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Theorem hoidmvlelem3 42340
Description: This is the contradiction proven in step (d) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
hoidmvlelem3.l 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
hoidmvlelem3.x (𝜑 → 𝑋 ∈ Fin)
hoidmvlelem3.y (𝜑 → 𝑌 ⊆ 𝑋)
hoidmvlelem3.z (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
hoidmvlelem3.w 𝑊 = (𝑌 ∪ {𝑍})
hoidmvlelem3.a (𝜑 → 𝐴:𝑊⟶ℝ)
hoidmvlelem3.b (𝜑 → 𝐵:𝑊⟶ℝ)
hoidmvlelem3.lt ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))
hoidmvlelem3.f 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)
hoidmvlelem3.c (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
hoidmvlelem3.j 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
hoidmvlelem3.d (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
hoidmvlelem3.k 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
hoidmvlelem3.r (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
hoidmvlelem3.h 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
hoidmvlelem3.g 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
hoidmvlelem3.e (𝜑 → 𝐸 ∈ ℝ+)
hoidmvlelem3.u 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
hoidmvlelem3.s (𝜑 → 𝑆 ∈ 𝑈)
hoidmvlelem3.sb (𝜑 → 𝑆 < (𝐵‘𝑍))
hoidmvlelem3.p 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
hoidmvlelem3.i (𝜑 → ∀𝑒 ∈ (ℝ ↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
hoidmvlelem3.i2 (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
hoidmvlelem3.o 𝑂 = (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↦ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
Assertion
Ref Expression
hoidmvlelem3 (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
Distinct variable groups:   𝐴,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥   𝐴,𝑒,𝑓,𝑔,ℎ,𝑗,𝑘   𝑧,𝐴,ℎ,𝑗   𝐵,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝐵,𝑐,ℎ,𝑗,𝑘,𝑥   𝐵,𝑓,𝑔   𝑢,𝐵,ℎ,𝑗   𝑧,𝐵   𝐶,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝐶,𝑐   𝑢,𝐶   𝑧,𝐶   𝐷,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝐷,𝑐   𝑢,𝐷   𝑧,𝐷   𝐸,𝑎,𝑏,ℎ,𝑘,𝑥,𝑦   𝐸,𝑐   𝑧,𝐸   𝑗,𝐹   𝐺,𝑎,𝑏,ℎ,𝑘,𝑥,𝑦   𝐺,𝑐   𝑧,𝐺   𝐻,𝑎,𝑏,𝑗,𝑘   𝑧,𝐻   𝐽,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥   𝑔,𝐽   𝐾,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥   𝐿,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥   𝑒,𝐿,𝑓,𝑔   𝑧,𝐿   𝑗,𝑂,𝑘   𝑃,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝑃,𝑐   𝑆,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝑆,𝑐   𝑢,𝑆   𝑧,𝑆   𝑢,𝑈   𝑊,𝑎,𝑏,𝑗,𝑘,𝑥   𝑊,𝑐   𝑧,𝑊   𝑌,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝑌,𝑐   𝑒,𝑌,𝑓,𝑔   𝑍,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝑍,𝑐   𝑢,𝑍   𝑧,𝑍   𝜑,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝜑,𝑐
Allowed substitution hints:   𝜑(𝑧,𝑢,𝑒,𝑓,𝑔)   𝐴(𝑦,𝑢,𝑐)   𝐵(𝑒)   𝐶(𝑒,𝑓,𝑔)   𝐷(𝑒,𝑓,𝑔)   𝑃(𝑧,𝑢,𝑒,𝑓,𝑔)   𝑆(𝑒,𝑓,𝑔)   𝑈(𝑥,𝑦,𝑧,𝑒,𝑓,𝑔,ℎ,𝑗,𝑘,𝑎,𝑏,𝑐)   𝐸(𝑢,𝑒,𝑓,𝑔,𝑗)   𝐹(𝑥,𝑦,𝑧,𝑢,𝑒,𝑓,𝑔,ℎ,𝑘,𝑎,𝑏,𝑐)   𝐺(𝑢,𝑒,𝑓,𝑔,𝑗)   𝐻(𝑥,𝑦,𝑢,𝑒,𝑓,𝑔,ℎ,𝑐)   𝐽(𝑦,𝑧,𝑢,𝑒,𝑓,𝑐)   𝐾(𝑦,𝑧,𝑢,𝑒,𝑓,𝑔,𝑐)   𝐿(𝑦,𝑢,𝑐)   𝑂(𝑥,𝑦,𝑧,𝑢,𝑒,𝑓,𝑔,ℎ,𝑎,𝑏,𝑐)   𝑊(𝑦,𝑢,𝑒,𝑓,𝑔,ℎ)   𝑋(𝑥,𝑦,𝑧,𝑢,𝑒,𝑓,𝑔,ℎ,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑌(𝑧,𝑢)   𝑍(𝑒,𝑓,𝑔)
Proof of Theorem hoidmvlelem3
Dummy variables 𝑖 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1nn 11451 . . . . 5 1 ∈ ℕ
21a1i 11 . . . 4 ((𝜑 ∧ 𝑌 = ∅) → 1 ∈ ℕ)
3 0le0 11547 . . . . . 6 0 ≤ 0
43a1i 11 . . . . 5 ((𝜑 ∧ 𝑌 = ∅) → 0 ≤ 0)
5 hoidmvlelem3.g . . . . . . . 8 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
65a1i 11 . . . . . . 7 ((𝜑 ∧ 𝑌 = ∅) → 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
7 fveq2 6497 . . . . . . . . 9 (𝑌 = ∅ → (𝐿‘𝑌) = (𝐿‘∅))
8 reseq2 5688 . . . . . . . . . 10 (𝑌 = ∅ → (𝐴 ↾ 𝑌) = (𝐴 ↾ ∅))
9 res0 5697 . . . . . . . . . . 11 (𝐴 ↾ ∅) = ∅
109a1i 11 . . . . . . . . . 10 (𝑌 = ∅ → (𝐴 ↾ ∅) = ∅)
118, 10eqtrd 2809 . . . . . . . . 9 (𝑌 = ∅ → (𝐴 ↾ 𝑌) = ∅)
12 reseq2 5688 . . . . . . . . . 10 (𝑌 = ∅ → (𝐵 ↾ 𝑌) = (𝐵 ↾ ∅))
13 res0 5697 . . . . . . . . . . 11 (𝐵 ↾ ∅) = ∅
1413a1i 11 . . . . . . . . . 10 (𝑌 = ∅ → (𝐵 ↾ ∅) = ∅)
1512, 14eqtrd 2809 . . . . . . . . 9 (𝑌 = ∅ → (𝐵 ↾ 𝑌) = ∅)
167, 11, 15oveq123d 6996 . . . . . . . 8 (𝑌 = ∅ → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = (∅(𝐿‘∅)∅))
1716adantl 474 . . . . . . 7 ((𝜑 ∧ 𝑌 = ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = (∅(𝐿‘∅)∅))
18 hoidmvlelem3.l . . . . . . . 8 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
19 f0 6387 . . . . . . . . 9 ∅:∅⟶ℝ
2019a1i 11 . . . . . . . 8 ((𝜑 ∧ 𝑌 = ∅) → ∅:∅⟶ℝ)
2118, 20, 20hoidmv0val 42326 . . . . . . 7 ((𝜑 ∧ 𝑌 = ∅) → (∅(𝐿‘∅)∅) = 0)
226, 17, 213eqtrd 2813 . . . . . 6 ((𝜑 ∧ 𝑌 = ∅) → 𝐺 = 0)
23 nfcvd 2928 . . . . . . . . . . 11 (𝜑 → Ⅎ𝑗(𝑃‘1))
24 nfv 1874 . . . . . . . . . . 11 Ⅎ𝑗𝜑
25 simpr 477 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑗 = 1) → 𝑗 = 1)
2625fveq2d 6501 . . . . . . . . . . 11 ((𝜑 ∧ 𝑗 = 1) → (𝑃‘𝑗) = (𝑃‘1))
27 1red 10439 . . . . . . . . . . 11 (𝜑 → 1 ∈ ℝ)
28 rge0ssre 12659 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
29 id 22 . . . . . . . . . . . . . 14 (𝜑 → 𝜑)
301a1i 11 . . . . . . . . . . . . . 14 (𝜑 → 1 ∈ ℕ)
311elexi 3429 . . . . . . . . . . . . . . 15 1 ∈ V
32 eleq1 2848 . . . . . . . . . . . . . . . . 17 (𝑗 = 1 → (𝑗 ∈ ℕ ↔ 1 ∈ ℕ))
3332anbi2d 620 . . . . . . . . . . . . . . . 16 (𝑗 = 1 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
34 fveq2 6497 . . . . . . . . . . . . . . . . 17 (𝑗 = 1 → (𝑃‘𝑗) = (𝑃‘1))
3534eleq1d 2845 . . . . . . . . . . . . . . . 16 (𝑗 = 1 → ((𝑃‘𝑗) ∈ (0[,)+∞) ↔ (𝑃‘1) ∈ (0[,)+∞)))
3633, 35imbi12d 337 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞)) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝑃‘1) ∈ (0[,)+∞))))
37 id 22 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
38 ovexd 7009 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℕ → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V)
39 hoidmvlelem3.p . . . . . . . . . . . . . . . . . . 19 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
4039fvmpt2 6604 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ ∧ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
4137, 38, 40syl2anc 576 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
4241adantl 474 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
43 hoidmvlelem3.x . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 𝑋 ∈ Fin)
44 hoidmvlelem3.w . . . . . . . . . . . . . . . . . . . . . 22 𝑊 = (𝑌 ∪ {𝑍})
4544a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 𝑊 = (𝑌 ∪ {𝑍}))
46 hoidmvlelem3.y . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → 𝑌 ⊆ 𝑋)
47 hoidmvlelem3.z . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
4847eldifad 3836 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → 𝑍 ∈ 𝑋)
49 snssi 4612 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑍 ∈ 𝑋 → {𝑍} ⊆ 𝑋)
5048, 49syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {𝑍} ⊆ 𝑋)
5146, 50unssd 4045 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑌 ∪ {𝑍}) ⊆ 𝑋)
5245, 51eqsstrd 3890 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 𝑊 ⊆ 𝑋)
5343, 52ssfid 8535 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝑊 ∈ Fin)
54 ssun1 4032 . . . . . . . . . . . . . . . . . . . . 21 𝑌 ⊆ (𝑌 ∪ {𝑍})
5544eqcomi 2782 . . . . . . . . . . . . . . . . . . . . 21 (𝑌 ∪ {𝑍}) = 𝑊
5654, 55sseqtri 3888 . . . . . . . . . . . . . . . . . . . 20 𝑌 ⊆ 𝑊
5756a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝑌 ⊆ 𝑊)
5853, 57ssfid 8535 . . . . . . . . . . . . . . . . . 18 (𝜑 → 𝑌 ∈ Fin)
5958adantr 473 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ Fin)
60 iftrue 4351 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
6160adantl 474 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
62 hoidmvlelem3.c . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
6362ffvelrnda 6675 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑𝑚 𝑊))
64 elmapi 8227 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐶‘𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐶‘𝑗):𝑊⟶ℝ)
6563, 64syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ)
6654, 44sseqtr4i 3889 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑌 ⊆ 𝑊
6766a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ⊆ 𝑊)
6865, 67fssresd 6372 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
69 reex 10425 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ℝ ∈ V
7069a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑗 ∈ ℕ) → ℝ ∈ V)
7153, 57ssexd 5081 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → 𝑌 ∈ V)
7271adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ V)
7370, 72elmapd 8219 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌) ↔ ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
7468, 73mpbird 249 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌))
7574adantr 473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌))
7661, 75eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚 𝑌))
