv 7868 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑒X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)))
180179sseq1d 3611 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑒 → (X𝑘𝑦 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
181 oveq1 6611 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑒 → (𝑎(𝐿𝑦)𝑏) = (𝑒(𝐿𝑦)𝑏))
182181breq1d 4623 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑒 → ((𝑎(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))) ↔ (𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))))
183180, 182imbi12d 334 . . . . . . . . . . . . . . . 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)))))))
184183ralbidv 2980 . . . . . . . . . . . . . . 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)))))))
185184ralbidv 2980 . . . . . . . . . . . . . 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)))))))
186185ralbidv 2980 . . . . . . . . . . . . 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)))))))
187133ixpeq2dv 7868 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑓X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) = X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)))
188187sseq1d 3611 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑓 → (X𝑘𝑦 ((𝑒𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
189 oveq2 6612 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑓 → (𝑒(𝐿𝑦)𝑏) = (𝑒(𝐿𝑦)𝑓))
190189breq1d 4623 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑓 → ((𝑒(𝐿𝑦)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))) ↔ (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))))))
191188, 190imbi12d 334 . . . . . . . . . . . . . . . . . 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)))))))
192191ralbidv 2980 . . . . . . . . . . . . . . . . 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)))))))
193192ralbidv 2980 . . . . . . . . . . . . . . . 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)))))))
194 fveq1 6147 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑐 = 𝑔 → (𝑐𝑗) = (𝑔𝑗))
195194fveq1d 6150 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = 𝑔 → ((𝑐𝑗)‘𝑘) = ((𝑔𝑗)‘𝑘))
196195oveq1d 6619 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = 𝑔 → (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
197196ixpeq2dv 7868 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = 𝑔X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
198197adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐 = 𝑔𝑗 ∈ ℕ) → X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
199198iuneq2dv 4508 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = 𝑔 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
200199sseq2d 3612 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑔 → (X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
201194oveq1d 6619 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = 𝑔 → ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)) = ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗)))
202201mpteq2dv 4705 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = 𝑔 → (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗))))
203202fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = 𝑔 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗)))))
204203breq2d 4625 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑔 → ((𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑦)(𝑑𝑗)))) ↔ (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗))))))
205200, 204imbi12d 334 . . . . . . . . . . . . . . . . . . . 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)))))))
206205ralbidv 2980 . . . . . . . . . . . . . . . . . . 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)))))))
207 fveq1 6147 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑑 = → (𝑑𝑗) = (𝑗))
208207fveq1d 6150 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑 = → ((𝑑𝑗)‘𝑘) = ((𝑗)‘𝑘))
209208oveq2d 6620 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = → (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)))
210209ixpeq2dv 7868 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)))
211210adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑑 = 𝑗 ∈ ℕ) → X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)))
212211iuneq2dv 4508 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)))
213212sseq2d 3612 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 = → (X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘))))
214207oveq2d 6620 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = → ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗)) = ((𝑔𝑗)(𝐿𝑦)(𝑗)))
215214mpteq2dv 4705 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = → (𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))
216215fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))))
217216breq2d 4625 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 = → ((𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑑𝑗)))) ↔ (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))))))
218213, 217imbi12d 334 . . . . . . . . . . . . . . . . . . . . 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)))))))
