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Theorem hoidmvlelem2 40147
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
hoidmvlelem2.l 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
hoidmvlelem2.x (𝜑𝑋 ∈ Fin)
hoidmvlelem2.y (𝜑𝑌𝑋)
hoidmvlelem2.z (𝜑𝑍 ∈ (𝑋𝑌))
hoidmvlelem2.w 𝑊 = (𝑌 ∪ {𝑍})
hoidmvlelem2.a (𝜑𝐴:𝑊⟶ℝ)
hoidmvlelem2.b (𝜑𝐵:𝑊⟶ℝ)
hoidmvlelem2.c (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
hoidmvlelem2.f 𝐹 = (𝑦𝑌 ↦ 0)
hoidmvlelem2.j 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))
hoidmvlelem2.d (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
hoidmvlelem2.k 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))
hoidmvlelem2.r (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
hoidmvlelem2.h 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))
hoidmvlelem2.g 𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))
hoidmvlelem2.e (𝜑𝐸 ∈ ℝ+)
hoidmvlelem2.u 𝑈 = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))}
hoidmvlelem2.su (𝜑𝑆𝑈)
hoidmvlelem2.sb (𝜑𝑆 < (𝐵𝑍))
hoidmvlelem2.p 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
hoidmvlelem2.m (𝜑𝑀 ∈ ℕ)
hoidmvlelem2.le (𝜑𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)))
hoidmvlelem2.O 𝑂 = ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍))
hoidmvlelem2.v 𝑉 = ({(𝐵𝑍)} ∪ 𝑂)
hoidmvlelem2.q 𝑄 = inf(𝑉, ℝ, < )
Assertion
Ref Expression
hoidmvlelem2 (𝜑 → ∃𝑢𝑈 𝑆 < 𝑢)
Distinct variable groups:   𝐴,𝑎,𝑏,𝑘   𝑧,𝐴   𝐵,𝑎,𝑏,𝑘   𝑦,𝐵   𝑧,𝐵   𝐶,𝑎,𝑏,𝑗,𝑘   𝐶,𝑖,𝑗   𝑧,𝐶,𝑗   𝐷,𝑎,𝑏,𝑗,𝑘   𝐷,𝑐,𝑗,𝑘   𝐷,𝑖   𝑦,𝐷,𝑗   𝑧,𝐷   𝑧,𝐸   𝐹,𝑎,𝑏,𝑘   𝑧,𝐺   𝐻,𝑎,𝑏,𝑘   𝑧,𝐻   𝐽,𝑎,𝑏,𝑘   𝐾,𝑎,𝑏,𝑘   𝑧,𝐿   𝑖,𝑀,𝑗   𝑖,𝑂   𝑃,𝑎,𝑏,𝑘,𝑥   𝑄,𝑎,𝑏,𝑗,𝑘,𝑥   𝑄,𝑐,𝑥   𝑢,𝑄   𝑧,𝑄   𝑆,𝑎,𝑏,𝑗,𝑘,𝑥   𝑆,𝑐   𝑆,𝑖,𝑥   𝑢,𝑆   𝑧,𝑆   𝑢,𝑈   𝑥,𝑉,𝑦   𝑊,𝑎,𝑏,𝑗,𝑘,𝑥   𝑊,𝑐   𝑧,𝑊   𝑌,𝑎,𝑏,𝑗,𝑘,𝑥   𝑌,𝑐   𝑦,𝑌   𝑍,𝑐,𝑗,𝑘,𝑥   𝑖,𝑍   𝑦,𝑍   𝑧,𝑍   𝜑,𝑎,𝑏,𝑗,𝑘,𝑥   𝜑,𝑐   𝜑,𝑖   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑧,𝑢)   𝐴(𝑥,𝑦,𝑢,𝑖,𝑗,𝑐)   𝐵(𝑥,𝑢,𝑖,𝑗,𝑐)   𝐶(𝑥,𝑦,𝑢,𝑐)   𝐷(𝑥,𝑢)   𝑃(𝑦,𝑧,𝑢,𝑖,𝑗,𝑐)   𝑄(𝑦,𝑖)   𝑆(𝑦)   𝑈(𝑥,𝑦,𝑧,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝐸(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝐹(𝑥,𝑦,𝑧,𝑢,𝑖,𝑗,𝑐)   𝐺(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝐻(𝑥,𝑦,𝑢,𝑖,𝑗,𝑐)   𝐽(𝑥,𝑦,𝑧,𝑢,𝑖,𝑗,𝑐)   𝐾(𝑥,𝑦,𝑧,𝑢,𝑖,𝑗,𝑐)   𝐿(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑀(𝑥,𝑦,𝑧,𝑢,𝑘,𝑎,𝑏,𝑐)   𝑂(𝑥,𝑦,𝑧,𝑢,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑉(𝑧,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑊(𝑦,𝑢,𝑖)   𝑋(𝑥,𝑦,𝑧,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑌(𝑧,𝑢,𝑖)   𝑍(𝑢,𝑎,𝑏)

Proof of Theorem hoidmvlelem2
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 hoidmvlelem2.a . . . . . . 7 (𝜑𝐴:𝑊⟶ℝ)
2 hoidmvlelem2.z . . . . . . . . . 10 (𝜑𝑍 ∈ (𝑋𝑌))
3 snidg 4184 . . . . . . . . . 10 (𝑍 ∈ (𝑋𝑌) → 𝑍 ∈ {𝑍})
42, 3syl 17 . . . . . . . . 9 (𝜑𝑍 ∈ {𝑍})
5 elun2 3765 . . . . . . . . 9 (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
64, 5syl 17 . . . . . . . 8 (𝜑𝑍 ∈ (𝑌 ∪ {𝑍}))
7 hoidmvlelem2.w . . . . . . . 8 𝑊 = (𝑌 ∪ {𝑍})
86, 7syl6eleqr 2709 . . . . . . 7 (𝜑𝑍𝑊)
91, 8ffvelrnd 6326 . . . . . 6 (𝜑 → (𝐴𝑍) ∈ ℝ)
10 hoidmvlelem2.b . . . . . . 7 (𝜑𝐵:𝑊⟶ℝ)
1110, 8ffvelrnd 6326 . . . . . 6 (𝜑 → (𝐵𝑍) ∈ ℝ)
12 hoidmvlelem2.v . . . . . . . 8 𝑉 = ({(𝐵𝑍)} ∪ 𝑂)
1311snssd 4316 . . . . . . . . 9 (𝜑 → {(𝐵𝑍)} ⊆ ℝ)
14 hoidmvlelem2.O . . . . . . . . . 10 𝑂 = ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍))
15 nfv 1840 . . . . . . . . . . 11 𝑖𝜑
16 eqid 2621 . . . . . . . . . . 11 (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍)) = (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍))
17 simpl 473 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → 𝜑)
18 fz1ssnn 12330 . . . . . . . . . . . . . 14 (1...𝑀) ⊆ ℕ
19 elrabi 3347 . . . . . . . . . . . . . 14 (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} → 𝑖 ∈ (1...𝑀))
2018, 19sseldi 3586 . . . . . . . . . . . . 13 (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} → 𝑖 ∈ ℕ)
2120adantl 482 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → 𝑖 ∈ ℕ)
22 eleq1 2686 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → (𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ))
2322anbi2d 739 . . . . . . . . . . . . . 14 (𝑗 = 𝑖 → ((𝜑𝑗 ∈ ℕ) ↔ (𝜑𝑖 ∈ ℕ)))
24 fveq2 6158 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑖 → (𝐷𝑗) = (𝐷𝑖))
2524fveq1d 6160 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → ((𝐷𝑗)‘𝑍) = ((𝐷𝑖)‘𝑍))
2625eleq1d 2683 . . . . . . . . . . . . . 14 (𝑗 = 𝑖 → (((𝐷𝑗)‘𝑍) ∈ ℝ ↔ ((𝐷𝑖)‘𝑍) ∈ ℝ))
2723, 26imbi12d 334 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → (((𝜑𝑗 ∈ ℕ) → ((𝐷𝑗)‘𝑍) ∈ ℝ) ↔ ((𝜑𝑖 ∈ ℕ) → ((𝐷𝑖)‘𝑍) ∈ ℝ)))
28 hoidmvlelem2.d . . . . . . . . . . . . . . . 16 (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
2928ffvelrnda 6325 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ (ℝ ↑𝑚 𝑊))
30 elmapi 7839 . . . . . . . . . . . . . . 15 ((𝐷𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐷𝑗):𝑊⟶ℝ)
3129, 30syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗):𝑊⟶ℝ)
328adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → 𝑍𝑊)
3331, 32ffvelrnd 6326 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝐷𝑗)‘𝑍) ∈ ℝ)
3427, 33chvarv 2262 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ℕ) → ((𝐷𝑖)‘𝑍) ∈ ℝ)
3517, 21, 34syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → ((𝐷𝑖)‘𝑍) ∈ ℝ)
3615, 16, 35rnmptssd 38894 . . . . . . . . . 10 (𝜑 → ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍)) ⊆ ℝ)
3714, 36syl5eqss 3634 . . . . . . . . 9 (𝜑𝑂 ⊆ ℝ)
3813, 37unssd 3773 . . . . . . . 8 (𝜑 → ({(𝐵𝑍)} ∪ 𝑂) ⊆ ℝ)
3912, 38syl5eqss 3634 . . . . . . 7 (𝜑𝑉 ⊆ ℝ)
40 hoidmvlelem2.q . . . . . . . 8 𝑄 = inf(𝑉, ℝ, < )
41 ltso 10078 . . . . . . . . . 10 < Or ℝ
4241a1i 11 . . . . . . . . 9 (𝜑 → < Or ℝ)
43 snfi 7998 . . . . . . . . . . . 12 {(𝐵𝑍)} ∈ Fin
4443a1i 11 . . . . . . . . . . 11 (𝜑 → {(𝐵𝑍)} ∈ Fin)
45 fzfi 12727 . . . . . . . . . . . . . . 15 (1...𝑀) ∈ Fin
46 rabfi 8145 . . . . . . . . . . . . . . 15 ((1...𝑀) ∈ Fin → {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ∈ Fin)
4745, 46ax-mp 5 . . . . . . . . . . . . . 14 {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ∈ Fin
4847a1i 11 . . . . . . . . . . . . 13 (𝜑 → {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ∈ Fin)
4916rnmptfi 38860 . . . . . . . . . . . . 13 ({𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ∈ Fin → ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍)) ∈ Fin)
5048, 49syl 17 . . . . . . . . . . . 12 (𝜑 → ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍)) ∈ Fin)
5114, 50syl5eqel 2702 . . . . . . . . . . 11 (𝜑𝑂 ∈ Fin)
52 unfi 8187 . . . . . . . . . . 11 (({(𝐵𝑍)} ∈ Fin ∧ 𝑂 ∈ Fin) → ({(𝐵𝑍)} ∪ 𝑂) ∈ Fin)
5344, 51, 52syl2anc 692 . . . . . . . . . 10 (𝜑 → ({(𝐵𝑍)} ∪ 𝑂) ∈ Fin)
5412, 53syl5eqel 2702 . . . . . . . . 9 (𝜑𝑉 ∈ Fin)
55 fvex 6168 . . . . . . . . . . . . . 14 (𝐵𝑍) ∈ V
5655snid 4186 . . . . . . . . . . . . 13 (𝐵𝑍) ∈ {(𝐵𝑍)}
57 elun1 3764 . . . . . . . . . . . . 13 ((𝐵𝑍) ∈ {(𝐵𝑍)} → (𝐵𝑍) ∈ ({(𝐵𝑍)} ∪ 𝑂))
5856, 57ax-mp 5 . . . . . . . . . . . 12 (𝐵𝑍) ∈ ({(𝐵𝑍)} ∪ 𝑂)
5912eqcomi 2630 . . . . . . . . . . . 12 ({(𝐵𝑍)} ∪ 𝑂) = 𝑉
6058, 59eleqtri 2696 . . . . . . . . . . 11 (𝐵𝑍) ∈ 𝑉
6160a1i 11 . . . . . . . . . 10 (𝜑 → (𝐵𝑍) ∈ 𝑉)
62 ne0i 3903 . . . . . . . . . 10 ((𝐵𝑍) ∈ 𝑉𝑉 ≠ ∅)
6361, 62syl 17 . . . . . . . . 9 (𝜑𝑉 ≠ ∅)
64 fiinfcl 8367 . . . . . . . . 9 (( < Or ℝ ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅ ∧ 𝑉 ⊆ ℝ)) → inf(𝑉, ℝ, < ) ∈ 𝑉)
6542, 54, 63, 39, 64syl13anc 1325 . . . . . . . 8 (𝜑 → inf(𝑉, ℝ, < ) ∈ 𝑉)
6640, 65syl5eqel 2702 . . . . . . 7 (𝜑𝑄𝑉)
6739, 66sseldd 3589 . . . . . 6 (𝜑𝑄 ∈ ℝ)
68 hoidmvlelem2.u . . . . . . . . . . . 12 𝑈 = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))}
69 ssrab2 3672 . . . . . . . . . . . 12 {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} ⊆ ((𝐴𝑍)[,](𝐵𝑍))
7068, 69eqsstri 3620 . . . . . . . . . . 11 𝑈 ⊆ ((𝐴𝑍)[,](𝐵𝑍))
7170a1i 11 . . . . . . . . . 10 (𝜑𝑈 ⊆ ((𝐴𝑍)[,](𝐵𝑍)))
729, 11iccssred 39173 . . . . . . . . . 10 (𝜑 → ((𝐴𝑍)[,](𝐵𝑍)) ⊆ ℝ)
7371, 72sstrd 3598 . . . . . . . . 9 (𝜑𝑈 ⊆ ℝ)
74 hoidmvlelem2.su . . . . . . . . 9 (𝜑𝑆𝑈)
7573, 74sseldd 3589 . . . . . . . 8 (𝜑𝑆 ∈ ℝ)
769rexrd 10049 . . . . . . . . 9 (𝜑 → (𝐴𝑍) ∈ ℝ*)
7711rexrd 10049 . . . . . . . . 9 (𝜑 → (𝐵𝑍) ∈ ℝ*)
7870, 74sseldi 3586 . . . . . . . . 9 (𝜑𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍)))
79 iccgelb 12188 . . . . . . . . 9 (((𝐴𝑍) ∈ ℝ* ∧ (𝐵𝑍) ∈ ℝ*𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍))) → (𝐴𝑍) ≤ 𝑆)
8076, 77, 78, 79syl3anc 1323 . . . . . . . 8 (𝜑 → (𝐴𝑍) ≤ 𝑆)
81 hoidmvlelem2.sb . . . . . . . . . . . . . . 15 (𝜑𝑆 < (𝐵𝑍))
8281adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑥 = (𝐵𝑍)) → 𝑆 < (𝐵𝑍))
83 id 22 . . . . . . . . . . . . . . . 16 (𝑥 = (𝐵𝑍) → 𝑥 = (𝐵𝑍))
8483eqcomd 2627 . . . . . . . . . . . . . . 15 (𝑥 = (𝐵𝑍) → (𝐵𝑍) = 𝑥)
8584adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑥 = (𝐵𝑍)) → (𝐵𝑍) = 𝑥)
8682, 85breqtrd 4649 . . . . . . . . . . . . 13 ((𝜑𝑥 = (𝐵𝑍)) → 𝑆 < 𝑥)
8786adantlr 750 . . . . . . . . . . . 12 (((𝜑𝑥𝑉) ∧ 𝑥 = (𝐵𝑍)) → 𝑆 < 𝑥)
88 simpll 789 . . . . . . . . . . . . 13 (((𝜑𝑥𝑉) ∧ ¬ 𝑥 = (𝐵𝑍)) → 𝜑)
89 id 22 . . . . . . . . . . . . . . . . 17 (𝑥𝑉𝑥𝑉)
9089, 12syl6eleq 2708 . . . . . . . . . . . . . . . 16 (𝑥𝑉𝑥 ∈ ({(𝐵𝑍)} ∪ 𝑂))
9190adantr 481 . . . . . . . . . . . . . . 15 ((𝑥𝑉 ∧ ¬ 𝑥 = (𝐵𝑍)) → 𝑥 ∈ ({(𝐵𝑍)} ∪ 𝑂))
92 elsni 4172 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {(𝐵𝑍)} → 𝑥 = (𝐵𝑍))
9392con3i 150 . . . . . . . . . . . . . . . 16 𝑥 = (𝐵𝑍) → ¬ 𝑥 ∈ {(𝐵𝑍)})