77 iffalse 4354 . . . . . . . . . . . . . . . . . . . . . . 23 (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
7877adantl 474 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
79 0red 10442 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ∈ ℝ)
80 hoidmvlelem3.f . . . . . . . . . . . . . . . . . . . . . . . . 25 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)
8179, 80fmptd 6700 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 𝐹:𝑌⟶ℝ)
8269a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ℝ ∈ V)
8382, 58elmapd 8219 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐹 ∈ (ℝ ↑𝑚 𝑌) ↔ 𝐹:𝑌⟶ℝ))
8481, 83mpbird 249 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → 𝐹 ∈ (ℝ ↑𝑚 𝑌))
8584ad2antrr 714 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹 ∈ (ℝ ↑𝑚 𝑌))
8678, 85eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚 𝑌))
8776, 86pm2.61dan 801 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚 𝑌))
88 hoidmvlelem3.j . . . . . . . . . . . . . . . . . . . 20 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
8987, 88fmptd 6700 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝐽:ℕ⟶(ℝ ↑𝑚 𝑌))
9089ffvelrnda 6675 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) ∈ (ℝ ↑𝑚 𝑌))
91 elmapi 8227 . . . . . . . . . . . . . . . . . 18 ((𝐽‘𝑗) ∈ (ℝ ↑𝑚 𝑌) → (𝐽‘𝑗):𝑌⟶ℝ)
9290, 91syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ)
93 iftrue 4351 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
9493adantl 474 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
95 hoidmvlelem3.d . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
9695ffvelrnda 6675 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑𝑚 𝑊))
97 elmapi 8227 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐷‘𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐷‘𝑗):𝑊⟶ℝ)
9896, 97syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
9998, 67fssresd 6372 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
10070, 72elmapd 8219 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌) ↔ ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
10199, 100mpbird 249 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌))
102101adantr 473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌))
10394, 102eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚 𝑌))
104 iffalse 4354 . . . . . . . . . . . . . . . . . . . . . . 23 (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
105104adantl 474 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
106105, 85eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚 𝑌))
107103, 106pm2.61dan 801 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚 𝑌))
108 hoidmvlelem3.k . . . . . . . . . . . . . . . . . . . 20 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
109107, 108fmptd 6700 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝐾:ℕ⟶(ℝ ↑𝑚 𝑌))
110109ffvelrnda 6675 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) ∈ (ℝ ↑𝑚 𝑌))
111 elmapi 8227 . . . . . . . . . . . . . . . . . 18 ((𝐾‘𝑗) ∈ (ℝ ↑𝑚 𝑌) → (𝐾‘𝑗):𝑌⟶ℝ)
112110, 111syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ)
11318, 59, 92, 112hoidmvcl 42325 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ (0[,)+∞))
11442, 113eqeltrd 2861 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞))
11531, 36, 114vtocl 3473 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 ∈ ℕ) → (𝑃‘1) ∈ (0[,)+∞))
11629, 30, 115syl2anc 576 . . . . . . . . . . . . 13 (𝜑 → (𝑃‘1) ∈ (0[,)+∞))
11728, 116sseldi 3851 . . . . . . . . . . . 12 (𝜑 → (𝑃‘1) ∈ ℝ)
118117recnd 10467 . . . . . . . . . . 11 (𝜑 → (𝑃‘1) ∈ ℂ)
11923, 24, 26, 27, 118sumsnd 40736 . . . . . . . . . 10 (𝜑 → Σ𝑗 ∈ {1} (𝑃‘𝑗) = (𝑃‘1))
120119adantr 473 . . . . . . . . 9 ((𝜑 ∧ 𝑌 = ∅) → Σ𝑗 ∈ {1} (𝑃‘𝑗) = (𝑃‘1))
121 fveq2 6497 . . . . . . . . . . . . 13 (𝑗 = 1 → (𝐽‘𝑗) = (𝐽‘1))
122 fveq2 6497 . . . . . . . . . . . . 13 (𝑗 = 1 → (𝐾‘𝑗) = (𝐾‘1))
123121, 122oveq12d 6993 . . . . . . . . . . . 12 (𝑗 = 1 → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)))
124 ovex 7007 . . . . . . . . . . . 12 ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) ∈ V
125123, 39, 124fvmpt 6594 . . . . . . . . . . 11 (1 ∈ ℕ → (𝑃‘1) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)))
1261, 125ax-mp 5 . . . . . . . . . 10 (𝑃‘1) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1))
127126a1i 11 . . . . . . . . 9 ((𝜑 ∧ 𝑌 = ∅) → (𝑃‘1) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)))
1287oveqd 6992 . . . . . . . . . . 11 (𝑌 = ∅ → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = ((𝐽‘1)(𝐿‘∅)(𝐾‘1)))
129128adantl 474 . . . . . . . . . 10 ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = ((𝐽‘1)(𝐿‘∅)(𝐾‘1)))
130121feq1d 6327 . . . . . . . . . . . . . . . 16 (𝑗 = 1 → ((𝐽‘𝑗):𝑌⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ))
13133, 130imbi12d 337 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝐽‘1):𝑌⟶ℝ)))
13268adantr 473 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
13361feq1d 6327 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
134132, 133mpbird 249 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
13581ad2antrr 714 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹:𝑌⟶ℝ)
13678feq1d 6327 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ 𝐹:𝑌⟶ℝ))
137135, 136mpbird 249 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
138134, 137pm2.61dan 801 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
139 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
140 fvex 6510 . . . . . . . . . . . . . . . . . . . . 21 (𝐶‘𝑗) ∈ V
141140resex 5742 . . . . . . . . . . . . . . . . . . . 20 ((𝐶‘𝑗) ↾ 𝑌) ∈ V
14261, 141syl6eqel 2869 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
14384elexd 3430 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → 𝐹 ∈ V)
144143adantr 473 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐹 ∈ V)
145144adantr 473 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹 ∈ V)
14678, 145eqeltrd 2861 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
147142, 146pm2.61dan 801 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
14888fvmpt2 6604 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
149139, 147, 148syl2anc 576 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
150149feq1d 6327 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗):𝑌⟶ℝ ↔ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ))
151138, 150mpbird 249 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ)
15231, 131, 151vtocl 3473 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 ∈ ℕ) → (𝐽‘1):𝑌⟶ℝ)
15329, 30, 152syl2anc 576 . . . . . . . . . . . . 13 (𝜑 → (𝐽‘1):𝑌⟶ℝ)
154153adantr 473 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 = ∅) → (𝐽‘1):𝑌⟶ℝ)
155 id 22 . . . . . . . . . . . . . . 15 (𝑌 = ∅ → 𝑌 = ∅)
156155eqcomd 2779 . . . . . . . . . . . . . 14 (𝑌 = ∅ → ∅ = 𝑌)
157156feq2d 6328 . . . . . . . . . . . . 13 (𝑌 = ∅ → ((𝐽‘1):∅⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ))
158157adantl 474 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1):∅⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ))
159154, 158mpbird 249 . . . . . . . . . . 11 ((𝜑 ∧ 𝑌 = ∅) → (𝐽‘1):∅⟶ℝ)
160122feq1d 6327 . . . . . . . . . . . . . . . 16 (𝑗 = 1 → ((𝐾‘𝑗):𝑌⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ))
16133, 160imbi12d 337 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝐾‘1):𝑌⟶ℝ)))
16231, 161, 112vtocl 3473 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 ∈ ℕ) → (𝐾‘1):𝑌⟶ℝ)
16329, 30, 162syl2anc 576 . . . . . . . . . . . . 13 (𝜑 → (𝐾‘1):𝑌⟶ℝ)
164163adantr 473 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 = ∅) → (𝐾‘1):𝑌⟶ℝ)
165156feq2d 6328 . . . . . . . . . . . . 13 (𝑌 = ∅ → ((𝐾‘1):∅⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ))
166165adantl 474 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 = ∅) → ((𝐾‘1):∅⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ))
167164, 166mpbird 249 . . . . . . . . . . 11 ((𝜑 ∧ 𝑌 = ∅) → (𝐾‘1):∅⟶ℝ)
16818, 159, 167hoidmv0val 42326 . . . . . . . . . 10 ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘∅)(𝐾‘1)) = 0)
169129, 168eqtrd 2809 . . . . . . . . 9 ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = 0)
170120, 127, 1693eqtrd 2813 . . . . . . . 8 ((𝜑 ∧ 𝑌 = ∅) → Σ𝑗 ∈ {1} (𝑃‘𝑗) = 0)
171170oveq2d 6991 . . . . . . 7 ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) = ((1 + 𝐸) · 0))
172 hoidmvlelem3.e . . . . . . . . . . . 12 (𝜑 → 𝐸 ∈ ℝ+)
173172rpred 12247 . . . . . . . . . . 11 (𝜑 → 𝐸 ∈ ℝ)
17427, 173readdcld 10468 . . . . . . . . . 10 (𝜑 → (1 + 𝐸) ∈ ℝ)
175174recnd 10467 . . . . . . . . 9 (𝜑 → (1 + 𝐸) ∈ ℂ)
176175mul01d 10638 . . . . . . . 8 (𝜑 → ((1 + 𝐸) · 0) = 0)
177176adantr 473 . . . . . . 7 ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · 0) = 0)
178 eqidd 2774 . . . . . . 7 ((𝜑 ∧ 𝑌 = ∅) → 0 = 0)
179171, 177, 1783eqtrd 2813 . . . . . 6 ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) = 0)