219218cbvralv 3159 . . . . . . . . . . . . . . . . . . . 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))))))
220219a1i 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)))))))
221206, 220bitrd 268 . . . . . . . . . . . . . . . . . 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)))))))
222221cbvralv 3159 . . . . . . . . . . . . . . . . 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))))))
223222a1i 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)))))))
224193, 223bitrd 268 . . . . . . . . . . . . . . 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)))))))
225224cbvralv 3159 . . . . . . . . . . . . . 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))))))
226225a1i 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)))))))
227186, 226bitrd 268 . . . . . . . . . . . 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)))))))
228227cbvralv 3159 . . . . . . . . . . 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))))))
229228biimpi 206 . . . . . . . . . 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))))))
230229adantl 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))))))
231 simplll 797 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ 𝑎 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝜑)
232 eldifi 3710 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ (𝑋𝑦) → 𝑧𝑋)
233232adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑧 ∈ (𝑋𝑦)) → 𝑧𝑋)
234233adantrl 751 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) → 𝑧𝑋)
235234ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ 𝑎 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝑧𝑋)
236 simpl 473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})))
237 uneq1 3738 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = (∅ ∪ {𝑧}))
238 0un 38700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∅ ∪ {𝑧}) = {𝑧}
239238a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → (∅ ∪ {𝑧}) = {𝑧})
240237, 239eqtr2d 2656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ∅ → {𝑧} = (𝑦 ∪ {𝑧}))
241240eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = {𝑧})
242241oveq2d 6620 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = ∅ → (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) = (ℝ ↑𝑚 {𝑧}))
243242adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) = (ℝ ↑𝑚 {𝑧}))
244236, 243eleqtrd 2700 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑𝑚 {𝑧}))
245244adantll 749 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ 𝑎 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑𝑚 {𝑧}))
246231, 235, 245jca31 556 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑦𝑋𝑧 ∈ (𝑋𝑦))) ∧ 𝑎 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → ((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚 {𝑧})))
247246adantllr 754 . . . . . . . . . . . . . . . . . . . . . 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})))
248247adantlr 750 . . . . . . . . . . . . . . . . . . . . 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})))
249248adantlr 750 . . . . . . . . . . . . . . . . . . . 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})))
250 simpl 473 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})))
251242adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) = (ℝ ↑𝑚 {𝑧}))
252250, 251eleqtrd 2700 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑𝑚 {𝑧}))
253252adantlr 750 . . . . . . . . . . . . . . . . . . . . 21 (((𝑏 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑𝑚 {𝑧}))
254253adantlll 753 . . . . . . . . . . . . . . . . . . . 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}))
255 simpl 473 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑐 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧ 𝑦 = ∅) → 𝑐 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
256242oveq1d 6619 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = ∅ → ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) = ((ℝ ↑𝑚 {𝑧}) ↑𝑚 ℕ))
257256adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑐 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧ 𝑦 = ∅) → ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) = ((ℝ ↑𝑚 {𝑧}) ↑𝑚 ℕ))
258255, 257eleqtrd 2700 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧ 𝑦 = ∅) → 𝑐 ∈ ((ℝ ↑𝑚 {𝑧}) ↑𝑚 ℕ))
259258adantll 749 . . . . . . . . . . . . . . . . . . . 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))
260249, 254, 259jca31 556 . . . . . . . . . . . . . . . . . . 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)))
261260adantlr 750 . . . . . . . . . . . . . . . . . 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)))
262261adantlr 750 . . . . . . . . . . . . . . . . 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)))
263 simpl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
264256adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧ 𝑦 = ∅) → ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) = ((ℝ ↑𝑚 {𝑧}) ↑𝑚 ℕ))
265263, 264eleqtrd 2700 . . . . . . . . . . . . . . . . . . 19 ((𝑑 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑𝑚 {𝑧}) ↑𝑚 ℕ))
266265adantlr 750 . . . . . . . . . . . . . . . . . 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))