9493adantl 482 . . . . . . . . . . . . . . 15 ((𝑥𝑉 ∧ ¬ 𝑥 = (𝐵𝑍)) → ¬ 𝑥 ∈ {(𝐵𝑍)})
95 elunnel1 3738 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ({(𝐵𝑍)} ∪ 𝑂) ∧ ¬ 𝑥 ∈ {(𝐵𝑍)}) → 𝑥𝑂)
9691, 94, 95syl2anc 692 . . . . . . . . . . . . . 14 ((𝑥𝑉 ∧ ¬ 𝑥 = (𝐵𝑍)) → 𝑥𝑂)
9796adantll 749 . . . . . . . . . . . . 13 (((𝜑𝑥𝑉) ∧ ¬ 𝑥 = (𝐵𝑍)) → 𝑥𝑂)
98 id 22 . . . . . . . . . . . . . . . . 17 (𝑥𝑂𝑥𝑂)
9998, 14syl6eleq 2708 . . . . . . . . . . . . . . . 16 (𝑥𝑂𝑥 ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍)))
100 vex 3193 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
10116elrnmpt 5342 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ V → (𝑥 ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍)) ↔ ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}𝑥 = ((𝐷𝑖)‘𝑍)))
102100, 101ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍)) ↔ ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}𝑥 = ((𝐷𝑖)‘𝑍))
10399, 102sylib 208 . . . . . . . . . . . . . . 15 (𝑥𝑂 → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}𝑥 = ((𝐷𝑖)‘𝑍))
104103adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑂) → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}𝑥 = ((𝐷𝑖)‘𝑍))
105 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = 𝑖 → (𝐶𝑗) = (𝐶𝑖))
106105fveq1d 6160 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑖 → ((𝐶𝑗)‘𝑍) = ((𝐶𝑖)‘𝑍))
107106eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 𝑖 → (((𝐶𝑗)‘𝑍) ∈ ℝ ↔ ((𝐶𝑖)‘𝑍) ∈ ℝ))
10823, 107imbi12d 334 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑖 → (((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)‘𝑍) ∈ ℝ) ↔ ((𝜑𝑖 ∈ ℕ) → ((𝐶𝑖)‘𝑍) ∈ ℝ)))
109 hoidmvlelem2.c . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
110109ffvelrnda 6325 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ (ℝ ↑𝑚 𝑊))
111 elmapi 7839 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐶𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐶𝑗):𝑊⟶ℝ)
112110, 111syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗):𝑊⟶ℝ)
113112, 32ffvelrnd 6326 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)‘𝑍) ∈ ℝ)
114108, 113chvarv 2262 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ ℕ) → ((𝐶𝑖)‘𝑍) ∈ ℝ)
115114rexrd 10049 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ ℕ) → ((𝐶𝑖)‘𝑍) ∈ ℝ*)
11617, 21, 115syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → ((𝐶𝑖)‘𝑍) ∈ ℝ*)
11734rexrd 10049 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ ℕ) → ((𝐷𝑖)‘𝑍) ∈ ℝ*)
11817, 21, 117syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → ((𝐷𝑖)‘𝑍) ∈ ℝ*)
119106, 25oveq12d 6633 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑖 → (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)) = (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)))
120119eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 𝑖 → (𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)) ↔ 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍))))
121120elrab 3351 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↔ (𝑖 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍))))
122121biimpi 206 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} → (𝑖 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍))))
123122simprd 479 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} → 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)))
124123adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)))
125 icoltub 39178 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶𝑖)‘𝑍) ∈ ℝ* ∧ ((𝐷𝑖)‘𝑍) ∈ ℝ*𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍))) → 𝑆 < ((𝐷𝑖)‘𝑍))
126116, 118, 124, 125syl3anc 1323 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → 𝑆 < ((𝐷𝑖)‘𝑍))
1271263adant3 1079 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ∧ 𝑥 = ((𝐷𝑖)‘𝑍)) → 𝑆 < ((𝐷𝑖)‘𝑍))
128 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = ((𝐷𝑖)‘𝑍) → 𝑥 = ((𝐷𝑖)‘𝑍))
129128eqcomd 2627 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ((𝐷𝑖)‘𝑍) → ((𝐷𝑖)‘𝑍) = 𝑥)
1301293ad2ant3 1082 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ∧ 𝑥 = ((𝐷𝑖)‘𝑍)) → ((𝐷𝑖)‘𝑍) = 𝑥)
131127, 130breqtrd 4649 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ∧ 𝑥 = ((𝐷𝑖)‘𝑍)) → 𝑆 < 𝑥)
1321313exp 1261 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} → (𝑥 = ((𝐷𝑖)‘𝑍) → 𝑆 < 𝑥)))
133132adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑂) → (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} → (𝑥 = ((𝐷𝑖)‘𝑍) → 𝑆 < 𝑥)))
134133rexlimdv 3025 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑂) → (∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}𝑥 = ((𝐷𝑖)‘𝑍) → 𝑆 < 𝑥))
135104, 134mpd 15 . . . . . . . . . . . . 13 ((𝜑𝑥𝑂) → 𝑆 < 𝑥)
13688, 97, 135syl2anc 692 . . . . . . . . . . . 12 (((𝜑𝑥𝑉) ∧ ¬ 𝑥 = (𝐵𝑍)) → 𝑆 < 𝑥)
13787, 136pm2.61dan 831 . . . . . . . . . . 11 ((𝜑𝑥𝑉) → 𝑆 < 𝑥)
138137ralrimiva 2962 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑉 𝑆 < 𝑥)
139 breq2 4627 . . . . . . . . . . 11 (𝑥 = inf(𝑉, ℝ, < ) → (𝑆 < 𝑥𝑆 < inf(𝑉, ℝ, < )))
140139rspcva 3297 . . . . . . . . . 10 ((inf(𝑉, ℝ, < ) ∈ 𝑉 ∧ ∀𝑥𝑉 𝑆 < 𝑥) → 𝑆 < inf(𝑉, ℝ, < ))
14165, 138, 140syl2anc 692 . . . . . . . . 9 (𝜑𝑆 < inf(𝑉, ℝ, < ))
14240eqcomi 2630 . . . . . . . . . 10 inf(𝑉, ℝ, < ) = 𝑄
143142a1i 11 . . . . . . . . 9 (𝜑 → inf(𝑉, ℝ, < ) = 𝑄)
144141, 143breqtrd 4649 . . . . . . . 8 (𝜑𝑆 < 𝑄)
1459, 75, 67, 80, 144lelttrd 10155 . . . . . . 7 (𝜑 → (𝐴𝑍) < 𝑄)
1469, 67, 145ltled 10145 . . . . . 6 (𝜑 → (𝐴𝑍) ≤ 𝑄)
147 fiminre 10932 . . . . . . . . 9 ((𝑉 ⊆ ℝ ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ∃𝑥𝑉𝑦𝑉 𝑥𝑦)
14839, 54, 63, 147syl3anc 1323 . . . . . . . 8 (𝜑 → ∃𝑥𝑉𝑦𝑉 𝑥𝑦)
149 lbinfle 10938 . . . . . . . 8 ((𝑉 ⊆ ℝ ∧ ∃𝑥𝑉𝑦𝑉 𝑥𝑦 ∧ (𝐵𝑍) ∈ 𝑉) → inf(𝑉, ℝ, < ) ≤ (𝐵𝑍))
15039, 148, 61, 149syl3anc 1323 . . . . . . 7 (𝜑 → inf(𝑉, ℝ, < ) ≤ (𝐵𝑍))
15140, 150syl5eqbr 4658 . . . . . 6 (𝜑𝑄 ≤ (𝐵𝑍))
1529, 11, 67, 146, 151eliccd 39172 . . . . 5 (𝜑𝑄 ∈ ((𝐴𝑍)[,](𝐵𝑍)))
15367recnd 10028 . . . . . . . . . 10 (𝜑𝑄 ∈ ℂ)
15475recnd 10028 . . . . . . . . . 10 (𝜑𝑆 ∈ ℂ)
1559recnd 10028 . . . . . . . . . 10 (𝜑 → (𝐴𝑍) ∈ ℂ)
156153, 154, 155npncand 10376 . . . . . . . . 9 (𝜑 → ((𝑄𝑆) + (𝑆 − (𝐴𝑍))) = (𝑄 − (𝐴𝑍)))
157156eqcomd 2627 . . . . . . . 8 (𝜑 → (𝑄 − (𝐴𝑍)) = ((𝑄𝑆) + (𝑆 − (𝐴𝑍))))
158157oveq2d 6631 . . . . . . 7 (𝜑 → (𝐺 · (𝑄 − (𝐴𝑍))) = (𝐺 · ((𝑄𝑆) + (𝑆 − (𝐴𝑍)))))
159 rge0ssre 12238 . . . . . . . . . 10 (0[,)+∞) ⊆ ℝ
160 hoidmvlelem2.g . . . . . . . . . . 11 𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))
161 hoidmvlelem2.l . . . . . . . . . . . 12 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
162 hoidmvlelem2.x . . . . . . . . . . . . 13 (𝜑𝑋 ∈ Fin)
163 hoidmvlelem2.y . . . . . . . . . . . . 13 (𝜑𝑌𝑋)
164162, 163ssfid 8143 . . . . . . . . . . . 12 (𝜑𝑌 ∈ Fin)
165 ssun1 3760 . . . . . . . . . . . . . . 15 𝑌 ⊆ (𝑌 ∪ {𝑍})
166165, 7sseqtr4i 3623 . . . . . . . . . . . . . 14 𝑌𝑊
167166a1i 11 . . . . . . . . . . . . 13 (𝜑𝑌𝑊)
1681, 167fssresd 6038 . . . . . . . . . . . 12 (𝜑 → (𝐴𝑌):𝑌⟶ℝ)
16910, 167fssresd 6038 . . . . . . . . . . . 12 (𝜑 → (𝐵𝑌):𝑌⟶ℝ)
170161, 164, 168, 169hoidmvcl 40133 . . . . . . . . . . 11 (𝜑 → ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) ∈ (0[,)+∞))
171160, 170syl5eqel 2702 . . . . . . . . . 10 (𝜑𝐺 ∈ (0[,)+∞))
172159, 171sseldi 3586 . . . . . . . . 9 (𝜑𝐺 ∈ ℝ)
173172recnd 10028 . . . . . . . 8 (𝜑𝐺 ∈ ℂ)
174153, 154subcld 10352 . . . . . . . 8 (𝜑 → (𝑄𝑆) ∈ ℂ)
175154, 155subcld 10352 . . . . . . . 8 (𝜑 → (𝑆 − (𝐴𝑍)) ∈ ℂ)
176173, 174, 175adddid 10024 . . . . . . 7 (𝜑 → (𝐺 · ((𝑄𝑆) + (𝑆 − (𝐴𝑍)))) = ((𝐺 · (𝑄𝑆)) + (𝐺 · (𝑆 − (𝐴𝑍)))))
177173, 174mulcld 10020 . . . . . . . 8 (𝜑 → (𝐺 · (𝑄𝑆)) ∈ ℂ)
178173, 175mulcld 10020 . . . . . . . 8 (𝜑 → (𝐺 · (𝑆 − (𝐴𝑍))) ∈ ℂ)
179177, 178addcomd 10198 . . . . . . 7 (𝜑 → ((𝐺 · (𝑄𝑆)) + (𝐺 · (𝑆 − (𝐴𝑍)))) = ((𝐺 · (𝑆 − (𝐴𝑍))) + (𝐺 · (𝑄𝑆))))
180158, 176, 1793eqtrd 2659 . . . . . 6 (𝜑 → (𝐺 · (𝑄 − (𝐴𝑍))) = ((𝐺 · (𝑆 − (𝐴𝑍))) + (𝐺 · (𝑄𝑆))))
18167, 75jca 554 . . . . . . . . . . . . 13 (𝜑 → (𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ))
182 resubcl 10305 . . . . . . . . . . . . 13 ((𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑄𝑆) ∈ ℝ)
183181, 182syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑄𝑆) ∈ ℝ)
184172, 183jca 554 . . . . . . . . . . 11 (𝜑 → (𝐺 ∈ ℝ ∧ (𝑄𝑆) ∈ ℝ))
185 remulcl 9981 . . . . . . . . . . 11 ((𝐺 ∈ ℝ ∧ (𝑄𝑆) ∈ ℝ) → (𝐺 · (𝑄𝑆)) ∈ ℝ)
186184, 185syl 17 . . . . . . . . . 10 (𝜑 → (𝐺 · (𝑄𝑆)) ∈ ℝ)
18775, 9jca 554 . . . . . . . . . . . . 13 (𝜑 → (𝑆 ∈ ℝ ∧ (𝐴𝑍) ∈ ℝ))
188 resubcl 10305 . . . . . . . . . . . . 13 ((𝑆 ∈ ℝ ∧ (𝐴𝑍) ∈ ℝ) → (𝑆 − (𝐴𝑍)) ∈ ℝ)
189187, 188syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑆 − (𝐴𝑍)) ∈ ℝ)
190172, 189jca 554 . . . . . . . . . . 11 (𝜑 → (𝐺 ∈ ℝ ∧ (𝑆 − (𝐴𝑍)) ∈ ℝ))
191 remulcl 9981 . . . . . . . . . . 11 ((𝐺 ∈ ℝ ∧ (𝑆 − (𝐴𝑍)) ∈ ℝ) → (𝐺 · (𝑆 − (𝐴𝑍))) ∈ ℝ)
192190, 191syl 17 . . . . . . . . . 10 (𝜑 → (𝐺 · (𝑆 − (𝐴𝑍))) ∈ ℝ)
193186, 192jca 554 . . . . . . . . 9 (𝜑 → ((𝐺 · (𝑄𝑆)) ∈ ℝ ∧ (𝐺 · (𝑆 − (𝐴𝑍))) ∈ ℝ))
194 readdcl 9979 . . . . . . . . 9 (((𝐺 · (𝑄𝑆)) ∈ ℝ ∧ (𝐺 · (𝑆 − (𝐴𝑍))) ∈ ℝ) → ((𝐺 · (𝑄𝑆)) + (𝐺 · (𝑆 − (𝐴𝑍)))) ∈ ℝ)
195193, 194syl 17 . . . . . . . 8 (𝜑 → ((𝐺 · (𝑄𝑆)) + (𝐺 · (𝑆 − (𝐴𝑍)))) ∈ ℝ)
196179, 195eqeltrrd 2699 . . . . . . 7 (𝜑 → ((𝐺 · (𝑆 − (𝐴𝑍))) + (𝐺 · (𝑄𝑆))) ∈ ℝ)
197 1red 10015 . . . . . . . . . 10 (𝜑 → 1 ∈ ℝ)
198 hoidmvlelem2.e . . . . . . . . . . 11 (𝜑𝐸 ∈ ℝ+)
199198rpred 11832 . . . . . . . . . 10 (𝜑𝐸 ∈ ℝ)
200197, 199readdcld 10029 . . . . . . . . 9 (𝜑 → (1 + 𝐸) ∈ ℝ)
2012eldifbd 3573 . . . . . . . . . . 11 (𝜑 → ¬ 𝑍𝑌)