18022, 179breq12d 4939 . . . . 5 ((𝜑 ∧ 𝑌 = ∅) → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) ↔ 0 ≤ 0))
1814, 180mpbird 249 . . . 4 ((𝜑 ∧ 𝑌 = ∅) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)))
182 oveq2 6983 . . . . . . . . 9 (𝑚 = 1 → (1...𝑚) = (1...1))
1831nnzi 11818 . . . . . . . . . . 11 1 ∈ ℤ
184 fzsn 12764 . . . . . . . . . . 11 (1 ∈ ℤ → (1...1) = {1})
185183, 184ax-mp 5 . . . . . . . . . 10 (1...1) = {1}
186185a1i 11 . . . . . . . . 9 (𝑚 = 1 → (1...1) = {1})
187182, 186eqtrd 2809 . . . . . . . 8 (𝑚 = 1 → (1...𝑚) = {1})
188187sumeq1d 14917 . . . . . . 7 (𝑚 = 1 → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) = Σ𝑗 ∈ {1} (𝑃‘𝑗))
189188oveq2d 6991 . . . . . 6 (𝑚 = 1 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)))
190189breq2d 4938 . . . . 5 (𝑚 = 1 → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) ↔ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗))))
191190rspcev 3530 . . . 4 ((1 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗))) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
1922, 181, 191syl2anc 576 . . 3 ((𝜑 ∧ 𝑌 = ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
193 simpl 475 . . . 4 ((𝜑 ∧ ¬ 𝑌 = ∅) → 𝜑)
194 neqne 2970 . . . . 5 (¬ 𝑌 = ∅ → 𝑌 ≠ ∅)
195194adantl 474 . . . 4 ((𝜑 ∧ ¬ 𝑌 = ∅) → 𝑌 ≠ ∅)
196 nfv 1874 . . . . . 6 Ⅎ𝑗(𝜑 ∧ 𝑌 ≠ ∅)
197183a1i 11 . . . . . 6 ((𝜑 ∧ 𝑌 ≠ ∅) → 1 ∈ ℤ)
198 nnuz 12094 . . . . . 6 ℕ = (ℤ≥‘1)
199114adantlr 703 . . . . . 6 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞))
200 hoidmvlelem3.a . . . . . . . . . . . 12 (𝜑 → 𝐴:𝑊⟶ℝ)
20166a1i 11 . . . . . . . . . . . 12 (𝜑 → 𝑌 ⊆ 𝑊)
202200, 201fssresd 6372 . . . . . . . . . . 11 (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
203 hoidmvlelem3.b . . . . . . . . . . . 12 (𝜑 → 𝐵:𝑊⟶ℝ)
204203, 201fssresd 6372 . . . . . . . . . . 11 (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
20518, 58, 202, 204hoidmvcl 42325 . . . . . . . . . 10 (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞))
20628, 205sseldi 3851 . . . . . . . . 9 (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ ℝ)
2075, 206syl5eqel 2865 . . . . . . . 8 (𝜑 → 𝐺 ∈ ℝ)
208 0red 10442 . . . . . . . . 9 (𝜑 → 0 ∈ ℝ)
209 1rp 12207 . . . . . . . . . . . . 13 1 ∈ ℝ+
210209a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℝ+)
211210, 172jca 504 . . . . . . . . . . 11 (𝜑 → (1 ∈ ℝ+ ∧ 𝐸 ∈ ℝ+))
212 rpaddcl 12227 . . . . . . . . . . 11 ((1 ∈ ℝ+ ∧ 𝐸 ∈ ℝ+) → (1 + 𝐸) ∈ ℝ+)
213211, 212syl 17 . . . . . . . . . 10 (𝜑 → (1 + 𝐸) ∈ ℝ+)
214 rpgt0 12217 . . . . . . . . . 10 ((1 + 𝐸) ∈ ℝ+ → 0 < (1 + 𝐸))
215213, 214syl 17 . . . . . . . . 9 (𝜑 → 0 < (1 + 𝐸))
216208, 215gtned 10574 . . . . . . . 8 (𝜑 → (1 + 𝐸) ≠ 0)
217207, 174, 216redivcld 11268 . . . . . . 7 (𝜑 → (𝐺 / (1 + 𝐸)) ∈ ℝ)
218217adantr 473 . . . . . 6 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
219217ltpnfd 12332 . . . . . . . . . 10 (𝜑 → (𝐺 / (1 + 𝐸)) < +∞)
220219adantr 473 . . . . . . . . 9 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < +∞)
221 id 22 . . . . . . . . . . 11 ((Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞ → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞)
222221eqcomd 2779 . . . . . . . . . 10 ((Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞ → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
223222adantl 474 . . . . . . . . 9 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
224220, 223breqtrd 4952 . . . . . . . 8 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
225224adantlr 703 . . . . . . 7 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
226 simpl 475 . . . . . . . 8 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝜑 ∧ 𝑌 ≠ ∅))
227 simpr 477 . . . . . . . . . 10 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞)
228 nnex 11445 . . . . . . . . . . . 12 ℕ ∈ V
229228a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ℕ ∈ V)
230 icossicc 12639 . . . . . . . . . . . . . 14 (0[,)+∞) ⊆ (0[,]+∞)
231230, 114sseldi 3851 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,]+∞))
232 eqid 2773 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) = (𝑗 ∈ ℕ ↦ (𝑃‘𝑗))
233231, 232fmptd 6700 . . . . . . . . . . . 12 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)):ℕ⟶(0[,]+∞))
234233adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)):ℕ⟶(0[,]+∞))
235229, 234sge0repnf 42129 . . . . . . . . . 10 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞))
236227, 235mpbird 249 . . . . . . . . 9 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
237236adantlr 703 . . . . . . . 8 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
238218adantr 473 . . . . . . . . 9 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
239207adantr 473 . . . . . . . . . 10 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ∈ ℝ)
240239adantlr 703 . . . . . . . . 9 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ∈ ℝ)
241 simpr 477 . . . . . . . . 9 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
24227, 172ltaddrpd 12280 . . . . . . . . . . . 12 (𝜑 → 1 < (1 + 𝐸))
243242adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ 𝑌 ≠ ∅) → 1 < (1 + 𝐸))
24458adantr 473 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑌 ∈ Fin)
245 simpr 477 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑌 ≠ ∅)
246202adantr 473 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
247204adantr 473 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
24818, 244, 245, 246, 247hoidmvn0val 42327 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑌 ≠ ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
2495a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
250 fvres 6516 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ 𝑌 → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘))
251 fvres 6516 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ 𝑌 → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘))
252250, 251oveq12d 6993 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ 𝑌 → (((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
253252fveq2d 6501 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ 𝑌 → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
254253adantl 474 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
255200adantr 473 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐴:𝑊⟶ℝ)
256 elun1 4036 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ 𝑌 → 𝑘 ∈ (𝑌 ∪ {𝑍}))
257256, 44syl6eleqr 2872 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ 𝑌 → 𝑘 ∈ 𝑊)
258257adantl 474 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
259255, 258ffvelrnd 6676 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) ∈ ℝ)
260203adantr 473 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵:𝑊⟶ℝ)
261260, 258ffvelrnd 6676 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐵‘𝑘) ∈ ℝ)
262 volico 41729 . . . . . . . . . . . . . . . . . . 19 (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
263259, 261, 262syl2anc 576 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
264 hoidmvlelem3.lt . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))
265258, 264syldan 583 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) < (𝐵‘𝑘))
266265iftrued 4353 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑘 ∈ 𝑌) → if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
267254, 263, 2663eqtrd 2813 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
268267prodeq2dv 15136 . . . . . . . . . . . . . . . 16 (𝜑 → ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
269268eqcomd 2779 . . . . . . . . . . . . . . 15 (𝜑 → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
270269adantr 473 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑌 ≠ ∅) → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
271248, 249, 2703eqtr4d 2819 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
272 difrp 12243 . . . . . . . . . . . . . . . . 17 (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+))
273259, 261, 272syl2anc 576 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+))
274265, 273mpbid 224 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+)
27558, 274fprodrpcl 15169 . . . . . . . . . . . . . 14 (𝜑 → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+)
276275adantr 473 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑌 ≠ ∅) → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+)
277271, 276eqeltrd 2861 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ∈ ℝ+)
278213adantr 473 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 ≠ ∅) → (1 + 𝐸) ∈ ℝ+)
279277, 278ltdivgt1 41083 . . . . . . . . . . 11 ((𝜑 ∧ 𝑌 ≠ ∅) → (1 < (1 + 𝐸) ↔ (𝐺 / (1 + 𝐸)) < 𝐺))
280243, 279mpbid 224 . . . . . . . . . 10 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) < 𝐺)
281280adantr 473 . . . . . . . . 9 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) < 𝐺)
282 hoidmvlelem3.i2 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