267266adantlll 753 . . . . . . . . . . . . . . . . 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))
268 simpl 473 . . . . . . . . . . . . . . . . . . 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)))
269240ixpeq1d 7864 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = ∅ → X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)))
270269adantl 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)))
271240ixpeq1d 7864 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = ∅ → X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)))
272271adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 = ∅ ∧ 𝑖 ∈ ℕ) → X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)))
273272iuneq2dv 4508 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = ∅ → 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)))
274 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 = 𝑗 → (𝑐𝑖) = (𝑐𝑗))
275274fveq1d 6150 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 = 𝑗 → ((𝑐𝑖)‘𝑘) = ((𝑐𝑗)‘𝑘))
276 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 = 𝑗 → (𝑑𝑖) = (𝑑𝑗))
277276fveq1d 6150 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 = 𝑗 → ((𝑑𝑖)‘𝑘) = ((𝑑𝑗)‘𝑘))
278275, 277oveq12d 6622 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑗 → (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
279278ixpeq2dv 7868 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑗X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
280279cbviunv 4525 . . . . . . . . . . . . . . . . . . . . . . 23 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))
281280a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = ∅ → 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
282273, 281eqtrd 2655 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = ∅ → 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
283282adantl 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)))
284270, 283sseq12d 3613 . . . . . . . . . . . . . . . . . . 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))))
285268, 284mpbird 247 . . . . . . . . . . . . . . . . . 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)))
286285adantll 749 . . . . . . . . . . . . . . . . 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)))
287262, 267, 286jca31 556 . . . . . . . . . . . . . . . 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))))
288 simpr 477 . . . . . . . . . . . . . . . 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 = ∅)
289 fveq1 6147 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑢 → (𝑎𝑘) = (𝑢𝑘))
290289oveq1d 6619 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑢 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑢𝑘)[,)(𝑏𝑘)))
291290fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑢 → (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = (vol‘((𝑢𝑘)[,)(𝑏𝑘))))
292291prodeq2ad 39228 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝑢 → ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑢𝑘)[,)(𝑏𝑘))))
293292ifeq2d 4077 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑢 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑢𝑘)[,)(𝑏𝑘)))))
294 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑙 → (𝑢𝑘) = (𝑢𝑙))
295 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑙 → (𝑏𝑘) = (𝑏𝑙))
296294, 295oveq12d 6622 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 = 𝑙 → ((𝑢𝑘)[,)(𝑏𝑘)) = ((𝑢𝑙)[,)(𝑏𝑙)))
297296fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 𝑙 → (vol‘((𝑢𝑘)[,)(𝑏𝑘))) = (vol‘((𝑢𝑙)[,)(𝑏𝑙))))
298297cbvprodv 14571 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘𝑥 (vol‘((𝑢𝑘)[,)(𝑏𝑘))) = ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑏𝑙)))
299298a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = 𝑣 → ∏𝑘𝑥 (vol‘((𝑢𝑘)[,)(𝑏𝑘))) = ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑏𝑙))))
300 fveq1 6147 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑏 = 𝑣 → (𝑏𝑙) = (𝑣𝑙))
301300oveq2d 6620 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 = 𝑣 → ((𝑢𝑙)[,)(𝑏𝑙)) = ((𝑢𝑙)[,)(𝑣𝑙)))
302301fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 = 𝑣 → (vol‘((𝑢𝑙)[,)(𝑏𝑙))) = (vol‘((𝑢𝑙)[,)(𝑣𝑙))))
303302prodeq2ad 39228 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = 𝑣 → ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑏𝑙))) = ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑣𝑙))))
304299, 303eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = 𝑣 → ∏𝑘𝑥 (vol‘((𝑢𝑘)[,)(𝑏𝑘))) = ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑣𝑙))))
305304ifeq2d 4077 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝑣 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑢𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑣𝑙)))))
306293, 305cbvmpt2v 6688 . . . . . . . . . . . . . . . . . . . . 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)))))
307306mpteq2i 4701 . . . . . . . . . . . . . . . . . . . 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))))))
308126, 307eqtri 2643 . . . . . . . . . . . . . . . . . . 19 𝐿 = (𝑥 ∈ Fin ↦ (𝑢 ∈ (ℝ ↑𝑚 𝑥), 𝑣 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑙𝑥 (vol‘((𝑢𝑙)[,)(𝑣𝑙))))))