2028, 201eldifd 3571 . . . . . . . . . 10 (𝜑𝑍 ∈ (𝑊𝑌))
203 hoidmvlelem2.r . . . . . . . . . 10 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
204 hoidmvlelem2.h . . . . . . . . . 10 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))
205161, 164, 202, 7, 109, 28, 203, 204, 75sge0hsphoire 40140 . . . . . . . . 9 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℝ)
206200, 205remulcld 10030 . . . . . . . 8 (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) ∈ ℝ)
207 fzfid 12728 . . . . . . . . . 10 (𝜑 → (1...𝑀) ∈ Fin)
208183adantr 481 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (𝑄𝑆) ∈ ℝ)
209 simpl 473 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → 𝜑)
210 elfznn 12328 . . . . . . . . . . . . 13 (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
211210adantl 482 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑗 ∈ ℕ)
212 id 22 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
213 ovexd 6645 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) ∈ V)
214 hoidmvlelem2.p . . . . . . . . . . . . . . . . 17 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
215214fvmpt2 6258 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ℕ ∧ ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) ∈ V) → (𝑃𝑗) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
216212, 213, 215syl2anc 692 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → (𝑃𝑗) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
217216adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (𝑃𝑗) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
218164adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑌 ∈ Fin)
219166a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → 𝑌𝑊)
220112, 219fssresd 6038 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗) ↾ 𝑌):𝑌⟶ℝ)
221220adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → ((𝐶𝑗) ↾ 𝑌):𝑌⟶ℝ)
222 iftrue 4070 . . . . . . . . . . . . . . . . . . . 20 (𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) = ((𝐶𝑗) ↾ 𝑌))
223222adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) = ((𝐶𝑗) ↾ 𝑌))
224223feq1d 5997 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ ((𝐶𝑗) ↾ 𝑌):𝑌⟶ℝ))
225221, 224mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
226 0red 10001 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦𝑌) → 0 ∈ ℝ)
227 hoidmvlelem2.f . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑦𝑌 ↦ 0)
228226, 227fmptd 6351 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹:𝑌⟶ℝ)
229228ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → 𝐹:𝑌⟶ℝ)
230 iffalse 4073 . . . . . . . . . . . . . . . . . . . 20 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) = 𝐹)
231230adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) = 𝐹)
232231feq1d 5997 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ 𝐹:𝑌⟶ℝ))
233229, 232mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
234225, 233pm2.61dan 831 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
235 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
236 fvex 6168 . . . . . . . . . . . . . . . . . . . . . 22 (𝐶𝑗) ∈ V
237236resex 5412 . . . . . . . . . . . . . . . . . . . . 21 ((𝐶𝑗) ↾ 𝑌) ∈ V
238237a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐶𝑗) ↾ 𝑌) ∈ V)
239162, 163ssexd 4775 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑌 ∈ V)
240 mptexg 6449 . . . . . . . . . . . . . . . . . . . . . 22 (𝑌 ∈ V → (𝑦𝑌 ↦ 0) ∈ V)
241239, 240syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑦𝑌 ↦ 0) ∈ V)
242227, 241syl5eqel 2702 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹 ∈ V)
243238, 242ifcld 4109 . . . . . . . . . . . . . . . . . . 19 (𝜑 → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) ∈ V)
244243adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) ∈ V)
245 hoidmvlelem2.j . . . . . . . . . . . . . . . . . . 19 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))
246245fvmpt2 6258 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐽𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))
247235, 244, 246syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐽𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))
248247feq1d 5997 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((𝐽𝑗):𝑌⟶ℝ ↔ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ))
249234, 248mpbird 247 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐽𝑗):𝑌⟶ℝ)
25031, 219fssresd 6038 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐷𝑗) ↾ 𝑌):𝑌⟶ℝ)
251250adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → ((𝐷𝑗) ↾ 𝑌):𝑌⟶ℝ)
252 iftrue 4070 . . . . . . . . . . . . . . . . . . . 20 (𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) = ((𝐷𝑗) ↾ 𝑌))
253252adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) = ((𝐷𝑗) ↾ 𝑌))
254253feq1d 5997 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ ((𝐷𝑗) ↾ 𝑌):𝑌⟶ℝ))
255251, 254mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
256 iffalse 4073 . . . . . . . . . . . . . . . . . . . 20 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) = 𝐹)
257256adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) = 𝐹)
258257feq1d 5997 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ 𝐹:𝑌⟶ℝ))
259229, 258mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
260255, 259pm2.61dan 831 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
261 fvex 6168 . . . . . . . . . . . . . . . . . . . . . 22 (𝐷𝑗) ∈ V
262261resex 5412 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷𝑗) ↾ 𝑌) ∈ V
263262a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐷𝑗) ↾ 𝑌) ∈ V)
264263, 242ifcld 4109 . . . . . . . . . . . . . . . . . . 19 (𝜑 → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) ∈ V)
265264adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) ∈ V)
266 hoidmvlelem2.k . . . . . . . . . . . . . . . . . . 19 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))
267266fvmpt2 6258 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐾𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))
268235, 265, 267syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐾𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))
269268feq1d 5997 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((𝐾𝑗):𝑌⟶ℝ ↔ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ))
270260, 269mpbird 247 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐾𝑗):𝑌⟶ℝ)
271161, 218, 249, 270hoidmvcl 40133 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) ∈ (0[,)+∞))
272217, 271eqeltrd 2698 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝑃𝑗) ∈ (0[,)+∞))
273159, 272sseldi 3586 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (𝑃𝑗) ∈ ℝ)
274209, 211, 273syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (𝑃𝑗) ∈ ℝ)
275208, 274remulcld 10030 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝑄𝑆) · (𝑃𝑗)) ∈ ℝ)
276207, 275fsumrecl 14414 . . . . . . . . 9 (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)) ∈ ℝ)
277200, 276remulcld 10030 . . . . . . . 8 (𝜑 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))) ∈ ℝ)
278206, 277readdcld 10029 . . . . . . 7 (𝜑 → (((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))) ∈ ℝ)
279161, 164, 202, 7, 109, 28, 203, 204, 67sge0hsphoire 40140 . . . . . . . 8 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) ∈ ℝ)
280200, 279remulcld 10030 . . . . . . 7 (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))) ∈ ℝ)
28174, 68syl6eleq 2708 . . . . . . . . . 10 (𝜑𝑆 ∈ {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))})
282 oveq1 6622 . . . . . . . . . . . . 13 (𝑧 = 𝑆 → (𝑧 − (𝐴𝑍)) = (𝑆 − (𝐴𝑍)))
283282oveq2d 6631 . . . . . . . . . . . 12 (𝑧 = 𝑆 → (𝐺 · (𝑧 − (𝐴𝑍))) = (𝐺 · (𝑆 − (𝐴𝑍))))
284 fveq2 6158 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑆 → (𝐻𝑧) = (𝐻𝑆))
285284fveq1d 6160 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑆 → ((𝐻𝑧)‘(𝐷𝑗)) = ((𝐻𝑆)‘(𝐷𝑗)))
286285oveq2d 6631 . . . . . . . . . . . . . . 15 (𝑧 = 𝑆 → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))) = ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))
287286mpteq2dv 4715 . . . . . . . . . . . . . 14 (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))
288287fveq2d 6162 . . . . . . . . . . . . 13 (𝑧 = 𝑆 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))))
289288oveq2d 6631 . . . . . . . . . . . 12 (𝑧 = 𝑆 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))))) = ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))))
290283, 289breq12d 4636 . . . . . . . . . . 11 (𝑧 = 𝑆 → ((𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))))) ↔ (𝐺 · (𝑆 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))))))
291290elrab 3351 . . . . . . . . . 10 (𝑆 ∈ {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} ↔ (𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∧ (𝐺 · (𝑆 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))))))
292281, 291sylib 208 . . . . . . . . 9 (𝜑 → (𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∧ (𝐺 · (𝑆 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))))))
293292simprd 479 . . . . . . . 8 (𝜑 → (𝐺 · (𝑆 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))))
294207, 274fsumrecl 14414 . . . . . . . . . . 11 (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝑃𝑗) ∈ ℝ)
295200, 294remulcld 10030 . . . . . . . . . 10 (𝜑 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)) ∈ ℝ)
296 0red 10001 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