283282adantr 473 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
284 fvexd 6512 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑥‘𝑘) ∈ V)
285 hoidmvlelem3.s . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → 𝑆 ∈ 𝑈)
286285elexd 3430 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → 𝑆 ∈ V)
287284, 286ifcld 4390 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
288287ralrimivw 3128 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
289288adantr 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
290 eqid 2773 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
291290fnmpt 6316 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊)
292289, 291syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊)
293 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
294 mptexg 6809 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑊 ∈ Fin → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V)
29553, 294syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V)
296295adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V)
297 hoidmvlelem3.o . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑂 = (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↦ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
298297fvmpt2 6604 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V) → (𝑂‘𝑥) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
299293, 296, 298syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
300299fneq1d 6277 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥) Fn 𝑊 ↔ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊))
301292, 300mpbird 249 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) Fn 𝑊)
302 nfv 1874 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘𝜑
303 nfcv 2927 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘𝑥
304 nfixp1 8278 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
305303, 304nfel 2939 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
306302, 305nfan 1863 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
307299fveq1d 6499 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘))
308307adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘))
309 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑊)
310287adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
311290fvmpt2 6604 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 ∈ 𝑊 ∧ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
312309, 310, 311syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
313312adantlr 703 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
314308, 313eqtrd 2809 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
315 iftrue 4351 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
316315adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
317 vex 3413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑥 ∈ V
318317elixp 8265 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
319318simprbi 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
320319adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
321 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑌)
322 rspa 3151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
323320, 321, 322syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
324323ad4ant24 742 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
325316, 324eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
326 snidg 4468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
32747, 326syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → 𝑍 ∈ {𝑍})
328 elun2 4037 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
329327, 328syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → 𝑍 ∈ (𝑌 ∪ {𝑍}))
33055a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → (𝑌 ∪ {𝑍}) = 𝑊)
331329, 330eleqtrd 2863 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → 𝑍 ∈ 𝑊)
332200, 331ffvelrnd 6676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑 → (𝐴‘𝑍) ∈ ℝ)
333332rexrd 10489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (𝐴‘𝑍) ∈ ℝ*)
334203, 331ffvelrnd 6676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑 → (𝐵‘𝑍) ∈ ℝ)
335334rexrd 10489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (𝐵‘𝑍) ∈ ℝ*)
336 iccssxr 12634 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ*
337 hoidmvlelem3.u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
338 ssrab2 3941 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
339337, 338eqsstri 3886 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
340339, 285sseldi 3851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
341336, 340sseldi 3851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → 𝑆 ∈ ℝ*)
342 iccgelb 12608 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝐴‘𝑍) ∈ ℝ* ∧ (𝐵‘𝑍) ∈ ℝ* ∧ 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑆)
343333, 335, 340, 342syl3anc 1352 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (𝐴‘𝑍) ≤ 𝑆)
344 hoidmvlelem3.sb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → 𝑆 < (𝐵‘𝑍))
345333, 335, 341, 343, 344elicod 12602 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
346345ad2antrr 714 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
347 iffalse 4354 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ 𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = 𝑆)
348347adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = 𝑆)
34944eleq2i 2852 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑘 ∈ 𝑊 ↔ 𝑘 ∈ (𝑌 ∪ {𝑍}))
350349biimpi 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑘 ∈ 𝑊 → 𝑘 ∈ (𝑌 ∪ {𝑍}))
351350adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
352 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → ¬ 𝑘 ∈ 𝑌)
353 elunnel1 4010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ {𝑍})
354351, 352, 353syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ {𝑍})
355 elsni 4453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 ∈ {𝑍} → 𝑘 = 𝑍)
356354, 355syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 = 𝑍)
357 fveq2 6497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
358 fveq2 6497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 = 𝑍 → (𝐵‘𝑘) = (𝐵‘𝑍))
359357, 358oveq12d 6993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
360356, 359syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
361360adantll 702 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
362348, 361eleq12d 2855 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → (if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍))))
363346, 362mpbird 249 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
364363adantllr 707 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
365325, 364pm2.61dan 801 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
366314, 365eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
367366ex 405 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 → ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
368306, 367ralrimi 3161 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
369301, 368jca 504 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
370 fvex 6510 . . . . . . . . . . . . . . . . . . . . . 22 (𝑂‘𝑥) ∈ V
371370elixp 8265 . . . . . . . . . . . . . . . . . . . . 21 ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
372369, 371sylibr 226 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
373283, 372sseldd 3854 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
374 eliun 4793 . . . . . . . . . . . . . . . . . . 19 ((𝑂‘𝑥) ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
375373, 374sylib 210 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
376 ixpfn 8264 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) → 𝑥 Fn 𝑌)
377376adantl 474 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 Fn 𝑌)
378377ad2antrr 714 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑥 Fn 𝑌)
379 nfv 1874 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘 𝑗 ∈ ℕ
380306, 379nfan 1863 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ)
381 nfcv 2927 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘(𝑂‘𝑥)
382 nfixp1 8278 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))
383381, 382nfel 2939 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘(𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))
384380, 383nfan 1863 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
3853073adant3 1113 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘))
386287adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
387258, 386, 311syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
3883873adant2 1112 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
3893153ad2ant3 1116 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
390385, 388, 3893eqtrrd 2814 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) = ((𝑂‘𝑥)‘𝑘))
