Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hoidmvlelem2 Structured version   Visualization version   GIF version

Theorem hoidmvlelem2 39286
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 4148 . . . . . . . . . 10 (𝑍 ∈ (𝑋𝑌) → 𝑍 ∈ {𝑍})
42, 3syl 17 . . . . . . . . 9 (𝜑𝑍 ∈ {𝑍})
5 elun2 3738 . . . . . . . . 9 (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
64, 5syl 17 . . . . . . . 8 (𝜑𝑍 ∈ (𝑌 ∪ {𝑍}))
7 hoidmvlelem2.w . . . . . . . 8 𝑊 = (𝑌 ∪ {𝑍})
86, 7syl6eleqr 2694 . . . . . . 7 (𝜑𝑍𝑊)
91, 8ffvelrnd 6249 . . . . . 6 (𝜑 → (𝐴𝑍) ∈ ℝ)
10 hoidmvlelem2.b . . . . . . 7 (𝜑𝐵:𝑊⟶ℝ)
1110, 8ffvelrnd 6249 . . . . . 6 (𝜑 → (𝐵𝑍) ∈ ℝ)
12 hoidmvlelem2.v . . . . . . . 8 𝑉 = ({(𝐵𝑍)} ∪ 𝑂)
1311snssd 4276 . . . . . . . . 9 (𝜑 → {(𝐵𝑍)} ⊆ ℝ)
14 hoidmvlelem2.O . . . . . . . . . 10 𝑂 = ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍))
15 nfv 1828 . . . . . . . . . . 11 𝑖𝜑
16 eqid 2605 . . . . . . . . . . 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 471 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → 𝜑)
18 fz1ssnn 12194 . . . . . . . . . . . . . 14 (1...𝑀) ⊆ ℕ
19 elrabi 3323 . . . . . . . . . . . . . 14 (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} → 𝑖 ∈ (1...𝑀))
2018, 19sseldi 3561 . . . . . . . . . . . . 13 (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} → 𝑖 ∈ ℕ)
2120adantl 480 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → 𝑖 ∈ ℕ)
22 eleq1 2671 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → (𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ))
2322anbi2d 735 . . . . . . . . . . . . . 14 (𝑗 = 𝑖 → ((𝜑𝑗 ∈ ℕ) ↔ (𝜑𝑖 ∈ ℕ)))
24 fveq2 6084 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑖 → (𝐷𝑗) = (𝐷𝑖))
2524fveq1d 6086 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → ((𝐷𝑗)‘𝑍) = ((𝐷𝑖)‘𝑍))
2625eleq1d 2667 . . . . . . . . . . . . . 14 (𝑗 = 𝑖 → (((𝐷𝑗)‘𝑍) ∈ ℝ ↔ ((𝐷𝑖)‘𝑍) ∈ ℝ))
2723, 26imbi12d 332 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → (((𝜑𝑗 ∈ ℕ) → ((𝐷𝑗)‘𝑍) ∈ ℝ) ↔ ((𝜑𝑖 ∈ ℕ) → ((𝐷𝑖)‘𝑍) ∈ ℝ)))
28 hoidmvlelem2.d . . . . . . . . . . . . . . . 16 (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
2928ffvelrnda 6248 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ (ℝ ↑𝑚 𝑊))
30 elmapi 7738 . . . . . . . . . . . . . . 15 ((𝐷𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐷𝑗):𝑊⟶ℝ)
3129, 30syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗):𝑊⟶ℝ)
328adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → 𝑍𝑊)
3331, 32ffvelrnd 6249 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝐷𝑗)‘𝑍) ∈ ℝ)
3427, 33chvarv 2245 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ℕ) → ((𝐷𝑖)‘𝑍) ∈ ℝ)
3517, 21, 34syl2anc 690 . . . . . . . . . . 11 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → ((𝐷𝑖)‘𝑍) ∈ ℝ)
3615, 16, 35rnmptssd 38179 . . . . . . . . . 10 (𝜑 → ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍)) ⊆ ℝ)
3714, 36syl5eqss 3607 . . . . . . . . 9 (𝜑𝑂 ⊆ ℝ)
3813, 37unssd 3746 . . . . . . . 8 (𝜑 → ({(𝐵𝑍)} ∪ 𝑂) ⊆ ℝ)
3912, 38syl5eqss 3607 . . . . . . 7 (𝜑𝑉 ⊆ ℝ)
40 hoidmvlelem2.q . . . . . . . 8 𝑄 = inf(𝑉, ℝ, < )
41 ltso 9965 . . . . . . . . . 10 < Or ℝ
4241a1i 11 . . . . . . . . 9 (𝜑 → < Or ℝ)
43 snfi 7896 . . . . . . . . . . . 12 {(𝐵𝑍)} ∈ Fin
4443a1i 11 . . . . . . . . . . 11 (𝜑 → {(𝐵𝑍)} ∈ Fin)
45 fzfi 12584 . . . . . . . . . . . . . . 15 (1...𝑀) ∈ Fin
46 rabfi 8043 . . . . . . . . . . . . . . 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 38145 . . . . . . . . . . . . 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 2687 . . . . . . . . . . 11 (𝜑𝑂 ∈ Fin)
52 unfi 8085 . . . . . . . . . . 11 (({(𝐵𝑍)} ∈ Fin ∧ 𝑂 ∈ Fin) → ({(𝐵𝑍)} ∪ 𝑂) ∈ Fin)
5344, 51, 52syl2anc 690 . . . . . . . . . 10 (𝜑 → ({(𝐵𝑍)} ∪ 𝑂) ∈ Fin)
5412, 53syl5eqel 2687 . . . . . . . . 9 (𝜑𝑉 ∈ Fin)
55 fvex 6094 . . . . . . . . . . . . . 14 (𝐵𝑍) ∈ V
5655snid 4150 . . . . . . . . . . . . 13 (𝐵𝑍) ∈ {(𝐵𝑍)}
57 elun1 3737 . . . . . . . . . . . . 13 ((𝐵𝑍) ∈ {(𝐵𝑍)} → (𝐵𝑍) ∈ ({(𝐵𝑍)} ∪ 𝑂))
5856, 57ax-mp 5 . . . . . . . . . . . 12 (𝐵𝑍) ∈ ({(𝐵𝑍)} ∪ 𝑂)
5912eqcomi 2614 . . . . . . . . . . . 12 ({(𝐵𝑍)} ∪ 𝑂) = 𝑉
6058, 59eleqtri 2681 . . . . . . . . . . 11 (𝐵𝑍) ∈ 𝑉
6160a1i 11 . . . . . . . . . 10 (𝜑 → (𝐵𝑍) ∈ 𝑉)
62 ne0i 3875 . . . . . . . . . 10 ((𝐵𝑍) ∈ 𝑉𝑉 ≠ ∅)
6361, 62syl 17 . . . . . . . . 9 (𝜑𝑉 ≠ ∅)
64 fiinfcl 8263 . . . . . . . . 9 (( < Or ℝ ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅ ∧ 𝑉 ⊆ ℝ)) → inf(𝑉, ℝ, < ) ∈ 𝑉)
6542, 54, 63, 39, 64syl13anc 1319 . . . . . . . 8 (𝜑 → inf(𝑉, ℝ, < ) ∈ 𝑉)
6640, 65syl5eqel 2687 . . . . . . 7 (𝜑𝑄𝑉)
6739, 66sseldd 3564 . . . . . 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 3645 . . . . . . . . . . . 12 {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} ⊆ ((𝐴𝑍)[,](𝐵𝑍))