309 simp-6r 810 . . . . . . . . . . . . . . . . . . 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)
310 eqid 2621 . . . . . . . . . . . . . . . . . . 19 {𝑧} = {𝑧}
311 elmapi 7823 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ (ℝ ↑𝑚 {𝑧}) → 𝑎:{𝑧}⟶ℝ)
312311ad2antlr 762 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚 {𝑧})) ∧ 𝑏 ∈ (ℝ ↑𝑚 {𝑧})) → 𝑎:{𝑧}⟶ℝ)
313312ad3antrrr 765 . . . . . . . . . . . . . . . . . . 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)
314 elmapi 7823 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 ∈ (ℝ ↑𝑚 {𝑧}) → 𝑏:{𝑧}⟶ℝ)
315314adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚 {𝑧})) ∧ 𝑏 ∈ (ℝ ↑𝑚 {𝑧})) → 𝑏:{𝑧}⟶ℝ)
316315ad3antrrr 765 . . . . . . . . . . . . . . . . . . 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)
317 elmapi 7823 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 ∈ ((ℝ ↑𝑚 {𝑧}) ↑𝑚 ℕ) → 𝑐:ℕ⟶(ℝ ↑𝑚 {𝑧}))
318317adantl 482 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑧𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚 {𝑧})) ∧ 𝑏 ∈ (ℝ ↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚 {𝑧}) ↑𝑚 ℕ)) → 𝑐:ℕ⟶(ℝ ↑𝑚 {𝑧}))
319318ad2antrr 761 . . . . . . . . . . . . . . . . . . 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}))
320 elmapi 7823 . . . . . . . . . . . . . . . . . . . 20 (𝑑 ∈ ((ℝ ↑𝑚 {𝑧}) ↑𝑚 ℕ) → 𝑑:ℕ⟶(ℝ ↑𝑚 {𝑧}))
321320ad2antlr 762 . . . . . . . . . . . . . . . . . . 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}))
322 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)))
323 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 𝑙 → (𝑎𝑘) = (𝑎𝑙))
324323, 295oveq12d 6622 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 𝑙 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑎𝑙)[,)(𝑏𝑙)))
325 eqcom 2628 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 = 𝑙𝑙 = 𝑘)
326325imbi1i 339 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 = 𝑙 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑎𝑙)[,)(𝑏𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑎𝑙)[,)(𝑏𝑙))))
327 eqcom 2628 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑎𝑙)[,)(𝑏𝑙)) ↔ ((𝑎𝑙)[,)(𝑏𝑙)) = ((𝑎𝑘)[,)(𝑏𝑘)))
328327imbi2i 326 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑙 = 𝑘 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑎𝑙)[,)(𝑏𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎𝑙)[,)(𝑏𝑙)) = ((𝑎𝑘)[,)(𝑏𝑘))))
329326, 328bitri 264 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 = 𝑙 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑎𝑙)[,)(𝑏𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎𝑙)[,)(𝑏𝑙)) = ((𝑎𝑘)[,)(𝑏𝑘))))
330324, 329mpbi 220 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = 𝑘 → ((𝑎𝑙)[,)(𝑏𝑙)) = ((𝑎𝑘)[,)(𝑏𝑘)))
331330cbvixpv 7870 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑙 ∈ {𝑧} ((𝑎𝑙)[,)(𝑏𝑙)) = X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘))
332331a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) → X𝑙 ∈ {𝑧} ((𝑎𝑙)[,)(𝑏𝑙)) = X𝑘 ∈ {𝑧} ((𝑎𝑘)[,)(𝑏𝑘)))
333278ixpeq2dv 7868 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑗X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = X𝑘 ∈ {𝑧} (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)))
334333cbviunv 4525 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘)) = 𝑗 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))
335 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑙 → ((𝑐𝑗)‘𝑘) = ((𝑐𝑗)‘𝑙))
336 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑙 → ((𝑑𝑗)‘𝑘) = ((𝑑𝑗)‘𝑙))
337335, 336oveq12d 6622 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 = 𝑙 → (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)))
338337cbvixpv 7870 . . . . . . . . . . . . . . . . . . . . . . . . . 26 X𝑘 ∈ {𝑧} (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑙 ∈ {𝑧} (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙))
339338a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ℕ → X𝑘 ∈ {𝑧} (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑙 ∈ {𝑧} (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)))
340339iuneq2i 4505 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑗 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙))
341334, 340eqtr2i 2644 . . . . . . . . . . . . . . . . . . . . . . 23 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)) = 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐𝑖)‘𝑘)[,)((𝑑𝑖)‘𝑘))
342341a1i 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)))
343332, 342sseq12d 3613 . . . . . . . . . . . . . . . . . . . . 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))))
344322, 343mpbird 247 . . . . . . . . . . . . . . . . . . . 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)))
345344adantl 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)))
346308, 309, 310, 313, 316, 319, 321, 345hoidmv1le 40115 . . . . . . . . . . . . . . . . . 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)))))
347346adantr 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)))))
348237, 239eqtrd 2655 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = {𝑧})
349348fveq2d 6152 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ∅ → (𝐿‘(𝑦 ∪ {𝑧})) = (𝐿‘{𝑧}))
350349oveqd 6621 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) = (𝑎(𝐿‘{𝑧})𝑏))