391390ad5ant125 1347 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) = ((𝑂‘𝑥)‘𝑘))
392 simpl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
393370elixp 8265 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
394392, 393sylib 210 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
395394simprd 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
396257adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
397 rspa 3151 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
398395, 396, 397syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
399398adantll 702 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
400391, 399eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
40129ad3antrrr 718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝜑)
40237ad2antlr 715 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑗 ∈ ℕ)
403299fveq1d 6499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥)‘𝑍) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍))
404 eqidd 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
405 eleq1 2848 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑘 = 𝑍 → (𝑘 ∈ 𝑌 ↔ 𝑍 ∈ 𝑌))
406 fveq2 6497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑘 = 𝑍 → (𝑥‘𝑘) = (𝑥‘𝑍))
407405, 406ifbieq1d 4368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑘 = 𝑍 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
408407adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ 𝑘 = 𝑍) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
409 fvexd 6512 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝜑 → (𝑥‘𝑍) ∈ V)
410409, 286ifcld 4390 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) ∈ V)
411404, 408, 331, 410fvmptd 6600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
412411adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
41347eldifbd 3837 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → ¬ 𝑍 ∈ 𝑌)
414413iffalsed 4356 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) = 𝑆)
415414adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) = 𝑆)
416403, 412, 4153eqtrrd 2814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑆 = ((𝑂‘𝑥)‘𝑍))
417416ad2antrr 714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑆 = ((𝑂‘𝑥)‘𝑍))
418401, 331syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑍 ∈ 𝑊)
419393simprbi 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
420419adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
421 fveq2 6497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑘 = 𝑍 → ((𝑂‘𝑥)‘𝑘) = ((𝑂‘𝑥)‘𝑍))
422 fveq2 6497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑘 = 𝑍 → ((𝐶‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑍))
423 fveq2 6497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑘 = 𝑍 → ((𝐷‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑍))
424422, 423oveq12d 6993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑘 = 𝑍 → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
425421, 424eleq12d 2855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑘 = 𝑍 → (((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))))
426425rspcva 3528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑍 ∈ 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
427418, 420, 426syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
428417, 427eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
4291493adant3 1113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
430603ad2ant3 1116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
431429, 430eqtrd 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = ((𝐶‘𝑗) ↾ 𝑌))
432431fveq1d 6499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
433401, 402, 428, 432syl3anc 1352 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
434433adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
435 fvres 6516 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ 𝑌 → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
436435adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
437434, 436eqtrd 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
438107elexd 3430 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
439108fvmpt2 6604 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
440139, 438, 439syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
4414403adant3 1113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
442933ad2ant3 1116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
443441, 442eqtrd 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = ((𝐷‘𝑗) ↾ 𝑌))
444443fveq1d 6499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
445401, 402, 428, 444syl3anc 1352 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
446445adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
447 fvres 6516 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ 𝑌 → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
448447adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
449446, 448eqtrd 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
450437, 449oveq12d 6993 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
451450eqcomd 2779 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
452400, 451eleqtrd 2863 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
453452ex 405 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑘 ∈ 𝑌 → (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
454384, 453ralrimi 3161 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
455378, 454jca 504 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
456317elixp 8265 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
457455, 456sylibr 226 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
458457ex 405 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) → ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
459458reximdva 3214 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
460375, 459mpd 15 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
461 eliun 4793 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
462460, 461sylibr 226 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
463462ralrimiva 3127 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ X 𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
464 dfss3 3842 . . . . . . . . . . . . . . 15 (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ ∀𝑥 ∈ X 𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
465463, 464sylibr 226 . . . . . . . . . . . . . 14 (𝜑 → X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
466 ovexd 7009 . . . . . . . . . . . . . . . . 17 (𝜑 → (ℝ ↑𝑚 𝑌) ∈ V)
467228a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ℕ ∈ V)
468466, 467elmapd 8219 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐾 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ) ↔ 𝐾:ℕ⟶(ℝ ↑𝑚 𝑌)))
469109, 468mpbird 249 . . . . . . . . . . . . . . 15 (𝜑 → 𝐾 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ))
470466, 467elmapd 8219 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐽 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ) ↔ 𝐽:ℕ⟶(ℝ ↑𝑚 𝑌)))
47189, 470mpbird 249 . . . . . . . . . . . . . . . 16 (𝜑 → 𝐽 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ))
47282, 71elmapd 8219 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝐵 ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌) ↔ (𝐵 ↾ 𝑌):𝑌⟶ℝ))
473204, 472mpbird 249 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐵 ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌))
47482, 71elmapd 8219 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐴 ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌) ↔ (𝐴 ↾ 𝑌):𝑌⟶ℝ))
475202, 474mpbird 249 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴 ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌))
476 hoidmvlelem3.i . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑒 ∈ (ℝ ↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
477 fveq1 6496 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑒 = (𝐴 ↾ 𝑌) → (𝑒‘𝑘) = ((𝐴 ↾ 𝑌)‘𝑘))
478477adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑒‘𝑘) = ((𝐴 ↾ 𝑌)‘𝑘))
479250adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘))
480478, 479eqtrd 2809 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑒‘𝑘) = (𝐴‘𝑘))
481480oveq1d 6990 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = ((𝐴‘𝑘)[,)(𝑓‘𝑘)))
482481ixpeq2dva 8273 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑒 = (𝐴 ↾ 𝑌) → X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)))
483482sseq1d 3883 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 = (𝐴 ↾ 𝑌) → (X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))))
484 oveq1 6982 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑒 = (𝐴 ↾ 𝑌) → (𝑒(𝐿‘𝑌)𝑓) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓))
485484breq1d 4936 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 = (𝐴 ↾ 𝑌) → ((𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