7068, 69eqsstri 3593 . . . . . . . . . . 11 𝑈 ⊆ ((𝐴𝑍)[,](𝐵𝑍))
7170a1i 11 . . . . . . . . . 10 (𝜑𝑈 ⊆ ((𝐴𝑍)[,](𝐵𝑍)))
729, 11iccssred 38374 . . . . . . . . . 10 (𝜑 → ((𝐴𝑍)[,](𝐵𝑍)) ⊆ ℝ)
7371, 72sstrd 3573 . . . . . . . . 9 (𝜑𝑈 ⊆ ℝ)
74 hoidmvlelem2.su . . . . . . . . 9 (𝜑𝑆𝑈)
7573, 74sseldd 3564 . . . . . . . 8 (𝜑𝑆 ∈ ℝ)
769rexrd 9941 . . . . . . . . 9 (𝜑 → (𝐴𝑍) ∈ ℝ*)
7711rexrd 9941 . . . . . . . . 9 (𝜑 → (𝐵𝑍) ∈ ℝ*)
7870, 74sseldi 3561 . . . . . . . . 9 (𝜑𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍)))
79 iccgelb 12053 . . . . . . . . 9 (((𝐴𝑍) ∈ ℝ* ∧ (𝐵𝑍) ∈ ℝ*𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍))) → (𝐴𝑍) ≤ 𝑆)
8076, 77, 78, 79syl3anc 1317 . . . . . . . 8 (𝜑 → (𝐴𝑍) ≤ 𝑆)
81 hoidmvlelem2.sb . . . . . . . . . . . . . . 15 (𝜑𝑆 < (𝐵𝑍))
8281adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑥 = (𝐵𝑍)) → 𝑆 < (𝐵𝑍))
83 id 22 . . . . . . . . . . . . . . . 16 (𝑥 = (𝐵𝑍) → 𝑥 = (𝐵𝑍))
8483eqcomd 2611 . . . . . . . . . . . . . . 15 (𝑥 = (𝐵𝑍) → (𝐵𝑍) = 𝑥)
8584adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑥 = (𝐵𝑍)) → (𝐵𝑍) = 𝑥)
8682, 85breqtrd 4599 . . . . . . . . . . . . 13 ((𝜑𝑥 = (𝐵𝑍)) → 𝑆 < 𝑥)
8786adantlr 746 . . . . . . . . . . . 12 (((𝜑𝑥𝑉) ∧ 𝑥 = (𝐵𝑍)) → 𝑆 < 𝑥)
88 simpll 785 . . . . . . . . . . . . 13 (((𝜑𝑥𝑉) ∧ ¬ 𝑥 = (𝐵𝑍)) → 𝜑)
89 id 22 . . . . . . . . . . . . . . . . 17 (𝑥𝑉𝑥𝑉)
9089, 12syl6eleq 2693 . . . . . . . . . . . . . . . 16 (𝑥𝑉𝑥 ∈ ({(𝐵𝑍)} ∪ 𝑂))
9190adantr 479 . . . . . . . . . . . . . . 15 ((𝑥𝑉 ∧ ¬ 𝑥 = (𝐵𝑍)) → 𝑥 ∈ ({(𝐵𝑍)} ∪ 𝑂))
92 elsni 4137 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {(𝐵𝑍)} → 𝑥 = (𝐵𝑍))
9392con3i 148 . . . . . . . . . . . . . . . 16 𝑥 = (𝐵𝑍) → ¬ 𝑥 ∈ {(𝐵𝑍)})
9493adantl 480 . . . . . . . . . . . . . . 15 ((𝑥𝑉 ∧ ¬ 𝑥 = (𝐵𝑍)) → ¬ 𝑥 ∈ {(𝐵𝑍)})
95 elunnel1 3711 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ({(𝐵𝑍)} ∪ 𝑂) ∧ ¬ 𝑥 ∈ {(𝐵𝑍)}) → 𝑥𝑂)
9691, 94, 95syl2anc 690 . . . . . . . . . . . . . 14 ((𝑥𝑉 ∧ ¬ 𝑥 = (𝐵𝑍)) → 𝑥𝑂)
9796adantll 745 . . . . . . . . . . . . 13 (((𝜑𝑥𝑉) ∧ ¬ 𝑥 = (𝐵𝑍)) → 𝑥𝑂)
98 id 22 . . . . . . . . . . . . . . . . 17 (𝑥𝑂𝑥𝑂)
9998, 14syl6eleq 2693 . . . . . . . . . . . . . . . 16 (𝑥𝑂𝑥 ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍)))
100 vex 3171 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
10116elrnmpt 5276 . . . . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . . . 15 (𝑥𝑂 → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}𝑥 = ((𝐷𝑖)‘𝑍))
104103adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑂) → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}𝑥 = ((𝐷𝑖)‘𝑍))
105 fveq2 6084 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = 𝑖 → (𝐶𝑗) = (𝐶𝑖))
106105fveq1d 6086 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑖 → ((𝐶𝑗)‘𝑍) = ((𝐶𝑖)‘𝑍))
107106eleq1d 2667 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 𝑖 → (((𝐶𝑗)‘𝑍) ∈ ℝ ↔ ((𝐶𝑖)‘𝑍) ∈ ℝ))
10823, 107imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑖 → (((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)‘𝑍) ∈ ℝ) ↔ ((𝜑𝑖 ∈ ℕ) → ((𝐶𝑖)‘𝑍) ∈ ℝ)))
109 hoidmvlelem2.c . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
110109ffvelrnda 6248 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ (ℝ ↑𝑚 𝑊))
111 elmapi 7738 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐶𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐶𝑗):𝑊⟶ℝ)
112110, 111syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗):𝑊⟶ℝ)
113112, 32ffvelrnd 6249 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)‘𝑍) ∈ ℝ)
114108, 113chvarv 2245 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ ℕ) → ((𝐶𝑖)‘𝑍) ∈ ℝ)
115114rexrd 9941 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ ℕ) → ((𝐶𝑖)‘𝑍) ∈ ℝ*)
11617, 21, 115syl2anc 690 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → ((𝐶𝑖)‘𝑍) ∈ ℝ*)
11734rexrd 9941 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ ℕ) → ((𝐷𝑖)‘𝑍) ∈ ℝ*)
11817, 21, 117syl2anc 690 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → ((𝐷𝑖)‘𝑍) ∈ ℝ*)
119106, 25oveq12d 6541 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑖 → (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)) = (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)))
120119eleq2d 2668 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 𝑖 → (𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)) ↔ 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍))))
121120elrab 3326 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↔ (𝑖 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍))))
122121biimpi 204 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} → (𝑖 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍))))