351350adantl 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))
352349oveqd 6621 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = ∅ → ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)) = ((𝑐𝑗)(𝐿‘{𝑧})(𝑑𝑗)))
353352mpteq2dv 4705 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ∅ → (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘{𝑧})(𝑑𝑗))))
354353fveq2d 6152 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘{𝑧})(𝑑𝑗)))))
355354adantl 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)))))
356351, 355breq12d 4626 . . . . . . . . . . . . . . . . 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))))))
357347, 356mpbird 247 . . . . . . . . . . . . . . . 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)))))
358287, 288, 357syl2anc 692 . . . . . . . . . . . . . . 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)))))
35917ad8antr 775 . . . . . . . . . . . . . . . . 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)
360 simplrl 799 . . . . . . . . . . . . . . . . . . 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)
361360ad5ant12 1297 . . . . . . . . . . . . . . . . . 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)
362361ad5ant12 1297 . . . . . . . . . . . . . . . . 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)
363 simplrr 800 . . . . . . . . . . . . . . . . . . 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))
364363ad5ant12 1297 . . . . . . . . . . . . . . . . . 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))
365364ad5ant12 1297 . . . . . . . . . . . . . . . . 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))
366 eqid 2621 . . . . . . . . . . . . . . . . 17 (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧})
367 elmapi 7823 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ)
368367adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑏 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧}))) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ)
369368ad4ant23 1294 . . . . . . . . . . . . . . . . . 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)
370369ad3antrrr 765 . . . . . . . . . . . . . . . . 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)
371 elmapi 7823 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ)
372371adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑏 ∈ (ℝ ↑𝑚 (𝑦 ∪ {𝑧}))) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ)
373372ad4ant23 1294 . . . . . . . . . . . . . . . . . 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)
374373ad3antrrr 765 . . . . . . . . . . . . . . . . 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)
375 elmapi 7823 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) → 𝑐:ℕ⟶(ℝ ↑𝑚 (𝑦 ∪ {𝑧})))
376375adantl 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})))
377376ad3antrrr 765 . . . . . . . . . . . . . . . . 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})))
378 elmapi 7823 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) → 𝑑:ℕ⟶(ℝ ↑𝑚 (𝑦 ∪ {𝑧})))
379378ad2antrr 761 . . . . . . . . . . . . . . . . . 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})))
380379adantlll 753 . . . . . . . . . . . . . . . . 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})))
381 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑙 → (𝑒𝑘) = (𝑒𝑙))
382 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑙 → (𝑓𝑘) = (𝑓𝑙))
383381, 382oveq12d 6622 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑙 → ((𝑒𝑘)[,)(𝑓𝑘)) = ((𝑒𝑙)[,)(𝑓𝑙)))
384383cbvixpv 7870 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) = X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙))
385384a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = 𝑜X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) = X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙)))
386 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 = 𝑖 → (𝑔𝑗) = (𝑔𝑖))
387386fveq1d 6150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑖 → ((𝑔𝑗)‘𝑘) = ((𝑔𝑖)‘𝑘))
388 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 = 𝑖 → (𝑗) = (𝑖))
389388fveq1d 6150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑖 → ((𝑗)‘𝑘) = ((𝑖)‘𝑘))
390387, 389oveq12d 6622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = 𝑖 → (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) = (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)))
391390ixpeq2dv 7868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑖X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) = X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)))
392391cbviunv 4525 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) = 𝑖 ∈ ℕ X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘))
393392a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = 𝑜 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) = 𝑖 ∈ ℕ X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)))
394 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑘 = 𝑙 → ((𝑔𝑖)‘𝑘) = ((𝑔𝑖)‘𝑙))
395 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑘 = 𝑙 → ((𝑖)‘𝑘) = ((𝑖)‘𝑙))
396394, 395oveq12d 6622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 = 𝑙 → (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)) = (((𝑔𝑖)‘𝑙)[,)((𝑖)‘𝑙)))