486483, 485imbi12d 337 . . . . . . . . . . . . . . . . . . . . . 22 (𝑒 = (𝐴 ↾ 𝑌) → ((X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
487486ralbidv 3142 . . . . . . . . . . . . . . . . . . . . 21 (𝑒 = (𝐴 ↾ 𝑌) → (∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
488487ralbidv 3142 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = (𝐴 ↾ 𝑌) → (∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
489488ralbidv 3142 . . . . . . . . . . . . . . . . . . 19 (𝑒 = (𝐴 ↾ 𝑌) → (∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
490489rspcva 3528 . . . . . . . . . . . . . . . . . 18 (((𝐴 ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌) ∧ ∀𝑒 ∈ (ℝ ↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → ∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
491475, 476, 490syl2anc 576 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
492 fveq1 6496 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝐵 ↾ 𝑌) → (𝑓‘𝑘) = ((𝐵 ↾ 𝑌)‘𝑘))
493492adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑓‘𝑘) = ((𝐵 ↾ 𝑌)‘𝑘))
494251adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘))
495493, 494eqtrd 2809 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑓‘𝑘) = (𝐵‘𝑘))
496495oveq2d 6991 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝑓‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
497496ixpeq2dva 8273 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = (𝐵 ↾ 𝑌) → X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
498497sseq1d 3883 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝐵 ↾ 𝑌) → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))))
499 oveq2 6983 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = (𝐵 ↾ 𝑌) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
500499breq1d 4936 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝐵 ↾ 𝑌) → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
501498, 500imbi12d 337 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝐵 ↾ 𝑌) → ((X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
502501ralbidv 3142 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝐵 ↾ 𝑌) → (∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
503502ralbidv 3142 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐵 ↾ 𝑌) → (∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
504503rspcva 3528 . . . . . . . . . . . . . . . . 17 (((𝐵 ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌) ∧ ∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → ∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
505473, 491, 504syl2anc 576 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
506 fveq1 6496 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔 = 𝐽 → (𝑔‘𝑗) = (𝐽‘𝑗))
507506fveq1d 6499 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 = 𝐽 → ((𝑔‘𝑗)‘𝑘) = ((𝐽‘𝑗)‘𝑘))
508507oveq1d 6990 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 = 𝐽 → (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
509508ixpeq2dv 8274 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝐽 → X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
510509iuneq2d 4817 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝐽 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
511510sseq2d 3884 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝐽 → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))))
512506oveq1d 6990 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 = 𝐽 → ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)) = ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))
513512mpteq2dv 5020 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝐽 → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))
514513fveq2d 6501 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝐽 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))
515514breq2d 4938 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝐽 → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
516511, 515imbi12d 337 . . . . . . . . . . . . . . . . . 18 (𝑔 = 𝐽 → ((X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
517516ralbidv 3142 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝐽 → (∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
518517rspcva 3528 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ) ∧ ∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → ∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
519471, 505, 518syl2anc 576 . . . . . . . . . . . . . . 15 (𝜑 → ∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
520 fveq1 6496 . . . . . . . . . . . . . . . . . . . . . 22 (ℎ = 𝐾 → (ℎ‘𝑗) = (𝐾‘𝑗))
521520fveq1d 6499 . . . . . . . . . . . . . . . . . . . . 21 (ℎ = 𝐾 → ((ℎ‘𝑗)‘𝑘) = ((𝐾‘𝑗)‘𝑘))
522521oveq2d 6991 . . . . . . . . . . . . . . . . . . . 20 (ℎ = 𝐾 → (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
523522ixpeq2dv 8274 . . . . . . . . . . . . . . . . . . 19 (ℎ = 𝐾 → X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
524523iuneq2d 4817 . . . . . . . . . . . . . . . . . 18 (ℎ = 𝐾 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
525524sseq2d 3884 . . . . . . . . . . . . . . . . 17 (ℎ = 𝐾 → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
526520oveq2d 6991 . . . . . . . . . . . . . . . . . . . 20 (ℎ = 𝐾 → ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
527526mpteq2dv 5020 . . . . . . . . . . . . . . . . . . 19 (ℎ = 𝐾 → (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))
528527fveq2d 6501 . . . . . . . . . . . . . . . . . 18 (ℎ = 𝐾 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
529528breq2d 4938 . . . . . . . . . . . . . . . . 17 (ℎ = 𝐾 → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
530525, 529imbi12d 337 . . . . . . . . . . . . . . . 16 (ℎ = 𝐾 → ((X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))))
531530rspcva 3528 . . . . . . . . . . . . . . 15 ((𝐾 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ) ∧ ∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
532469, 519, 531syl2anc 576 . . . . . . . . . . . . . 14 (𝜑 → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
533465, 532mpd 15 . . . . . . . . . . . . 13 (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
534 idd 24 . . . . . . . . . . . . 13 (𝜑 → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
535533, 534mpd 15 . . . . . . . . . . . 12 (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
536535adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ 𝑌 ≠ ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
53741adantl 474 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
538537mpteq2dva 5019 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))
539538fveq2d 6501 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
540249, 539breq12d 4939 . . . . . . . . . . 11 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
541536, 540mpbird 249 . . . . . . . . . 10 ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
542541adantr 473 . . . . . . . . 9 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
543238, 240, 241, 281, 542ltletrd 10599 . . . . . . . 8 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
544226, 237, 543syl2anc 576 . . . . . . 7 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
545225, 544pm2.61dan 801 . . . . . 6 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
546196, 197, 198, 199, 218, 545sge0uzfsumgt 42187 . . . . 5 ((𝜑 ∧ 𝑌 ≠ ∅) → ∃𝑚 ∈ ℕ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
547217adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
548 fzfid 13155 . . . . . . . . . . . . 13 (𝜑 → (1...𝑚) ∈ Fin)
549 simpl 475 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → 𝜑)
550 elfznn 12751 . . . . . . . . . . . . . . 15 (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
551550adantl 474 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → 𝑗 ∈ ℕ)
55228, 114sseldi 3851 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ ℝ)
553549, 551, 552syl2anc 576 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → (𝑃‘𝑗) ∈ ℝ)
554548, 553fsumrecl 14950 . . . . . . . . . . . 12 (𝜑 → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ∈ ℝ)
555554adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ∈ ℝ)
556 simpr 477 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
557547, 555, 556ltled 10587 . . . . . . . . . 10 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) ≤ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
558207adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ∈ ℝ)
559213adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (1 + 𝐸) ∈ ℝ+)
560558, 555, 559ledivmuld 12300 . . . . . . . . . 10 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ((𝐺 / (1 + 𝐸)) ≤ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ↔ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
561557, 560mpbid 224 . . . . . . . . 9 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