123122simprd 477 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} → 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)))
124123adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)))
125 icoltub 38379 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶𝑖)‘𝑍) ∈ ℝ* ∧ ((𝐷𝑖)‘𝑍) ∈ ℝ*𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍))) → 𝑆 < ((𝐷𝑖)‘𝑍))
126116, 118, 124, 125syl3anc 1317 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))}) → 𝑆 < ((𝐷𝑖)‘𝑍))
1271263adant3 1073 . . . . . . . . . . . . . . . . . 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 2611 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ((𝐷𝑖)‘𝑍) → ((𝐷𝑖)‘𝑍) = 𝑥)
1301293ad2ant3 1076 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ∧ 𝑥 = ((𝐷𝑖)‘𝑍)) → ((𝐷𝑖)‘𝑍) = 𝑥)
131127, 130breqtrd 4599 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ∧ 𝑥 = ((𝐷𝑖)‘𝑍)) → 𝑆 < 𝑥)
1321313exp 1255 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} → (𝑥 = ((𝐷𝑖)‘𝑍) → 𝑆 < 𝑥)))
133132adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑂) → (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} → (𝑥 = ((𝐷𝑖)‘𝑍) → 𝑆 < 𝑥)))
134133rexlimdv 3007 . . . . . . . . . . . . . 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 690 . . . . . . . . . . . 12 (((𝜑𝑥𝑉) ∧ ¬ 𝑥 = (𝐵𝑍)) → 𝑆 < 𝑥)
13787, 136pm2.61dan 827 . . . . . . . . . . 11 ((𝜑𝑥𝑉) → 𝑆 < 𝑥)
138137ralrimiva 2944 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑉 𝑆 < 𝑥)
139 breq2 4577 . . . . . . . . . . 11 (𝑥 = inf(𝑉, ℝ, < ) → (𝑆 < 𝑥𝑆 < inf(𝑉, ℝ, < )))
140139rspcva 3275 . . . . . . . . . 10 ((inf(𝑉, ℝ, < ) ∈ 𝑉 ∧ ∀𝑥𝑉 𝑆 < 𝑥) → 𝑆 < inf(𝑉, ℝ, < ))
14165, 138, 140syl2anc 690 . . . . . . . . 9 (𝜑𝑆 < inf(𝑉, ℝ, < ))
14240eqcomi 2614 . . . . . . . . . 10 inf(𝑉, ℝ, < ) = 𝑄
143142a1i 11 . . . . . . . . 9 (𝜑 → inf(𝑉, ℝ, < ) = 𝑄)
144141, 143breqtrd 4599 . . . . . . . 8 (𝜑𝑆 < 𝑄)
1459, 75, 67, 80, 144lelttrd 10042 . . . . . . 7 (𝜑 → (𝐴𝑍) < 𝑄)
1469, 67, 145ltled 10032 . . . . . 6 (𝜑 → (𝐴𝑍) ≤ 𝑄)
147 fiminre 10817 . . . . . . . . 9 ((𝑉 ⊆ ℝ ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ∃𝑥𝑉𝑦𝑉 𝑥𝑦)
14839, 54, 63, 147syl3anc 1317 . . . . . . . 8 (𝜑 → ∃𝑥𝑉𝑦𝑉 𝑥𝑦)
149 lbinfle 10823 . . . . . . . 8 ((𝑉 ⊆ ℝ ∧ ∃𝑥𝑉𝑦𝑉 𝑥𝑦 ∧ (𝐵𝑍) ∈ 𝑉) → inf(𝑉, ℝ, < ) ≤ (𝐵𝑍))
15039, 148, 61, 149syl3anc 1317 . . . . . . 7 (𝜑 → inf(𝑉, ℝ, < ) ≤ (𝐵𝑍))
15140, 150syl5eqbr 4608 . . . . . 6 (𝜑𝑄 ≤ (𝐵𝑍))
1529, 11, 67, 146, 151eliccd 38373 . . . . 5 (𝜑𝑄 ∈ ((𝐴𝑍)[,](𝐵𝑍)))
15367recnd 9920 . . . . . . . . . 10 (𝜑𝑄 ∈ ℂ)
15475recnd 9920 . . . . . . . . . 10 (𝜑𝑆 ∈ ℂ)
1559recnd 9920 . . . . . . . . . 10 (𝜑 → (𝐴𝑍) ∈ ℂ)
156153, 154, 155npncand 10263 . . . . . . . . 9 (𝜑 → ((𝑄𝑆) + (𝑆 − (𝐴𝑍))) = (𝑄 − (𝐴𝑍)))
157156eqcomd 2611 . . . . . . . 8 (𝜑 → (𝑄 − (𝐴𝑍)) = ((𝑄𝑆) + (𝑆 − (𝐴𝑍))))
158157oveq2d 6539 . . . . . . 7 (𝜑 → (𝐺 · (𝑄 − (𝐴𝑍))) = (𝐺 · ((𝑄𝑆) + (𝑆 − (𝐴𝑍)))))
159 rge0ssre 12103 . . . . . . . . . 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 8041 . . . . . . . . . . . 12 (𝜑𝑌 ∈ Fin)
165 ssun1 3733 . . . . . . . . . . . . . . 15 𝑌 ⊆ (𝑌 ∪ {𝑍})
166165, 7sseqtr4i 3596 . . . . . . . . . . . . . 14 𝑌𝑊
167166a1i 11 . . . . . . . . . . . . 13 (𝜑𝑌𝑊)
1681, 167fssresd 5965 . . . . . . . . . . . 12 (𝜑 → (𝐴𝑌):𝑌⟶ℝ)
16910, 167fssresd 5965 . . . . . . . . . . . 12 (𝜑 → (𝐵𝑌):𝑌⟶ℝ)
170161, 164, 168, 169hoidmvcl 39272 . . . . . . . . . . 11 (𝜑 → ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) ∈ (0[,)+∞))
171160, 170syl5eqel 2687 . . . . . . . . . 10 (𝜑𝐺 ∈ (0[,)+∞))
172159, 171sseldi 3561 . . . . . . . . 9 (𝜑𝐺 ∈ ℝ)
173172recnd 9920 . . . . . . . 8 (𝜑𝐺 ∈ ℂ)
174153, 154subcld 10239 . . . . . . . 8 (𝜑 → (𝑄𝑆) ∈ ℂ)
175154, 155subcld 10239 . . . . . . . 8 (𝜑 → (𝑆 − (𝐴𝑍)) ∈ ℂ)
176173, 174, 175adddid 9916 . . . . . . 7 (𝜑 → (𝐺 · ((𝑄𝑆) + (𝑆 − (𝐴𝑍)))) = ((𝐺 · (𝑄𝑆)) + (𝐺 · (𝑆 − (𝐴𝑍)))))
177173, 174mulcld 9912 . . . . . . . 8 (𝜑 → (𝐺 · (𝑄𝑆)) ∈ ℂ)
178173, 175mulcld 9912 . . . . . . . 8 (𝜑 → (𝐺 · (𝑆 − (𝐴𝑍))) ∈ ℂ)
179177, 178addcomd 10085 . . . . . . 7 (𝜑 → ((𝐺 · (𝑄𝑆)) + (𝐺 · (𝑆 − (𝐴𝑍)))) = ((𝐺 · (𝑆 − (𝐴𝑍))) + (𝐺 · (𝑄𝑆))))
180158, 176, 1793eqtrd 2643 . . . . . 6 (𝜑 → (𝐺 · (𝑄 − (𝐴𝑍))) = ((𝐺 · (𝑆 − (𝐴𝑍))) + (𝐺 · (𝑄𝑆))))
18167, 75jca 552 . . . . . . . . . . . . 13 (𝜑 → (𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ))
182 resubcl 10192 . . . . . . . . . . . . 13 ((𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑄𝑆) ∈ ℝ)
183181, 182syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑄𝑆) ∈ ℝ)
184172, 183jca 552 . . . . . . . . . . 11 (𝜑 → (𝐺 ∈ ℝ ∧ (𝑄𝑆) ∈ ℝ))
185 remulcl 9873 . . . . . . . . . . 11 ((𝐺 ∈ ℝ ∧ (𝑄𝑆) ∈ ℝ) → (𝐺 · (𝑄𝑆)) ∈ ℝ)
186184, 185syl 17 . . . . . . . . . 10 (𝜑 → (𝐺 · (𝑄𝑆)) ∈ ℝ)
18775, 9jca 552 . . . . . . . . . . . . 13 (𝜑 → (𝑆 ∈ ℝ ∧ (𝐴𝑍) ∈ ℝ))
188 resubcl 10192 . . . . . . . . . . . . 13 ((𝑆 ∈ ℝ ∧ (𝐴𝑍) ∈ ℝ) → (𝑆 − (𝐴𝑍)) ∈ ℝ)