397396cbvixpv 7870 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)) = X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑖)‘𝑙))
398397a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ( = 𝑜X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)) = X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑖)‘𝑙)))
399 fveq1 6147 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ( = 𝑜 → (𝑖) = (𝑜𝑖))
400399fveq1d 6150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ( = 𝑜 → ((𝑖)‘𝑙) = ((𝑜𝑖)‘𝑙))
401400oveq2d 6620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ( = 𝑜 → (((𝑔𝑖)‘𝑙)[,)((𝑖)‘𝑙)) = (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)))
402401ixpeq2dv 7868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ( = 𝑜X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑖)‘𝑙)) = X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)))
403398, 402eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ( = 𝑜X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)) = X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)))
404403adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (( = 𝑜𝑖 ∈ ℕ) → X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)) = X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)))
405404iuneq2dv 4508 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = 𝑜 𝑖 ∈ ℕ X𝑘𝑦 (((𝑔𝑖)‘𝑘)[,)((𝑖)‘𝑘)) = 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)))
406393, 405eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = 𝑜 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) = 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙)))
407385, 406sseq12d 3613 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( = 𝑜 → (X𝑘𝑦 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑦 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) ↔ X𝑙𝑦 ((𝑒𝑙)[,)(𝑓𝑙)) ⊆ 𝑖 ∈ ℕ X𝑙𝑦 (((𝑔𝑖)‘𝑙)[,)((𝑜𝑖)‘𝑙))))
408386, 388oveq12d 6622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = 𝑖 → ((𝑔𝑗)(𝐿𝑦)(𝑗)) = ((𝑔𝑖)(𝐿𝑦)(𝑖)))
409408cbvmptv 4710 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑖)))
410409a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( = 𝑜 → (𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑖))))
411399oveq2d 6620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ( = 𝑜 → ((𝑔𝑖)(𝐿𝑦)(𝑖)) = ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖)))
412411mpteq2dv 4705 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( = 𝑜 → (𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑖))) = (𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))
413410, 412eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = 𝑜 → (𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))
414413fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = 𝑜 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))) = (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖)))))
415414breq2d 4625 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( = 𝑜 → ((𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑦)(𝑗)))) ↔ (𝑒(𝐿𝑦)𝑓) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔𝑖)(𝐿𝑦)(𝑜𝑖))))))
416407, 415imbi12d 334 . . . . . . . . . . . . . . . . . . . . . . . 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)))))))
417416cbvralv 3159 . . . . . . . . . . . . . . . . . . . . . . 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))))))
418417ralbii 2974 . . . . . . . . . . . . . . . . . . . . . 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))))))
419418ralbii 2974 . . . . . . . . . . . . . . . . . . . . 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))))))
420419ralbii 2974 . . . . . . . . . . . . . . . . . . . 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))))))
421420biimpi 206 . . . . . . . . . . . . . . . . . . 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))))))
422421adantl 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))))))
423422ad6antr 771 . . . . . . . . . . . . . . . . 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))))))
424324cbvixpv 7870 . . . . . . . . . . . . . . . . . . . 20 X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎𝑘)[,)(𝑏𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎𝑙)[,)(𝑏𝑙))
425337cbvixpv 7870 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙))
426425a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑖X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)))
427 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑖 → (𝑐𝑗) = (𝑐𝑖))
428427fveq1d 6150 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 𝑖 → ((𝑐𝑗)‘𝑙) = ((𝑐𝑖)‘𝑙))
429 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑖 → (𝑑𝑗) = (𝑑𝑖))
430429fveq1d 6150 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 𝑖 → ((𝑑𝑗)‘𝑙) = ((𝑑𝑖)‘𝑙))
431428, 430oveq12d 6622 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑖 → (((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)) = (((𝑐𝑖)‘𝑙)[,)((𝑑𝑖)‘𝑙)))
432431ixpeq2dv 7868 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑖X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑙)[,)((𝑑𝑗)‘𝑙)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑙)[,)((𝑑𝑖)‘𝑙)))