29775, 67posdifd 10574 . . . . . . . . . . . 12 (𝜑 → (𝑆 < 𝑄 ↔ 0 < (𝑄𝑆)))
298144, 297mpbid 222 . . . . . . . . . . 11 (𝜑 → 0 < (𝑄𝑆))
299296, 183, 298ltled 10145 . . . . . . . . . 10 (𝜑 → 0 ≤ (𝑄𝑆))
300 hoidmvlelem2.le . . . . . . . . . 10 (𝜑𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)))
301172, 295, 183, 299, 300lemul1ad 10923 . . . . . . . . 9 (𝜑 → (𝐺 · (𝑄𝑆)) ≤ (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)) · (𝑄𝑆)))
302200recnd 10028 . . . . . . . . . . 11 (𝜑 → (1 + 𝐸) ∈ ℂ)
303294recnd 10028 . . . . . . . . . . 11 (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝑃𝑗) ∈ ℂ)
304302, 303, 174mulassd 10023 . . . . . . . . . 10 (𝜑 → (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)) · (𝑄𝑆)) = ((1 + 𝐸) · (Σ𝑗 ∈ (1...𝑀)(𝑃𝑗) · (𝑄𝑆))))
305274recnd 10028 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (𝑃𝑗) ∈ ℂ)
306207, 174, 305fsummulc1 14464 . . . . . . . . . . . 12 (𝜑 → (Σ𝑗 ∈ (1...𝑀)(𝑃𝑗) · (𝑄𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑃𝑗) · (𝑄𝑆)))
307174adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → (𝑄𝑆) ∈ ℂ)
308305, 307mulcomd 10021 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝑃𝑗) · (𝑄𝑆)) = ((𝑄𝑆) · (𝑃𝑗)))
309308sumeq2dv 14383 . . . . . . . . . . . 12 (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑃𝑗) · (𝑄𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))
310306, 309eqtrd 2655 . . . . . . . . . . 11 (𝜑 → (Σ𝑗 ∈ (1...𝑀)(𝑃𝑗) · (𝑄𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))
311310oveq2d 6631 . . . . . . . . . 10 (𝜑 → ((1 + 𝐸) · (Σ𝑗 ∈ (1...𝑀)(𝑃𝑗) · (𝑄𝑆))) = ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))))
312304, 311eqtrd 2655 . . . . . . . . 9 (𝜑 → (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)) · (𝑄𝑆)) = ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))))
313301, 312breqtrd 4649 . . . . . . . 8 (𝜑 → (𝐺 · (𝑄𝑆)) ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))))
314192, 186, 206, 277, 293, 313leadd12dd 39031 . . . . . . 7 (𝜑 → ((𝐺 · (𝑆 − (𝐴𝑍))) + (𝐺 · (𝑄𝑆))) ≤ (((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))))
315 hoidmvlelem2.m . . . . . . . . . . . . . . . . 17 (𝜑𝑀 ∈ ℕ)
316 nnsplit 39073 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℕ → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
317315, 316syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
318 uncom 3741 . . . . . . . . . . . . . . . . 17 ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))) = ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀))
319318a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))) = ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)))
320317, 319eqtr2d 2656 . . . . . . . . . . . . . . 15 (𝜑 → ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)) = ℕ)
321320eqcomd 2627 . . . . . . . . . . . . . 14 (𝜑 → ℕ = ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)))
322321mpteq1d 4708 . . . . . . . . . . . . 13 (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))) = (𝑗 ∈ ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))
323322fveq2d 6162 . . . . . . . . . . . 12 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) = (Σ^‘(𝑗 ∈ ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))))
324 nfv 1840 . . . . . . . . . . . . 13 𝑗𝜑
325 fvexd 6170 . . . . . . . . . . . . 13 (𝜑 → (ℤ‘(𝑀 + 1)) ∈ V)
326 ovexd 6645 . . . . . . . . . . . . 13 (𝜑 → (1...𝑀) ∈ V)
327 incom 3789 . . . . . . . . . . . . . . 15 ((ℤ‘(𝑀 + 1)) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ‘(𝑀 + 1)))
328 nnuzdisj 39070 . . . . . . . . . . . . . . 15 ((1...𝑀) ∩ (ℤ‘(𝑀 + 1))) = ∅
329327, 328eqtri 2643 . . . . . . . . . . . . . 14 ((ℤ‘(𝑀 + 1)) ∩ (1...𝑀)) = ∅
330329a1i 11 . . . . . . . . . . . . 13 (𝜑 → ((ℤ‘(𝑀 + 1)) ∩ (1...𝑀)) = ∅)
331 icossicc 12218 . . . . . . . . . . . . . 14 (0[,)+∞) ⊆ (0[,]+∞)
332 ssid 3609 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ (0[,)+∞)
333 simpl 473 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → 𝜑)
334315peano2nnd 10997 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 + 1) ∈ ℕ)
335 uznnssnn 11695 . . . . . . . . . . . . . . . . . . 19 ((𝑀 + 1) ∈ ℕ → (ℤ‘(𝑀 + 1)) ⊆ ℕ)
336334, 335syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℤ‘(𝑀 + 1)) ⊆ ℕ)
337336adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → (ℤ‘(𝑀 + 1)) ⊆ ℕ)
338 simpr 477 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → 𝑗 ∈ (ℤ‘(𝑀 + 1)))
339337, 338sseldd 3589 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → 𝑗 ∈ ℕ)
340 snfi 7998 . . . . . . . . . . . . . . . . . . . . 21 {𝑍} ∈ Fin
341340a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → {𝑍} ∈ Fin)
342 unfi 8187 . . . . . . . . . . . . . . . . . . . 20 ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
343164, 341, 342syl2anc 692 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
3447, 343syl5eqel 2702 . . . . . . . . . . . . . . . . . 18 (𝜑𝑊 ∈ Fin)
345344adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
346 eleq1 2686 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑙 → (𝑗𝑌𝑙𝑌))
347 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑙 → (𝑐𝑗) = (𝑐𝑙))
348347breq1d 4633 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 𝑙 → ((𝑐𝑗) ≤ 𝑥 ↔ (𝑐𝑙) ≤ 𝑥))
349348, 347ifbieq1d 4087 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑙 → if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥) = if((𝑐𝑙) ≤ 𝑥, (𝑐𝑙), 𝑥))
350346, 347, 349ifbieq12d 4091 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑙 → if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)) = if(𝑙𝑌, (𝑐𝑙), if((𝑐𝑙) ≤ 𝑥, (𝑐𝑙), 𝑥)))
351350cbvmptv 4720 . . . . . . . . . . . . . . . . . . . . 21 (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))) = (𝑙𝑊 ↦ if(𝑙𝑌, (𝑐𝑙), if((𝑐𝑙) ≤ 𝑥, (𝑐𝑙), 𝑥)))
352351mpteq2i 4711 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))) = (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑙𝑊 ↦ if(𝑙𝑌, (𝑐𝑙), if((𝑐𝑙) ≤ 𝑥, (𝑐𝑙), 𝑥))))
353352mpteq2i 4711 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))))) = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑙𝑊 ↦ if(𝑙𝑌, (𝑐𝑙), if((𝑐𝑙) ≤ 𝑥, (𝑐𝑙), 𝑥)))))
354204, 353eqtri 2643 . . . . . . . . . . . . . . . . . 18 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑙𝑊 ↦ if(𝑙𝑌, (𝑐𝑙), if((𝑐𝑙) ≤ 𝑥, (𝑐𝑙), 𝑥)))))
35575adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
356354, 355, 345, 31hsphoif 40127 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → ((𝐻𝑆)‘(𝐷𝑗)):𝑊⟶ℝ)
357161, 345, 112, 356hoidmvcl 40133 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,)+∞))
358333, 339, 357syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,)+∞))
359332, 358sseldi 3586 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,)+∞))
360331, 359sseldi 3586 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,]+∞))
361209, 211, 357syl2anc 692 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,)+∞))
362331, 361sseldi 3586 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,]+∞))
363324, 325, 326, 330, 360, 362sge0splitmpt 39965 . . . . . . . . . . . 12 (𝜑 → (Σ^‘(𝑗 ∈ ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) = ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) +𝑒^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))))
364 nnex 10986 . . . . . . . . . . . . . . 15 ℕ ∈ V
365364a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ℕ ∈ V)
366331, 357sseldi 3586 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,]+∞))
367324, 365, 366, 205, 336sge0ssrempt 39959 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℝ)
36818a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (1...𝑀) ⊆ ℕ)
369324, 365, 366, 205, 368sge0ssrempt 39959 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℝ)
370 rexadd 12022 . . . . . . . . . . . . 13 (((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℝ ∧ (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℝ) → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) +𝑒^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) = ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))))
371367, 369, 370syl2anc 692 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) +𝑒^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) = ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))))
372323, 363, 3713eqtrd 2659 . . . . . . . . . . 11 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) = ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))))
373372oveq2d 6631 . . . . . . . . . 10 (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) = ((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))))))
374373oveq1d 6630 . . . . . . . . 9 (𝜑 → (((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))) = (((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))))
375372, 205eqeltrrd 2699 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) ∈ ℝ)
376375recnd 10028 . . . . . . . . . . 11 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) ∈ ℂ)
377276recnd 10028 . . . . . . . . . . 11 (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)) ∈ ℂ)
378302, 376, 377adddid 10024 . . . . . . . . . 10 (𝜑 → ((1 + 𝐸) · (((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))) = (((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))))
379378eqcomd 2627 . . . . . . . . 9 (𝜑 → (((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))) = ((1 + 𝐸) · (((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))))
380367recnd 10028 . . . . . . . . . . . 12 (𝜑 → (Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℂ)