189187, 188syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑆 − (𝐴𝑍)) ∈ ℝ)
190172, 189jca 552 . . . . . . . . . . 11 (𝜑 → (𝐺 ∈ ℝ ∧ (𝑆 − (𝐴𝑍)) ∈ ℝ))
191 remulcl 9873 . . . . . . . . . . 11 ((𝐺 ∈ ℝ ∧ (𝑆 − (𝐴𝑍)) ∈ ℝ) → (𝐺 · (𝑆 − (𝐴𝑍))) ∈ ℝ)
192190, 191syl 17 . . . . . . . . . 10 (𝜑 → (𝐺 · (𝑆 − (𝐴𝑍))) ∈ ℝ)
193186, 192jca 552 . . . . . . . . 9 (𝜑 → ((𝐺 · (𝑄𝑆)) ∈ ℝ ∧ (𝐺 · (𝑆 − (𝐴𝑍))) ∈ ℝ))
194 readdcl 9871 . . . . . . . . 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 2684 . . . . . . 7 (𝜑 → ((𝐺 · (𝑆 − (𝐴𝑍))) + (𝐺 · (𝑄𝑆))) ∈ ℝ)
197 1red 9907 . . . . . . . . . 10 (𝜑 → 1 ∈ ℝ)
198 hoidmvlelem2.e . . . . . . . . . . 11 (𝜑𝐸 ∈ ℝ+)
199198rpred 11700 . . . . . . . . . 10 (𝜑𝐸 ∈ ℝ)
200197, 199readdcld 9921 . . . . . . . . 9 (𝜑 → (1 + 𝐸) ∈ ℝ)
2012eldifbd 3548 . . . . . . . . . . 11 (𝜑 → ¬ 𝑍𝑌)
2028, 201eldifd 3546 . . . . . . . . . 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 39279 . . . . . . . . 9 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℝ)
206200, 205remulcld 9922 . . . . . . . 8 (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) ∈ ℝ)
207 fzfid 12585 . . . . . . . . . 10 (𝜑 → (1...𝑀) ∈ Fin)
208183adantr 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (𝑄𝑆) ∈ ℝ)
209 simpl 471 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → 𝜑)
210 elfznn 12192 . . . . . . . . . . . . 13 (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
211210adantl 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑗 ∈ ℕ)
212 id 22 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
213 ovex 6551 . . . . . . . . . . . . . . . . 17 ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) ∈ V
214213a1i 11 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) ∈ V)
215 hoidmvlelem2.p . . . . . . . . . . . . . . . . 17 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
216215fvmpt2 6181 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ℕ ∧ ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) ∈ V) → (𝑃𝑗) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
217212, 214, 216syl2anc 690 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → (𝑃𝑗) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
218217adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (𝑃𝑗) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
219164adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑌 ∈ Fin)
220166a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → 𝑌𝑊)
221112, 220fssresd 5965 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗) ↾ 𝑌):𝑌⟶ℝ)
222221adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → ((𝐶𝑗) ↾ 𝑌):𝑌⟶ℝ)
223 iftrue 4037 . . . . . . . . . . . . . . . . . . . 20 (𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) = ((𝐶𝑗) ↾ 𝑌))
224223adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) = ((𝐶𝑗) ↾ 𝑌))
225224feq1d 5925 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ ((𝐶𝑗) ↾ 𝑌):𝑌⟶ℝ))
226222, 225mpbird 245 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
227 0red 9893 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦𝑌) → 0 ∈ ℝ)
228 hoidmvlelem2.f . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑦𝑌 ↦ 0)
229227, 228fmptd 6273 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹:𝑌⟶ℝ)
230229ad2antrr 757 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → 𝐹:𝑌⟶ℝ)
231 iffalse 4040 . . . . . . . . . . . . . . . . . . . 20 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) = 𝐹)
232231adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) = 𝐹)
233232feq1d 5925 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ 𝐹:𝑌⟶ℝ))
234230, 233mpbird 245 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
235226, 234pm2.61dan 827 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
236 simpr 475 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
237 fvex 6094 . . . . . . . . . . . . . . . . . . . . . 22 (𝐶𝑗) ∈ V
238237resex 5346 . . . . . . . . . . . . . . . . . . . . 21 ((𝐶𝑗) ↾ 𝑌) ∈ V
239238a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐶𝑗) ↾ 𝑌) ∈ V)
240162, 163ssexd 4724 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑌 ∈ V)
241 mptexg 6363 . . . . . . . . . . . . . . . . . . . . . 22 (𝑌 ∈ V → (𝑦𝑌 ↦ 0) ∈ V)
242240, 241syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑦𝑌 ↦ 0) ∈ V)
243228, 242syl5eqel 2687 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹 ∈ V)
244239, 243ifcld 4076 . . . . . . . . . . . . . . . . . . 19 (𝜑 → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) ∈ V)
245244adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) ∈ V)
246 hoidmvlelem2.j . . . . . . . . . . . . . . . . . . 19 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))