433426, 432eqtrd 2655 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑖X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑙)[,)((𝑑𝑖)‘𝑙)))
434433cbviunv 4525 . . . . . . . . . . . . . . . . . . . 20 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) = 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐𝑖)‘𝑙)[,)((𝑑𝑖)‘𝑙))
435424, 434sseq12i 3610 . . . . . . . . . . . . . . . . . . 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)))
436435biimpi 206 . . . . . . . . . . . . . . . . . 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)))
437436ad2antlr 762 . . . . . . . . . . . . . . . . 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)))
438 neqne 2798 . . . . . . . . . . . . . . . . . 18 𝑦 = ∅ → 𝑦 ≠ ∅)
439438adantl 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 ≠ ∅)
440308, 359, 362, 365, 366, 370, 374, 377, 380, 423, 437, 439hoidmvlelem5 40120 . . . . . . . . . . . . . . . 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)))))
441274, 276oveq12d 6622 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑗 → ((𝑐𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑖)) = ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))
442441cbvmptv 4710 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ ℕ ↦ ((𝑐𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑖))) = (𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗)))
443442fveq2i 6151 . . . . . . . . . . . . . . . . 17 ^‘(𝑖 ∈ ℕ ↦ ((𝑐𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑖)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑𝑗))))
444443breq2i 4621 . . . . . . . . . . . . . . . 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)))))
445440, 444sylib 208 . . . . . . . . . . . . . . 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)))))
446358, 445pm2.61dan 831 . . . . . . . . . . . . . 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)))))
447446ex 450 . . . . . . . . . . . . 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))))))
448447ralrimiva 2960 . . . . . . . . . . . 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))))))
449448ralrimiva 2960 . . . . . . . . . . 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))))))
450449ralrimiva 2960 . . . . . . . . . 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))))))
451450ralrimiva 2960 . . . . . . . . 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))))))
452178, 230, 451syl2anc 692 . . . . . . . 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))))))
453452ex 450 . . . . . . 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)))))))
45450, 75, 100, 125, 177, 453, 17findcard2d 8146 . . . . . 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))))))
455 fveq1 6147 . . . . . . . . . . . . . 14 (𝑎 = 𝐴 → (𝑎𝑘) = (𝐴𝑘))
456455oveq1d 6619 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝐴𝑘)[,)(𝑏𝑘)))
457456ixpeq2dv 7868 . . . . . . . . . . . 12 (𝑎 = 𝐴X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)))
458457sseq1d 3611 . . . . . . . . . . 11 (𝑎 = 𝐴 → (X𝑘𝑋 ((𝑎𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
459 oveq1 6611 . . . . . . . . . . . 12 (𝑎 = 𝐴 → (𝑎(𝐿𝑋)𝑏) = (𝐴(𝐿𝑋)𝑏))
460459breq1d 4623 . . . . . . . . . . 11 (𝑎 = 𝐴 → ((𝑎(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))) ↔ (𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))
461458, 460imbi12d 334 . . . . . . . . . 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)))))))
462461ralbidv 2980 . . . . . . . . 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)))))))
463462ralbidv 2980 . . . . . . . 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)))))))
464463ralbidv 2980 . . . . . . 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)))))))
465464rspcva 3293 . . . . . 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))))))
46625, 454, 465syl2anc 692 . . . . 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))))))
467 fveq1 6147 . . . . . . . . . . . 12 (𝑏 = 𝐵 → (𝑏𝑘) = (𝐵𝑘))
468467oveq2d 6620 . . . . . . . . . . 11 (𝑏 = 𝐵 → ((𝐴𝑘)[,)(𝑏𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
469468ixpeq2dv 7868 . . . . . . . . . 10 (𝑏 = 𝐵X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
470469sseq1d 3611 . . . . . . . . 9 (𝑏 = 𝐵 → (X𝑘𝑋 ((𝐴𝑘)[,)(𝑏𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘)) ↔ X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝑐𝑗)‘𝑘)[,)((𝑑𝑗)‘𝑘))))
471 oveq2 6612 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴(𝐿𝑋)𝑏) = (𝐴(𝐿𝑋)𝐵))
472471breq1d 4623 . . . . . . . . 9 (𝑏 = 𝐵 → ((𝐴(𝐿𝑋)𝑏) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗)))) ↔ (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐𝑗)(𝐿𝑋)(𝑑𝑗))))))
473470, 472imbi12d 334 . . . . . . . 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)))))))
474473ralbidv 2980 . . . . . . 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)))))))
475474ralbidv 2980 . . . . . 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)))))))
476475rspcva 3293 . . . . 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))))))
47721, 466, 476syl2anc 692 . . . 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))))))
478