381369recnd 10028 . . . . . . . . . . . 12 (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℂ)
382380, 381, 377addassd 10022 . . . . . . . . . . 11 (𝜑 → (((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))) = ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + ((Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))))
383207, 361sge0fsummpt 39944 . . . . . . . . . . . . . 14 (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) = Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))
384383oveq1d 6630 . . . . . . . . . . . . 13 (𝜑 → ((Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))) = (Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))))
385 ax-resscn 9953 . . . . . . . . . . . . . . . . . 18 ℝ ⊆ ℂ
386159, 385sstri 3597 . . . . . . . . . . . . . . . . 17 (0[,)+∞) ⊆ ℂ
387386, 357sseldi 3586 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ ℂ)
388209, 211, 387syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ ℂ)
389183adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (𝑄𝑆) ∈ ℝ)
390389, 273remulcld 10030 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → ((𝑄𝑆) · (𝑃𝑗)) ∈ ℝ)
391390recnd 10028 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((𝑄𝑆) · (𝑃𝑗)) ∈ ℂ)
392211, 391syldan 487 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝑄𝑆) · (𝑃𝑗)) ∈ ℂ)
393207, 388, 392fsumadd 14419 . . . . . . . . . . . . . 14 (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = (Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))))
394393eqcomd 2627 . . . . . . . . . . . . 13 (𝜑 → (Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))) = Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))))
395384, 394eqtrd 2655 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))) = Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))))
396395oveq2d 6631 . . . . . . . . . . 11 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + ((Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))) = ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗)))))
397382, 396eqtrd 2655 . . . . . . . . . 10 (𝜑 → (((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))) = ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗)))))
398397oveq2d 6631 . . . . . . . . 9 (𝜑 → ((1 + 𝐸) · (((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))) = ((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))))))
399374, 379, 3983eqtrd 2659 . . . . . . . 8 (𝜑 → (((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))) = ((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))))))
400159, 357sseldi 3586 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ ℝ)
401400, 390readdcld 10029 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ∈ ℝ)
402209, 211, 401syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ∈ ℝ)
403207, 402fsumrecl 14414 . . . . . . . . . 10 (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ∈ ℝ)
404367, 403readdcld 10029 . . . . . . . . 9 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗)))) ∈ ℝ)
405 0le1 10511 . . . . . . . . . . 11 0 ≤ 1
406405a1i 11 . . . . . . . . . 10 (𝜑 → 0 ≤ 1)
407198rpge0d 11836 . . . . . . . . . 10 (𝜑 → 0 ≤ 𝐸)
408197, 199, 406, 407addge0d 10563 . . . . . . . . 9 (𝜑 → 0 ≤ (1 + 𝐸))
40967adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑄 ∈ ℝ)
410354, 409, 345, 31hsphoif 40127 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝐻𝑄)‘(𝐷𝑗)):𝑊⟶ℝ)
411161, 345, 112, 410hoidmvcl 40133 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ (0[,)+∞))
412331, 411sseldi 3586 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ (0[,]+∞))
413324, 365, 412, 279, 336sge0ssrempt 39959 . . . . . . . . . . 11 (𝜑 → (Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) ∈ ℝ)
414159, 411sseldi 3586 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ ℝ)
415209, 211, 414syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ ℝ)
416207, 415fsumrecl 14414 . . . . . . . . . . 11 (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ ℝ)
417333, 339, 412syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ (0[,]+∞))
418202adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊𝑌))
41975, 67, 144ltled 10145 . . . . . . . . . . . . . . 15 (𝜑𝑆𝑄)
420419adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → 𝑆𝑄)
421161, 345, 418, 7, 355, 409, 420, 354, 112, 31hsphoidmvle2 40136 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
422333, 339, 421syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
423324, 325, 360, 417, 422sge0lempt 39964 . . . . . . . . . . 11 (𝜑 → (Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ≤ (Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))))
424209adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) = 0) → 𝜑)
425211adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) = 0) → 𝑗 ∈ ℕ)
426 simpr 477 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) = 0) → (𝑃𝑗) = 0)
427 oveq2 6623 . . . . . . . . . . . . . . . . . . 19 ((𝑃𝑗) = 0 → ((𝑄𝑆) · (𝑃𝑗)) = ((𝑄𝑆) · 0))
428427adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → ((𝑄𝑆) · (𝑃𝑗)) = ((𝑄𝑆) · 0))
429174mul01d 10195 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑄𝑆) · 0) = 0)
430429ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → ((𝑄𝑆) · 0) = 0)
431428, 430eqtrd 2655 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → ((𝑄𝑆) · (𝑃𝑗)) = 0)
432431oveq2d 6631 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + 0))
433387addid1d 10196 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + 0) = ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))
434433adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + 0) = ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))
435432, 434eqtrd 2655 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))
436421adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
437435, 436eqbrtrd 4645 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
438424, 425, 426, 437syl21anc 1322 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) = 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
439 simpl 473 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃𝑗) = 0) → (𝜑𝑗 ∈ (1...𝑀)))
440 neqne 2798 . . . . . . . . . . . . . . 15 (¬ (𝑃𝑗) = 0 → (𝑃𝑗) ≠ 0)
441440adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃𝑗) = 0) → (𝑃𝑗) ≠ 0)
442402adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ∈ ℝ)
443209adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝜑)
444211adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑗 ∈ ℕ)
445 simpr 477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (𝑃𝑗) ≠ 0)
4462adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ ℕ) → 𝑍 ∈ (𝑋𝑌))
447201adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ ℕ) → ¬ 𝑍𝑌)
448 eqid 2621 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘)))
449161, 218, 446, 447, 7, 112, 356, 448hoiprodp1 40139 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) = (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍)))))
450449adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) = (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍)))))
451217adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝑃𝑗) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
452218adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → 𝑌 ∈ Fin)
453217adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝑃𝑗) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
454 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑌 = ∅ → (𝐿𝑌) = (𝐿‘∅))
455454oveqd 6632 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑌 = ∅ → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) = ((𝐽𝑗)(𝐿‘∅)(𝐾𝑗)))
456455adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) = ((𝐽𝑗)(𝐿‘∅)(𝐾𝑗)))
457249adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐽𝑗):𝑌⟶ℝ)
458 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑌 = ∅ → 𝑌 = ∅)
459458eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑌 = ∅ → ∅ = 𝑌)
460459adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ∅ = 𝑌)
461460feq2d 5998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽𝑗):∅⟶ℝ ↔ (𝐽𝑗):𝑌⟶ℝ))
462457, 461mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐽𝑗):∅⟶ℝ)
463270adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐾𝑗):𝑌⟶ℝ)
464460feq2d 5998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐾𝑗):∅⟶ℝ ↔ (𝐾𝑗):𝑌⟶ℝ))
465463, 464mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐾𝑗):∅⟶ℝ)
466161, 462, 465hoidmv0val 40134 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽𝑗)(𝐿‘∅)(𝐾𝑗)) = 0)
467453, 456, 4663eqtrd 2659 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝑃𝑗) = 0)
468467adantlr 750 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑌 = ∅) → (𝑃𝑗) = 0)
469 neneq 2796 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃𝑗) ≠ 0 → ¬ (𝑃𝑗) = 0)
470469ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑌 = ∅) → ¬ (𝑃𝑗) = 0)
471468, 470pm2.65da 599 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ¬ 𝑌 = ∅)
472471neqned 2797 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → 𝑌 ≠ ∅)
473249adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝐽𝑗):𝑌⟶ℝ)
474270adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝐾𝑗):𝑌⟶ℝ)
475161, 452, 472, 473, 474hoidmvn0val 40135 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) = ∏𝑘𝑌 (vol‘(((𝐽𝑗)‘𝑘)[,)((𝐾𝑗)‘𝑘))))