247246fvmpt2 6181 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐽𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))
248236, 245, 247syl2anc 690 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐽𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))
249248feq1d 5925 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((𝐽𝑗):𝑌⟶ℝ ↔ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ))
250235, 249mpbird 245 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐽𝑗):𝑌⟶ℝ)
25131, 220fssresd 5965 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐷𝑗) ↾ 𝑌):𝑌⟶ℝ)
252251adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → ((𝐷𝑗) ↾ 𝑌):𝑌⟶ℝ)
253 iftrue 4037 . . . . . . . . . . . . . . . . . . . 20 (𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) = ((𝐷𝑗) ↾ 𝑌))
254253adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) = ((𝐷𝑗) ↾ 𝑌))
255254feq1d 5925 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ ((𝐷𝑗) ↾ 𝑌):𝑌⟶ℝ))
256252, 255mpbird 245 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
257 iffalse 4040 . . . . . . . . . . . . . . . . . . . 20 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) = 𝐹)
258257adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) = 𝐹)
259258feq1d 5925 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ 𝐹:𝑌⟶ℝ))
260230, 259mpbird 245 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
261256, 260pm2.61dan 827 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
262 fvex 6094 . . . . . . . . . . . . . . . . . . . . . 22 (𝐷𝑗) ∈ V
263262resex 5346 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷𝑗) ↾ 𝑌) ∈ V
264263a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐷𝑗) ↾ 𝑌) ∈ V)
265264, 243ifcld 4076 . . . . . . . . . . . . . . . . . . 19 (𝜑 → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) ∈ V)
266265adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) ∈ V)
267 hoidmvlelem2.k . . . . . . . . . . . . . . . . . . 19 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))
268267fvmpt2 6181 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐾𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))
269236, 266, 268syl2anc 690 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐾𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))
270269feq1d 5925 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((𝐾𝑗):𝑌⟶ℝ ↔ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ))
271261, 270mpbird 245 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐾𝑗):𝑌⟶ℝ)
272161, 219, 250, 271hoidmvcl 39272 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) ∈ (0[,)+∞))
273218, 272eqeltrd 2683 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝑃𝑗) ∈ (0[,)+∞))
274159, 273sseldi 3561 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (𝑃𝑗) ∈ ℝ)
275209, 211, 274syl2anc 690 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (𝑃𝑗) ∈ ℝ)
276208, 275remulcld 9922 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝑄𝑆) · (𝑃𝑗)) ∈ ℝ)
277207, 276fsumrecl 14254 . . . . . . . . 9 (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)) ∈ ℝ)
278200, 277remulcld 9922 . . . . . . . 8 (𝜑 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))) ∈ ℝ)
279206, 278readdcld 9921 . . . . . . 7 (𝜑 → (((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))) ∈ ℝ)
280161, 164, 202, 7, 109, 28, 203, 204, 67sge0hsphoire 39279 . . . . . . . 8 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) ∈ ℝ)
281200, 280remulcld 9922 . . . . . . 7 (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))) ∈ ℝ)
28274, 68syl6eleq 2693 . . . . . . . . . 10 (𝜑𝑆 ∈ {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))})
283 oveq1 6530 . . . . . . . . . . . . 13 (𝑧 = 𝑆 → (𝑧 − (𝐴𝑍)) = (𝑆 − (𝐴𝑍)))
284283oveq2d 6539 . . . . . . . . . . . 12 (𝑧 = 𝑆 → (𝐺 · (𝑧 − (𝐴𝑍))) = (𝐺 · (𝑆 − (𝐴𝑍))))
285 fveq2 6084 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑆 → (𝐻𝑧) = (𝐻𝑆))
286285fveq1d 6086 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑆 → ((𝐻𝑧)‘(𝐷𝑗)) = ((𝐻𝑆)‘(𝐷𝑗)))
287286oveq2d 6539 . . . . . . . . . . . . . . 15 (𝑧 = 𝑆 → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))) = ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))
288287mpteq2dv 4663 . . . . . . . . . . . . . 14 (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))
289288fveq2d 6088 . . . . . . . . . . . . 13 (𝑧 = 𝑆 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))))
290289oveq2d 6539 . . . . . . . . . . . 12 (𝑧 = 𝑆 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))))) = ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))))
291284, 290breq12d 4586 . . . . . . . . . . 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))))))))