562561ex 405 . . . . . . . 8 (𝜑 → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
563562adantr 473 . . . . . . 7 ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
564563adantlr 703 . . . . . 6 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑚 ∈ ℕ) → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
565564reximdva 3214 . . . . 5 ((𝜑 ∧ 𝑌 ≠ ∅) → (∃𝑚 ∈ ℕ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
566546, 565mpd 15 . . . 4 ((𝜑 ∧ 𝑌 ≠ ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
567193, 195, 566syl2anc 576 . . 3 ((𝜑 ∧ ¬ 𝑌 = ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
568192, 567pm2.61dan 801 . 2 (𝜑 → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
569433ad2ant1 1114 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑋 ∈ Fin)
570463ad2ant1 1114 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑌 ⊆ 𝑋)
571473ad2ant1 1114 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑍 ∈ (𝑋 ∖ 𝑌))
5722003ad2ant1 1114 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐴:𝑊⟶ℝ)
5732033ad2ant1 1114 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐵:𝑊⟶ℝ)
574623ad2ant1 1114 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
575 eqid 2773 . . . . 5 (𝑦 ∈ 𝑌 ↦ 0) = (𝑦 ∈ 𝑌 ↦ 0)
576 eqid 2773 . . . . 5 (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
577953ad2ant1 1114 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
578 eqid 2773 . . . . 5 (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
579 fveq2 6497 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
580 fveq2 6497 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
581579, 580oveq12d 6993 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)) = ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
582581cbvmptv 5025 . . . . . . . 8 (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
583582fveq2i 6500 . . . . . . 7 (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))
584 hoidmvlelem3.r . . . . . . 7 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
585583, 584syl5eqel 2865 . . . . . 6 (𝜑 → (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ)
5865853ad2ant1 1114 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ)
587 hoidmvlelem3.h . . . . . 6 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
588 eleq1w 2843 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝑗 ∈ 𝑌 ↔ 𝑖 ∈ 𝑌))
589 fveq2 6497 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝑐‘𝑗) = (𝑐‘𝑖))
590589breq1d 4936 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘𝑖) ≤ 𝑥))
591590, 589ifbieq1d 4368 . . . . . . . . . 10 (𝑗 = 𝑖 → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))
592588, 589, 591ifbieq12d 4372 . . . . . . . . 9 (𝑗 = 𝑖 → if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)) = if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))
593592cbvmptv 5025 . . . . . . . 8 (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))) = (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))
594593mpteq2i 5016 . . . . . . 7 (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))) = (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))))
595594mpteq2i 5016 . . . . . 6 (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))))) = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))))
596587, 595eqtri 2797 . . . . 5 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))))
5971723ad2ant1 1114 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐸 ∈ ℝ+)
598 fveq2 6497 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝐶‘𝑗) = (𝐶‘𝑖))
599 fveq2 6497 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → (𝐷‘𝑗) = (𝐷‘𝑖))
600599fveq2d 6501 . . . . . . . . . . . 12 (𝑗 = 𝑖 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑧)‘(𝐷‘𝑖)))
601598, 600oveq12d 6993 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))
602601cbvmptv 5025 . . . . . . . . . 10 (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))
603602fveq2i 6500 . . . . . . . . 9 (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))) = (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖)))))
604603oveq2i 6986 . . . . . . . 8 ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) · (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))))
605604breq2i 4934 . . . . . . 7 ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖)))))))
606605rabbii 3394 . . . . . 6 {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))))}
607337, 606eqtri 2797 . . . . 5 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))))}
6082853ad2ant1 1114 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑆 ∈ 𝑈)
6093443ad2ant1 1114 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑆 < (𝐵‘𝑍))
610 eqid 2773 . . . . 5 (𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖))) = (𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))
611 simp2 1118 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑚 ∈ ℕ)
612 id 22 . . . . . . . 8 (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
613 fveq2 6497 . . . . . . . . . . 11 (𝑗 = 𝑖 → (𝑃‘𝑗) = (𝑃‘𝑖))
614613cbvsumv 14912 . . . . . . . . . 10 Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) = Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)
615614oveq2i 6986 . . . . . . . . 9 ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖))
616615a1i 11 . . . . . . . 8 (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
617612, 616breqtrd 4952 . . . . . . 7 (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
6186173ad2ant3 1116 . . . . . 6 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
619 simpl 475 . . . . . . . . . 10 ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → 𝜑)
620 elfznn 12751 . . . . . . . . . . 11 (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℕ)
621620adantl 474 . . . . . . . . . 10 ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℕ)
622 eleq1w 2843 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → (𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ))
623 fveq2 6497 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → (𝐽‘𝑗) = (𝐽‘𝑖))
624 fveq2 6497 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → (𝐾‘𝑗) = (𝐾‘𝑖))
625623, 624oveq12d 6993 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑖 → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
626613, 625eqeq12d 2788 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → ((𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ↔ (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖))))
627622, 626imbi12d 337 . . . . . . . . . . . . . 14 (𝑗 = 𝑖 → ((𝑗 ∈ ℕ → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))) ↔ (𝑖 ∈ ℕ → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))))
628627, 41chvarv 2328 . . . . . . . . . . . . 13 (𝑖 ∈ ℕ → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
629628adantl 474 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
630622anbi2d 620 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 𝑖 ∈ ℕ)))
631598fveq1d 6499 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → ((𝐶‘𝑗)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
632599fveq1d 6499 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
633631, 632oveq12d 6993 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
634633eleq2d 2846 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
635598reseq1d 5692 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → ((𝐶‘𝑗) ↾ 𝑌) = ((𝐶‘𝑖) ↾ 𝑌))
636634, 635ifbieq1d 4368 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑖 → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
637623, 636eqeq12d 2788 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → ((𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ↔ (𝐽‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)))
638630, 637imbi12d 337 . . . . . . . . . . . . . 14 (𝑗 = 𝑖 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹)) ↔ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐽‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))))
639638, 149chvarv 2328 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐽‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
640599reseq1d 5692 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → ((𝐷‘𝑗) ↾ 𝑌) = ((𝐷‘𝑖) ↾ 𝑌))
641634, 640ifbieq1d 4368 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑖 → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))
642624, 641eqeq12d 2788 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → ((𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ↔ (𝐾‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)))
643630, 642imbi12d 337 . . . . . . . . . . . . . 14 (𝑗 = 𝑖 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹)) ↔ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐾‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))))
644643, 440chvarv 2328 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐾‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))