476247adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝐽𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))
477217adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝑃𝑗) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
478247adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝐽𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))
479478, 231eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝐽𝑗) = 𝐹)
480268adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝐾𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))
481480, 257eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝐾𝑗) = 𝐹)
482479, 481oveq12d 6633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) = (𝐹(𝐿𝑌)𝐹))
483161, 164, 228hoidmvval0b 40141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → (𝐹(𝐿𝑌)𝐹) = 0)
484483ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝐹(𝐿𝑌)𝐹) = 0)
485477, 482, 4843eqtrd 2659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝑃𝑗) = 0)
486485adantlr 750 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝑃𝑗) = 0)
487469ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → ¬ (𝑃𝑗) = 0)
488486, 487condan 834 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)))
489488iftrued 4072 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) = ((𝐶𝑗) ↾ 𝑌))
490476, 489eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝐽𝑗) = ((𝐶𝑗) ↾ 𝑌))
491490fveq1d 6160 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐽𝑗)‘𝑘) = (((𝐶𝑗) ↾ 𝑌)‘𝑘))
492491adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → ((𝐽𝑗)‘𝑘) = (((𝐶𝑗) ↾ 𝑌)‘𝑘))
493 fvres 6174 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘𝑌 → (((𝐶𝑗) ↾ 𝑌)‘𝑘) = ((𝐶𝑗)‘𝑘))
494493adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → (((𝐶𝑗) ↾ 𝑌)‘𝑘) = ((𝐶𝑗)‘𝑘))
495492, 494eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → ((𝐽𝑗)‘𝑘) = ((𝐶𝑗)‘𝑘))
496268adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝐾𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))
497488, 252syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) = ((𝐷𝑗) ↾ 𝑌))
498496, 497eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝐾𝑗) = ((𝐷𝑗) ↾ 𝑌))
499498fveq1d 6160 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐾𝑗)‘𝑘) = (((𝐷𝑗) ↾ 𝑌)‘𝑘))
500499adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → ((𝐾𝑗)‘𝑘) = (((𝐷𝑗) ↾ 𝑌)‘𝑘))
501 fvres 6174 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘𝑌 → (((𝐷𝑗) ↾ 𝑌)‘𝑘) = ((𝐷𝑗)‘𝑘))
502501adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → (((𝐷𝑗) ↾ 𝑌)‘𝑘) = ((𝐷𝑗)‘𝑘))
503500, 502eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → ((𝐾𝑗)‘𝑘) = ((𝐷𝑗)‘𝑘))
504495, 503oveq12d 6633 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → (((𝐽𝑗)‘𝑘)[,)((𝐾𝑗)‘𝑘)) = (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
505504fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → (vol‘(((𝐽𝑗)‘𝑘)[,)((𝐾𝑗)‘𝑘))) = (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
506505prodeq2dv 14597 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐽𝑗)‘𝑘)[,)((𝐾𝑗)‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
507475, 506eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
508355adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → 𝑆 ∈ ℝ)
509345adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → 𝑊 ∈ Fin)
51031adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (𝐷𝑗):𝑊⟶ℝ)
511 elun1 3764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘𝑌𝑘 ∈ (𝑌 ∪ {𝑍}))
512511, 7syl6eleqr 2709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘𝑌𝑘𝑊)
513512adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → 𝑘𝑊)
514354, 508, 509, 510, 513hsphoival 40130 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (((𝐻𝑆)‘(𝐷𝑗))‘𝑘) = if(𝑘𝑌, ((𝐷𝑗)‘𝑘), if(((𝐷𝑗)‘𝑘) ≤ 𝑆, ((𝐷𝑗)‘𝑘), 𝑆)))
515 iftrue 4070 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘𝑌 → if(𝑘𝑌, ((𝐷𝑗)‘𝑘), if(((𝐷𝑗)‘𝑘) ≤ 𝑆, ((𝐷𝑗)‘𝑘), 𝑆)) = ((𝐷𝑗)‘𝑘))
516515adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → if(𝑘𝑌, ((𝐷𝑗)‘𝑘), if(((𝐷𝑗)‘𝑘) ≤ 𝑆, ((𝐷𝑗)‘𝑘), 𝑆)) = ((𝐷𝑗)‘𝑘))
517514, 516eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (((𝐻𝑆)‘(𝐷𝑗))‘𝑘) = ((𝐷𝑗)‘𝑘))
518517oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘)) = (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
519518fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) = (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
520519prodeq2dv 14597 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
521520eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))))
522521adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))))
523451, 507, 5223eqtrrd 2660 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) = (𝑃𝑗))
524354, 355, 345, 31, 32hsphoival 40130 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑗 ∈ ℕ) → (((𝐻𝑆)‘(𝐷𝑗))‘𝑍) = if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆)))
525201iffalsed 4075 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆)) = if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆))
526525adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑗 ∈ ℕ) → if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆)) = if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆))
527524, 526eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ ℕ) → (((𝐻𝑆)‘(𝐷𝑗))‘𝑍) = if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆))
528527oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ ℕ) → (((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍)) = (((𝐶𝑗)‘𝑍)[,)if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆)))
529528adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍)) = (((𝐶𝑗)‘𝑍)[,)if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆)))
530113rexrd 10049 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)‘𝑍) ∈ ℝ*)
531530adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)‘𝑍) ∈ ℝ*)
53233rexrd 10049 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑗 ∈ ℕ) → ((𝐷𝑗)‘𝑍) ∈ ℝ*)
533532adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ ℝ*)
534 icoltub 39178 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝐶𝑗)‘𝑍) ∈ ℝ* ∧ ((𝐷𝑗)‘𝑍) ∈ ℝ*𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → 𝑆 < ((𝐷𝑗)‘𝑍))
535531, 533, 488, 534syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → 𝑆 < ((𝐷𝑗)‘𝑍))
536355adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → 𝑆 ∈ ℝ)
53733adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ ℝ)
538536, 537ltnled 10144 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝑆 < ((𝐷𝑗)‘𝑍) ↔ ¬ ((𝐷𝑗)‘𝑍) ≤ 𝑆))
539535, 538mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ¬ ((𝐷𝑗)‘𝑍) ≤ 𝑆)
540539iffalsed 4075 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆) = 𝑆)
541540oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)‘𝑍)[,)if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆)) = (((𝐶𝑗)‘𝑍)[,)𝑆))
542529, 541eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍)) = (((𝐶𝑗)‘𝑍)[,)𝑆))
543542fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍))) = (vol‘(((𝐶𝑗)‘𝑍)[,)𝑆)))
544 volico 39537 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐶𝑗)‘𝑍) ∈ ℝ ∧ 𝑆 ∈ ℝ) → (vol‘(((𝐶𝑗)‘𝑍)[,)𝑆)) = if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0))
545113, 536, 544syl2an 494 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ℕ) ∧ ((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0)) → (vol‘(((𝐶𝑗)‘𝑍)[,)𝑆)) = if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0))
546545anabss5 856 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (vol‘(((𝐶𝑗)‘𝑍)[,)𝑆)) = if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0))
547 iftrue 4070 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐶𝑗)‘𝑍) < 𝑆 → if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶𝑗)‘𝑍)))
548547adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ((𝐶𝑗)‘𝑍) < 𝑆) → if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶𝑗)‘𝑍)))
549 iffalse 4073 . . . . . . . . . . . . . . . . . . . . . . . . 25 (¬ ((𝐶𝑗)‘𝑍) < 𝑆 → if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0) = 0)
550549adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0) = 0)
551 simpll 789 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → (𝜑𝑗 ∈ ℕ))
552 icogelb 12183 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝐶𝑗)‘𝑍) ∈ ℝ* ∧ ((𝐷𝑗)‘𝑍) ∈ ℝ*𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → ((𝐶𝑗)‘𝑍) ≤ 𝑆)
553531, 533, 488, 552syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)‘𝑍) ≤ 𝑆)
554553adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → ((𝐶𝑗)‘𝑍) ≤ 𝑆)
555 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → ¬ ((𝐶𝑗)‘𝑍) < 𝑆)
556554, 555jca 554 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → (((𝐶𝑗)‘𝑍) ≤ 𝑆 ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆))
557551, 113syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → ((𝐶𝑗)‘𝑍) ∈ ℝ)
558551, 355syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → 𝑆 ∈ ℝ)
559557, 558eqleltd 10141 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → (((𝐶𝑗)‘𝑍) = 𝑆 ↔ (((𝐶𝑗)‘𝑍) ≤ 𝑆 ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆)))
560556, 559mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → ((𝐶𝑗)‘𝑍) = 𝑆)
561 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐶𝑗)‘𝑍) = 𝑆 → ((𝐶𝑗)‘𝑍) = 𝑆)
562561eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐶𝑗)‘𝑍) = 𝑆𝑆 = ((𝐶𝑗)‘𝑍))
563562oveq1d 6630 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐶𝑗)‘𝑍) = 𝑆 → (𝑆 − ((𝐶𝑗)‘𝑍)) = (((𝐶𝑗)‘𝑍) − ((𝐶𝑗)‘𝑍)))