292291elrab 3326 . . . . . . . . . 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))))))))
293282, 292sylib 206 . . . . . . . . 9 (𝜑 → (𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∧ (𝐺 · (𝑆 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))))))
294293simprd 477 . . . . . . . 8 (𝜑 → (𝐺 · (𝑆 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))))
295207, 275fsumrecl 14254 . . . . . . . . . . 11 (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝑃𝑗) ∈ ℝ)
296200, 295remulcld 9922 . . . . . . . . . 10 (𝜑 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)) ∈ ℝ)
297 0red 9893 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
29875, 67posdifd 10459 . . . . . . . . . . . 12 (𝜑 → (𝑆 < 𝑄 ↔ 0 < (𝑄𝑆)))
299144, 298mpbid 220 . . . . . . . . . . 11 (𝜑 → 0 < (𝑄𝑆))
300297, 183, 299ltled 10032 . . . . . . . . . 10 (𝜑 → 0 ≤ (𝑄𝑆))
301 hoidmvlelem2.le . . . . . . . . . 10 (𝜑𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)))
302172, 296, 183, 300, 301lemul1ad 10808 . . . . . . . . 9 (𝜑 → (𝐺 · (𝑄𝑆)) ≤ (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)) · (𝑄𝑆)))
303200recnd 9920 . . . . . . . . . . 11 (𝜑 → (1 + 𝐸) ∈ ℂ)
304295recnd 9920 . . . . . . . . . . 11 (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝑃𝑗) ∈ ℂ)
305303, 304, 174mulassd 9915 . . . . . . . . . 10 (𝜑 → (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)) · (𝑄𝑆)) = ((1 + 𝐸) · (Σ𝑗 ∈ (1...𝑀)(𝑃𝑗) · (𝑄𝑆))))
306275recnd 9920 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (𝑃𝑗) ∈ ℂ)
307207, 174, 306fsummulc1 14301 . . . . . . . . . . . 12 (𝜑 → (Σ𝑗 ∈ (1...𝑀)(𝑃𝑗) · (𝑄𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑃𝑗) · (𝑄𝑆)))
308174adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → (𝑄𝑆) ∈ ℂ)
309306, 308mulcomd 9913 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝑃𝑗) · (𝑄𝑆)) = ((𝑄𝑆) · (𝑃𝑗)))
310309sumeq2dv 14223 . . . . . . . . . . . 12 (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑃𝑗) · (𝑄𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))
311307, 310eqtrd 2639 . . . . . . . . . . 11 (𝜑 → (Σ𝑗 ∈ (1...𝑀)(𝑃𝑗) · (𝑄𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))
312311oveq2d 6539 . . . . . . . . . 10 (𝜑 → ((1 + 𝐸) · (Σ𝑗 ∈ (1...𝑀)(𝑃𝑗) · (𝑄𝑆))) = ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))))
313305, 312eqtrd 2639 . . . . . . . . 9 (𝜑 → (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)) · (𝑄𝑆)) = ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))))
314302, 313breqtrd 4599 . . . . . . . 8 (𝜑 → (𝐺 · (𝑄𝑆)) ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))))
315192, 186, 206, 278, 294, 314leadd12dd 38273 . . . . . . 7 (𝜑 → ((𝐺 · (𝑆 − (𝐴𝑍))) + (𝐺 · (𝑄𝑆))) ≤ (((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)))))
316 hoidmvlelem2.m . . . . . . . . . . . . . . . . 17 (𝜑𝑀 ∈ ℕ)
317 nnsplit 38315 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℕ → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
318316, 317syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
319 uncom 3714 . . . . . . . . . . . . . . . . 17 ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))) = ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀))
320319a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))) = ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)))
321318, 320eqtr2d 2640 . . . . . . . . . . . . . . 15 (𝜑 → ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)) = ℕ)
322321eqcomd 2611 . . . . . . . . . . . . . 14 (𝜑 → ℕ = ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)))
323322mpteq1d 4656 . . . . . . . . . . . . 13 (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))) = (𝑗 ∈ ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))
324323fveq2d 6088 . . . . . . . . . . . 12 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) = (Σ^‘(𝑗 ∈ ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))))
325 nfv 1828 . . . . . . . . . . . . 13 𝑗𝜑
326 fvex 6094 . . . . . . . . . . . . . 14 (ℤ‘(𝑀 + 1)) ∈ V
327326a1i 11 . . . . . . . . . . . . 13 (𝜑 → (ℤ‘(𝑀 + 1)) ∈ V)
328 ovex 6551 . . . . . . . . . . . . . 14 (1...𝑀) ∈ V
329328a1i 11 . . . . . . . . . . . . 13 (𝜑 → (1...𝑀) ∈ V)
330 incom 3762 . . . . . . . . . . . . . . 15 ((ℤ‘(𝑀 + 1)) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ‘(𝑀 + 1)))
331 nnuzdisj 38312 . . . . . . . . . . . . . . 15 ((1...𝑀) ∩ (ℤ‘(𝑀 + 1))) = ∅
332330, 331eqtri 2627 . . . . . . . . . . . . . 14 ((ℤ‘(𝑀 + 1)) ∩ (1...𝑀)) = ∅
333332a1i 11 . . . . . . . . . . . . 13 (𝜑 → ((ℤ‘(𝑀 + 1)) ∩ (1...𝑀)) = ∅)
334 icossicc 12083 . . . . . . . . . . . . . 14 (0[,)+∞) ⊆ (0[,]+∞)
335 ssid 3582 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ (0[,)+∞)