645639, 644oveq12d 6993 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)) = (if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)(𝐿‘𝑌)if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)))
646629, 645eqtrd 2809 . . . . . . . . . . 11 ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑃‘𝑖) = (if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)(𝐿‘𝑌)if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)))
647 simpr 477 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
648 ovexd 7009 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ ℕ) → (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)) ∈ V)
649610fvmpt2 6604 . . . . . . . . . . . . 13 ((𝑖 ∈ ℕ ∧ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)) ∈ V) → ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖) = (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))
650647, 648, 649syl2anc 576 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖) = (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))
651 fvex 6510 . . . . . . . . . . . . . . . . . . 19 (𝐶‘𝑖) ∈ V
652651resex 5742 . . . . . . . . . . . . . . . . . 18 ((𝐶‘𝑖) ↾ 𝑌) ∈ V
653652a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐶‘𝑖) ↾ 𝑌) ∈ V)
65480, 143syl5eqelr 2866 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑦 ∈ 𝑌 ↦ 0) ∈ V)
655653, 654ifcld 4390 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
656655adantr 473 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
657576fvmpt2 6604 . . . . . . . . . . . . . . 15 ((𝑖 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
658647, 656, 657syl2anc 576 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
65980eqcomi 2782 . . . . . . . . . . . . . . . 16 (𝑦 ∈ 𝑌 ↦ 0) = 𝐹
660 ifeq2 4350 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ 𝑌 ↦ 0) = 𝐹 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
661659, 660ax-mp 5 . . . . . . . . . . . . . . 15 if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)
662661a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
663658, 662eqtrd 2809 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
664 fvex 6510 . . . . . . . . . . . . . . . . . . 19 (𝐷‘𝑖) ∈ V
665664resex 5742 . . . . . . . . . . . . . . . . . 18 ((𝐷‘𝑖) ↾ 𝑌) ∈ V
666665a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐷‘𝑖) ↾ 𝑌) ∈ V)
667666, 654ifcld 4390 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
668667adantr 473 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
669578fvmpt2 6604 . . . . . . . . . . . . . . 15 ((𝑖 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
670647, 668, 669syl2anc 576 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
671 biid 253 . . . . . . . . . . . . . . . 16 (𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
672671, 659ifbieq2i 4369 . . . . . . . . . . . . . . 15 if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)
673672a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))
674670, 673eqtrd 2809 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))
675663, 674oveq12d 6993 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑖 ∈ ℕ) → (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)) = (if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)(𝐿‘𝑌)if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)))
676650, 675eqtrd 2809 . . . . . . . . . . 11 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖) = (if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)(𝐿‘𝑌)if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)))
677646, 676eqtr4d 2812 . . . . . . . . . 10 ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑃‘𝑖) = ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖))
678619, 621, 677syl2anc 576 . . . . . . . . 9 ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → (𝑃‘𝑖) = ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖))
6796783ad2antl1 1166 . . . . . . . 8 (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) ∧ 𝑖 ∈ (1...𝑚)) → (𝑃‘𝑖) = ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖))
680679sumeq2dv 14919 . . . . . . 7 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖) = Σ𝑖 ∈ (1...𝑚)((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖))
681680oveq2d 6991 . . . . . 6 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)) = ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖)))
682618, 681breqtrd 4952 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖)))
683 fveq2 6497 . . . . . . . 8 (𝑗 = ℎ → (𝐷‘𝑗) = (𝐷‘ℎ))
684683fveq1d 6499 . . . . . . 7 (𝑗 = ℎ → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘ℎ)‘𝑍))
685684cbvmptv 5025 . . . . . 6 (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) = (ℎ ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘ℎ)‘𝑍))
686685rneqi 5648 . . . . 5 ran (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) = ran (ℎ ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘ℎ)‘𝑍))
687 fveq2 6497 . . . . . . . . . . . 12 (ℎ = 𝑖 → (𝐶‘ℎ) = (𝐶‘𝑖))
688687fveq1d 6499 . . . . . . . . . . 11 (ℎ = 𝑖 → ((𝐶‘ℎ)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
689 fveq2 6497 . . . . . . . . . . . 12 (ℎ = 𝑖 → (𝐷‘ℎ) = (𝐷‘𝑖))
690689fveq1d 6499 . . . . . . . . . . 11 (ℎ = 𝑖 → ((𝐷‘ℎ)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
691688, 690oveq12d 6993 . . . . . . . . . 10 (ℎ = 𝑖 → (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
692691eleq2d 2846 . . . . . . . . 9 (ℎ = 𝑖 → (𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
693692cbvrabv 3407 . . . . . . . 8 {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} = {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))}
694693mpteq1i 5014 . . . . . . 7 (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) = (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))
695694rneqi 5648 . . . . . 6 ran (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) = ran (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))
696695uneq2i 4020 . . . . 5 ({(𝐵‘𝑍)} ∪ ran (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))) = ({(𝐵‘𝑍)} ∪ ran (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)))
697 eqid 2773 . . . . 5 inf(({(𝐵‘𝑍)} ∪ ran (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))), ℝ, < ) = inf(({(𝐵‘𝑍)} ∪ ran (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))), ℝ, < )
69818, 569, 570, 571, 44, 572, 573, 574, 575, 576, 577, 578, 586, 596, 5, 597, 607, 608, 609, 610, 611, 682, 686, 696, 697hoidmvlelem2 42339 . . . 4 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
6996983exp 1100 . . 3 (𝜑 → (𝑚 ∈ ℕ → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)))
700699rexlimdv 3223 . 2 (𝜑 → (∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢))
701568, 700mpd 15 1 (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051   ≠ wne 2962  ∀wral 3083  ∃wrex 3084  {crab 3087  Vcvv 3410   ∖ cdif 3821   ∪ cun 3822   ⊆ wss 3824  ∅c0 4173  ifcif 4345  {csn 4436  ∪ ciun 4789   class class class wbr 4926   ↦ cmpt 5005  ran crn 5405   ↾ cres 5406   Fn wfn 6181  ⟶wf 6182  ‘cfv 6186  (class class class)co 6975   ∈ cmpo 6977   ↑𝑚 cmap 8205  Xcixp 8258  Fincfn 8305  infcinf 8699  ℝcr 10333  0cc0 10334  1c1 10335   + caddc 10337   · cmul 10339  +∞cpnf 10470  ℝ*cxr 10472   < clt 10473   ≤ cle 10474   − cmin 10669   / cdiv 11097  ℕcn 11438  ℤcz 11792  ℝ+crp 12203  [,)cico 12555  [,]cicc 12556  ...cfz 12707  Σcsu 14902  ∏cprod 15118  volcvol 23783  Σ^csumge0 42105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-inf2 8897  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411  ax-pre-sup 10412
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-int 4747  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-se 5364  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-isom 6195  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-of 7226  df-om 7396  df-1st 7500  df-2nd 7501  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-1o 7904  df-2o 7905  df-oadd 7908  df-er 8088  df-map 8207  df-pm 8208  df-ixp 8259  df-en 8306  df-dom 8307  df-sdom 8308  df-fin 8309  df-fi 8669  df-sup 8700  df-inf 8701  df-oi 8768  df-dju 9123  df-card 9161  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-div 11098  df-nn 11439  df-2 11502  df-3 11503  df-n0 11707  df-z 11793  df-uz 12058  df-q 12162  df-rp 12204  df-xneg 12323  df-xadd 12324  df-xmul 12325  df-ioo 12557  df-ico 12559  df-icc 12560  df-fz 12708  df-fzo 12849  df-fl 12976  df-seq 13184  df-exp 13244  df-hash 13505  df-cj 14318  df-re 14319  df-im 14320  df-sqrt 14454  df-abs 14455  df-clim 14705  df-rlim 14706  df-sum 14903  df-prod 15119  df-rest 16551  df-topgen 16572  df-psmet 20255  df-xmet 20256  df-met 20257  df-bl 20258  df-mopn 20259  df-top 21222  df-topon 21239  df-bases 21274  df-cmp 21715  df-ovol 23784  df-vol 23785  df-sumge0 42106
This theorem is referenced by:  hoidmvlelem4  42341
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