564563adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ ℕ) ∧ ((𝐶𝑗)‘𝑍) = 𝑆) → (𝑆 − ((𝐶𝑗)‘𝑍)) = (((𝐶𝑗)‘𝑍) − ((𝐶𝑗)‘𝑍)))
565385, 113sseldi 3586 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)‘𝑍) ∈ ℂ)
566565subidd 10340 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑗 ∈ ℕ) → (((𝐶𝑗)‘𝑍) − ((𝐶𝑗)‘𝑍)) = 0)
567566adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ ℕ) ∧ ((𝐶𝑗)‘𝑍) = 𝑆) → (((𝐶𝑗)‘𝑍) − ((𝐶𝑗)‘𝑍)) = 0)
568564, 567eqtr2d 2656 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ ℕ) ∧ ((𝐶𝑗)‘𝑍) = 𝑆) → 0 = (𝑆 − ((𝐶𝑗)‘𝑍)))
569551, 560, 568syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → 0 = (𝑆 − ((𝐶𝑗)‘𝑍)))
570550, 569eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶𝑗)‘𝑍)))
571548, 570pm2.61dan 831 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶𝑗)‘𝑍)))
572543, 546, 5713eqtrd 2659 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍))) = (𝑆 − ((𝐶𝑗)‘𝑍)))
573523, 572oveq12d 6633 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍)))) = ((𝑃𝑗) · (𝑆 − ((𝐶𝑗)‘𝑍))))
574386, 272sseldi 3586 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ ℕ) → (𝑃𝑗) ∈ ℂ)
575355, 113resubcld 10418 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ ℕ) → (𝑆 − ((𝐶𝑗)‘𝑍)) ∈ ℝ)
576575recnd 10028 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ ℕ) → (𝑆 − ((𝐶𝑗)‘𝑍)) ∈ ℂ)
577574, 576mulcomd 10021 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → ((𝑃𝑗) · (𝑆 − ((𝐶𝑗)‘𝑍))) = ((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
578577adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝑃𝑗) · (𝑆 − ((𝐶𝑗)‘𝑍))) = ((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
579450, 573, 5783eqtrd 2659 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) = ((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
580579oveq1d 6630 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = (((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)) + ((𝑄𝑆) · (𝑃𝑗))))
581174adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → (𝑄𝑆) ∈ ℂ)
582576, 581, 574adddird 10025 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) · (𝑃𝑗)) = (((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)) + ((𝑄𝑆) · (𝑃𝑗))))
583582eqcomd 2627 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → (((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)) + ((𝑄𝑆) · (𝑃𝑗))) = (((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) · (𝑃𝑗)))
584583adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)) + ((𝑄𝑆) · (𝑃𝑗))) = (((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) · (𝑃𝑗)))
585576, 581addcomd 10198 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → ((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) = ((𝑄𝑆) + (𝑆 − ((𝐶𝑗)‘𝑍))))
586153adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ ℕ) → 𝑄 ∈ ℂ)
587154adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ ℕ) → 𝑆 ∈ ℂ)
588586, 587, 565npncand 10376 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → ((𝑄𝑆) + (𝑆 − ((𝐶𝑗)‘𝑍))) = (𝑄 − ((𝐶𝑗)‘𝑍)))
589585, 588eqtrd 2655 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → ((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) = (𝑄 − ((𝐶𝑗)‘𝑍)))
590589oveq1d 6630 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → (((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) · (𝑃𝑗)) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
591590adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) · (𝑃𝑗)) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
592580, 584, 5913eqtrd 2659 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
593443, 444, 445, 592syl21anc 1322 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
594 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘)))
595161, 218, 32, 447, 7, 112, 410, 594hoiprodp1 40139 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) = (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍)))))
596209, 211, 595syl2anc 692 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) = (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍)))))
597596adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) = (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍)))))
598507eqcomd 2627 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
599409adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → 𝑄 ∈ ℝ)
600354, 599, 509, 510, 513hsphoival 40130 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (((𝐻𝑄)‘(𝐷𝑗))‘𝑘) = if(𝑘𝑌, ((𝐷𝑗)‘𝑘), if(((𝐷𝑗)‘𝑘) ≤ 𝑄, ((𝐷𝑗)‘𝑘), 𝑄)))
601 iftrue 4070 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘𝑌 → if(𝑘𝑌, ((𝐷𝑗)‘𝑘), if(((𝐷𝑗)‘𝑘) ≤ 𝑄, ((𝐷𝑗)‘𝑘), 𝑄)) = ((𝐷𝑗)‘𝑘))
602601adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → if(𝑘𝑌, ((𝐷𝑗)‘𝑘), if(((𝐷𝑗)‘𝑘) ≤ 𝑄, ((𝐷𝑗)‘𝑘), 𝑄)) = ((𝐷𝑗)‘𝑘))
603600, 602eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (((𝐻𝑄)‘(𝐷𝑗))‘𝑘) = ((𝐷𝑗)‘𝑘))
604603oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘)) = (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
605604fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) = (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
606605prodeq2dv 14597 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
607606adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
608598, 607, 4513eqtr4d 2665 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) = (𝑃𝑗))
609443, 444, 445, 608syl21anc 1322 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) = (𝑃𝑗))
610354, 409, 345, 31, 32hsphoival 40130 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → (((𝐻𝑄)‘(𝐷𝑗))‘𝑍) = if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄)))
611211, 610syldan 487 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (1...𝑀)) → (((𝐻𝑄)‘(𝐷𝑗))‘𝑍) = if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄)))
612611adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐻𝑄)‘(𝐷𝑗))‘𝑍) = if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄)))
613201iffalsed 4075 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄)) = if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄))
614613ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄)) = if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄))
615211, 33syldan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐷𝑗)‘𝑍) ∈ ℝ)
616615adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ((𝐷𝑗)‘𝑍) = 𝑄) → ((𝐷𝑗)‘𝑍) ∈ ℝ)
617 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ((𝐷𝑗)‘𝑍) = 𝑄) → ((𝐷𝑗)‘𝑍) = 𝑄)
618616, 617eqled 10100 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ((𝐷𝑗)‘𝑍) = 𝑄) → ((𝐷𝑗)‘𝑍) ≤ 𝑄)
619618iftrued 4072 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ((𝐷𝑗)‘𝑍) = 𝑄) → if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄) = ((𝐷𝑗)‘𝑍))
620619, 617eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ((𝐷𝑗)‘𝑍) = 𝑄) → if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄) = 𝑄)
621620adantlr 750 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ((𝐷𝑗)‘𝑍) = 𝑄) → if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄) = 𝑄)
62267adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑄 ∈ ℝ)
623622adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → 𝑄 ∈ ℝ)
624623adantlr 750 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → 𝑄 ∈ ℝ)
625615adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → ((𝐷𝑗)‘𝑍) ∈ ℝ)
626625adantlr 750 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → ((𝐷𝑗)‘𝑍) ∈ ℝ)
62740a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑄 = inf(𝑉, ℝ, < ))
628443, 39syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑉 ⊆ ℝ)
629148ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ∃𝑥𝑉𝑦𝑉 𝑥𝑦)
630 simplr 791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑗 ∈ (1...𝑀))
631210, 488sylanl2 682 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)))
632630, 631jca 554 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (𝑗 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))))
633 rabid 3110 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑗 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↔ (𝑗 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))))
634632, 633sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑗 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))})
635 eqidd 2622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) = ((𝐷𝑗)‘𝑍))
636 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑖 = 𝑗 → (𝐷𝑖) = (𝐷𝑗))
637636fveq1d 6160 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑖 = 𝑗 → ((𝐷𝑖)‘𝑍) = ((𝐷𝑗)‘𝑍))
638637eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑖 = 𝑗 → (((𝐷𝑗)‘𝑍) = ((𝐷𝑖)‘𝑍) ↔ ((𝐷𝑗)‘𝑍) = ((𝐷𝑗)‘𝑍)))
639638rspcev 3299 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑗 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ∧ ((𝐷𝑗)‘𝑍) = ((𝐷𝑗)‘𝑍)) → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ((𝐷𝑗)‘𝑍) = ((𝐷𝑖)‘𝑍))
640634, 635, 639syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ((𝐷𝑗)‘𝑍) = ((𝐷𝑖)‘𝑍))
641 fvexd 6170 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ V)
64216, 640, 641elrnmptd 38875 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍)))
643642, 14syl6eleqr 2709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ 𝑂)
644 elun2 3765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝐷𝑗)‘𝑍) ∈ 𝑂 → ((𝐷𝑗)‘𝑍) ∈ ({(𝐵𝑍)} ∪ 𝑂))
645643, 644syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ ({(𝐵𝑍)} ∪ 𝑂))
64659a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ({(𝐵𝑍)} ∪ 𝑂) = 𝑉)
647645, 646eleqtrd 2700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ 𝑉)