336 simpl 471 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → 𝜑)
337316peano2nnd 10880 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 + 1) ∈ ℕ)
338 uznnssnn 11563 . . . . . . . . . . . . . . . . . . 19 ((𝑀 + 1) ∈ ℕ → (ℤ‘(𝑀 + 1)) ⊆ ℕ)
339337, 338syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℤ‘(𝑀 + 1)) ⊆ ℕ)
340339adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → (ℤ‘(𝑀 + 1)) ⊆ ℕ)
341 simpr 475 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → 𝑗 ∈ (ℤ‘(𝑀 + 1)))
342340, 341sseldd 3564 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → 𝑗 ∈ ℕ)
343 snfi 7896 . . . . . . . . . . . . . . . . . . . . 21 {𝑍} ∈ Fin
344343a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → {𝑍} ∈ Fin)
345 unfi 8085 . . . . . . . . . . . . . . . . . . . 20 ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
346164, 344, 345syl2anc 690 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
3477, 346syl5eqel 2687 . . . . . . . . . . . . . . . . . 18 (𝜑𝑊 ∈ Fin)
348347adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
349 eleq1 2671 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑙 → (𝑗𝑌𝑙𝑌))
350 fveq2 6084 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑙 → (𝑐𝑗) = (𝑐𝑙))
351350breq1d 4583 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 𝑙 → ((𝑐𝑗) ≤ 𝑥 ↔ (𝑐𝑙) ≤ 𝑥))
352351, 350ifbieq1d 4054 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑙 → if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥) = if((𝑐𝑙) ≤ 𝑥, (𝑐𝑙), 𝑥))
353349, 350, 352ifbieq12d 4058 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑙 → if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)) = if(𝑙𝑌, (𝑐𝑙), if((𝑐𝑙) ≤ 𝑥, (𝑐𝑙), 𝑥)))
354353cbvmptv 4668 . . . . . . . . . . . . . . . . . . . . 21 (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))) = (𝑙𝑊 ↦ if(𝑙𝑌, (𝑐𝑙), if((𝑐𝑙) ≤ 𝑥, (𝑐𝑙), 𝑥)))
355354mpteq2i 4659 . . . . . . . . . . . . . . . . . . . 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))))
356355mpteq2i 4659 . . . . . . . . . . . . . . . . . . 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)))))
357204, 356eqtri 2627 . . . . . . . . . . . . . . . . . 18 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑙𝑊 ↦ if(𝑙𝑌, (𝑐𝑙), if((𝑐𝑙) ≤ 𝑥, (𝑐𝑙), 𝑥)))))
35875adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
359357, 358, 348, 31hsphoif 39266 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → ((𝐻𝑆)‘(𝐷𝑗)):𝑊⟶ℝ)
360161, 348, 112, 359hoidmvcl 39272 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,)+∞))
361336, 342, 360syl2anc 690 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,)+∞))
362335, 361sseldi 3561 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,)+∞))
363334, 362sseldi 3561 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,]+∞))
364209, 211, 360syl2anc 690 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,)+∞))
365334, 364sseldi 3561 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,]+∞))
366325, 327, 329, 333, 363, 365sge0splitmpt 39104 . . . . . . . . . . . 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)))))))
367 nnex 10869 . . . . . . . . . . . . . . 15 ℕ ∈ V
368367a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ℕ ∈ V)
369334, 360sseldi 3561 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ (0[,]+∞))
370325, 368, 369, 205, 339sge0ssrempt 39098 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℝ)
37118a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (1...𝑀) ⊆ ℕ)
372325, 368, 369, 205, 371sge0ssrempt 39098 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℝ)
373 rexadd 11892 . . . . . . . . . . . . 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)))))))
374370, 372, 373syl2anc 690 . . . . . . . . . . . 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)))))))
375324, 366, 3743eqtrd 2643 . . . . . . . . . . 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)))))))
376375oveq2d 6539 . . . . . . . . . 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))))))))
377376oveq1d 6538 . . . . . . . . 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)))))
378375, 205eqeltrrd 2684 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) ∈ ℝ)
379378recnd 9920 . . . . . . . . . . 11 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗)))))) ∈ ℂ)
380277recnd 9920 . . . . . . . . . . 11 (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗)) ∈ ℂ)
381303, 379, 380adddid 9916 . . . . . . . . . 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)))))
382381eqcomd 2611 . . . . . . . . 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)))))
383370recnd 9920 . . . . . . . . . . . 12 (𝜑 → (Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℂ)