648 lbinfle 10938 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑉 ⊆ ℝ ∧ ∃𝑥𝑉𝑦𝑉 𝑥𝑦 ∧ ((𝐷𝑗)‘𝑍) ∈ 𝑉) → inf(𝑉, ℝ, < ) ≤ ((𝐷𝑗)‘𝑍))
649628, 629, 647, 648syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → inf(𝑉, ℝ, < ) ≤ ((𝐷𝑗)‘𝑍))
650627, 649eqbrtrd 4645 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑄 ≤ ((𝐷𝑗)‘𝑍))
651650adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → 𝑄 ≤ ((𝐷𝑗)‘𝑍))
652 neqne 2798 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (¬ ((𝐷𝑗)‘𝑍) = 𝑄 → ((𝐷𝑗)‘𝑍) ≠ 𝑄)
653652adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → ((𝐷𝑗)‘𝑍) ≠ 𝑄)
654624, 626, 651, 653leneltd 10151 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → 𝑄 < ((𝐷𝑗)‘𝑍))
655624, 626ltnled 10144 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → (𝑄 < ((𝐷𝑗)‘𝑍) ↔ ¬ ((𝐷𝑗)‘𝑍) ≤ 𝑄))
656654, 655mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → ¬ ((𝐷𝑗)‘𝑍) ≤ 𝑄)
657656iffalsed 4075 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄) = 𝑄)
658621, 657pm2.61dan 831 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄) = 𝑄)
659612, 614, 6583eqtrd 2659 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐻𝑄)‘(𝐷𝑗))‘𝑍) = 𝑄)
660659oveq2d 6631 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍)) = (((𝐶𝑗)‘𝑍)[,)𝑄))
661660fveq2d 6162 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍))) = (vol‘(((𝐶𝑗)‘𝑍)[,)𝑄)))
662209, 211, 113syl2anc 692 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)‘𝑍) ∈ ℝ)
663662adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)‘𝑍) ∈ ℝ)
664443, 67syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑄 ∈ ℝ)
665 volico 39537 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶𝑗)‘𝑍) ∈ ℝ ∧ 𝑄 ∈ ℝ) → (vol‘(((𝐶𝑗)‘𝑍)[,)𝑄)) = if(((𝐶𝑗)‘𝑍) < 𝑄, (𝑄 − ((𝐶𝑗)‘𝑍)), 0))
666663, 664, 665syl2anc 692 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (vol‘(((𝐶𝑗)‘𝑍)[,)𝑄)) = if(((𝐶𝑗)‘𝑍) < 𝑄, (𝑄 − ((𝐶𝑗)‘𝑍)), 0))
667443, 75syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑆 ∈ ℝ)
668443, 444, 445, 553syl21anc 1322 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)‘𝑍) ≤ 𝑆)
669443, 144syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑆 < 𝑄)
670663, 667, 664, 668, 669lelttrd 10155 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)‘𝑍) < 𝑄)
671670iftrued 4072 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → if(((𝐶𝑗)‘𝑍) < 𝑄, (𝑄 − ((𝐶𝑗)‘𝑍)), 0) = (𝑄 − ((𝐶𝑗)‘𝑍)))
672661, 666, 6713eqtrd 2659 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍))) = (𝑄 − ((𝐶𝑗)‘𝑍)))
673609, 672oveq12d 6633 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍)))) = ((𝑃𝑗) · (𝑄 − ((𝐶𝑗)‘𝑍))))
674209, 153syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑄 ∈ ℂ)
675385, 662sseldi 3586 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)‘𝑍) ∈ ℂ)
676674, 675subcld 10352 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (1...𝑀)) → (𝑄 − ((𝐶𝑗)‘𝑍)) ∈ ℂ)
677305, 676mulcomd 10021 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝑃𝑗) · (𝑄 − ((𝐶𝑗)‘𝑍))) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
678677adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝑃𝑗) · (𝑄 − ((𝐶𝑗)‘𝑍))) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
679597, 673, 6783eqtrd 2659 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
680593, 679eqtr4d 2658 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
681442, 680eqled 10100 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
682439, 441, 681syl2anc 692 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃𝑗) = 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
683438, 682pm2.61dan 831 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
684207, 402, 415, 683fsumle 14477 . . . . . . . . . . 11 (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ≤ Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
685367, 403, 413, 416, 423, 684leadd12dd 39031 . . . . . . . . . 10 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗)))) ≤ ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))
686321mpteq1d 4708 . . . . . . . . . . . . 13 (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))) = (𝑗 ∈ ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))
687686fveq2d 6162 . . . . . . . . . . . 12 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) = (Σ^‘(𝑗 ∈ ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))))
688211, 412syldan 487 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ (0[,]+∞))
689324, 325, 326, 330, 417, 688sge0splitmpt 39965 . . . . . . . . . . . 12 (𝜑 → (Σ^‘(𝑗 ∈ ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) = ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) +𝑒^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))))
690687, 689eqtrd 2655 . . . . . . . . . . 11 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) = ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) +𝑒^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))))
691209, 211, 411syl2anc 692 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ (0[,)+∞))
692207, 691sge0fsummpt 39944 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) = Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
693692, 416eqeltrd 2698 . . . . . . . . . . . 12 (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) ∈ ℝ)
694 rexadd 12022 . . . . . . . . . . . 12 (((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) ∈ ℝ ∧ (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) ∈ ℝ) → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) +𝑒^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))) = ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))))
695413, 693, 694syl2anc 692 . . . . . . . . . . 11 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) +𝑒^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))) = ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))))
696692oveq2d 6631 . . . . . . . . . . 11 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))) = ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))
697690, 695, 6963eqtrrd 2660 . . . . . . . . . 10 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))))
698685, 697breqtrd 4649 . . . . . . . . 9 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗)))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))))
699404, 279, 200, 408, 698lemul2ad 10924 . . . . . . . 8 (𝜑 → ((1 + 𝐸) · ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))))
700399, 699eqbrtrd 4645 . . . . . . 7 (𝜑 → (((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))))
701196, 278, 280, 314, 700letrd 10154 . . . . . 6 (𝜑 → ((𝐺 · (𝑆 − (𝐴𝑍))) + (𝐺 · (𝑄𝑆))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))))
702180, 701eqbrtrd 4645 . . . . 5 (𝜑 → (𝐺 · (𝑄 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))))
703152, 702jca 554 . . . 4 (𝜑 → (𝑄 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∧ (𝐺 · (𝑄 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))))))
704 oveq1 6622 . . . . . . 7 (𝑧 = 𝑄 → (𝑧 − (𝐴𝑍)) = (𝑄 − (𝐴𝑍)))
705704oveq2d 6631 . . . . . 6 (𝑧 = 𝑄 → (𝐺 · (𝑧 − (𝐴𝑍))) = (𝐺 · (𝑄 − (𝐴𝑍))))
706 fveq2 6158 . . . . . . . . . . 11 (𝑧 = 𝑄 → (𝐻𝑧) = (𝐻𝑄))
707706fveq1d 6160 . . . . . . . . . 10 (𝑧 = 𝑄 → ((𝐻𝑧)‘(𝐷𝑗)) = ((𝐻𝑄)‘(𝐷𝑗)))
708707oveq2d 6631 . . . . . . . . 9 (𝑧 = 𝑄 → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))) = ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
709708mpteq2dv 4715 . . . . . . . 8 (𝑧 = 𝑄 → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))
710709fveq2d 6162 . . . . . . 7 (𝑧 = 𝑄 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))))
711710oveq2d 6631 . . . . . 6 (𝑧 = 𝑄 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))))) = ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))))
712705, 711breq12d 4636 . . . . 5 (𝑧 = 𝑄 → ((𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))))) ↔ (𝐺 · (𝑄 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))))))
713712elrab 3351 . . . 4 (𝑄 ∈ {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} ↔ (𝑄 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∧ (𝐺 · (𝑄 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))))))
714703, 713sylibr 224 . . 3 (𝜑𝑄 ∈ {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))})
715714, 68syl6eleqr 2709 . 2 (𝜑𝑄𝑈)
716 breq2 4627 . . 3 (𝑢 = 𝑄 → (𝑆 < 𝑢𝑆 < 𝑄))
717716rspcev 3299 . 2 ((𝑄𝑈𝑆 < 𝑄) → ∃𝑢𝑈 𝑆 < 𝑢)
718715, 144, 717syl2anc 692 1 (𝜑 → ∃𝑢𝑈 𝑆 < 𝑢)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  {crab 2912  Vcvv 3190  cdif 3557  cun 3558  cin 3559  wss 3560  c0 3897  ifcif 4064  {csn 4155   class class class wbr 4623  cmpt 4683   Or wor 5004  ran crn 5085  cres 5086  wf 5853  cfv 5857  (class class class)co 6615  cmpt2 6617  𝑚 cmap 7817  Fincfn 7915  infcinf 8307  cc 9894  cr 9895  0cc0 9896  1c1 9897   + caddc 9899   · cmul 9901  +∞cpnf 10031  *cxr 10033   < clt 10034  cle 10035  cmin 10226  cn 10980  cuz 11647  +crp 11792   +𝑒 cxad 11904  [,)cico 12135  [,]cicc 12136  ...cfz 12284  Σcsu 14366  cprod 14579  volcvol 23172  Σ^csumge0 39916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fi 8277  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ioo 12137  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-rlim 14170  df-sum 14367  df-prod 14580  df-rest 16023  df-topgen 16044  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-top 20639  df-topon 20656  df-bases 20690  df-cmp 21130  df-ovol 23173  df-vol 23174  df-sumge0 39917
This theorem is referenced by:  hoidmvlelem3  40148
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