384372recnd 9920 . . . . . . . . . . . 12 (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℂ)
385383, 384, 380addassd 9914 . . . . . . . . . . 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)))))
386207, 364sge0fsummpt 39083 . . . . . . . . . . . . . 14 (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) = Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))
387386oveq1d 6538 . . . . . . . . . . . . 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))))
388 ax-resscn 9845 . . . . . . . . . . . . . . . . . 18 ℝ ⊆ ℂ
389159, 388sstri 3572 . . . . . . . . . . . . . . . . 17 (0[,)+∞) ⊆ ℂ
390389, 360sseldi 3561 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ ℂ)
391209, 211, 390syl2anc 690 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ ℂ)
392183adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (𝑄𝑆) ∈ ℝ)
393392, 274remulcld 9922 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → ((𝑄𝑆) · (𝑃𝑗)) ∈ ℝ)
394393recnd 9920 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((𝑄𝑆) · (𝑃𝑗)) ∈ ℂ)
395211, 394syldan 485 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝑄𝑆) · (𝑃𝑗)) ∈ ℂ)
396207, 391, 395fsumadd 14259 . . . . . . . . . . . . . 14 (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = (Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))))
397396eqcomd 2611 . . . . . . . . . . . . 13 (𝜑 → (Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))) = Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))))
398387, 397eqtrd 2639 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄𝑆) · (𝑃𝑗))) = Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))))
399398oveq2d 6539 . . . . . . . . . . 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)))))
400385, 399eqtrd 2639 . . . . . . . . . 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)))))
401400oveq2d 6539 . . . . . . . . 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))))))
402377, 382, 4013eqtrd 2643 . . . . . . . 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))))))
403159, 360sseldi 3561 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ∈ ℝ)
404403, 393readdcld 9921 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ∈ ℝ)
405209, 211, 404syl2anc 690 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ∈ ℝ)
406207, 405fsumrecl 14254 . . . . . . . . . 10 (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ∈ ℝ)
407370, 406readdcld 9921 . . . . . . . . 9 (𝜑 → ((Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗)))) ∈ ℝ)
408 0le1 10396 . . . . . . . . . . 11 0 ≤ 1
409408a1i 11 . . . . . . . . . 10 (𝜑 → 0 ≤ 1)
410198rpge0d 11704 . . . . . . . . . 10 (𝜑 → 0 ≤ 𝐸)
411197, 199, 409, 410addge0d 10448 . . . . . . . . 9 (𝜑 → 0 ≤ (1 + 𝐸))
41267adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑄 ∈ ℝ)
413357, 412, 348, 31hsphoif 39266 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝐻𝑄)‘(𝐷𝑗)):𝑊⟶ℝ)
414161, 348, 112, 413hoidmvcl 39272 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ (0[,)+∞))
415334, 414sseldi 3561 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ (0[,]+∞))
416325, 368, 415, 280, 339sge0ssrempt 39098 . . . . . . . . . . 11 (𝜑 → (Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) ∈ ℝ)
417159, 414sseldi 3561 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ ℝ)
418209, 211, 417syl2anc 690 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ ℝ)
419207, 418fsumrecl 14254 . . . . . . . . . . 11 (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ ℝ)
420336, 342, 415syl2anc 690 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ (0[,]+∞))
421202adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊𝑌))
42275, 67, 144ltled 10032 . . . . . . . . . . . . . . 15 (𝜑𝑆𝑄)
423422adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → 𝑆𝑄)
424161, 348, 421, 7, 358, 412, 423, 357, 112, 31hsphoidmvle2 39275 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
425336, 342, 424syl2anc 690 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (ℤ‘(𝑀 + 1))) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
426325, 327, 363, 420, 425sge0lempt 39103 . . . . . . . . . . 11 (𝜑 → (Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ≤ (Σ^‘(𝑗 ∈ (ℤ‘(𝑀 + 1)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))))
427209adantr 479 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) = 0) → 𝜑)
428211adantr 479 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) = 0) → 𝑗 ∈ ℕ)
429 simpr 475 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) = 0) → (𝑃𝑗) = 0)
430 oveq2 6531 . . . . . . . . . . . . . . . . . . 19 ((𝑃𝑗) = 0 → ((𝑄𝑆) · (𝑃𝑗)) = ((𝑄𝑆) · 0))
431430adantl 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → ((𝑄𝑆) · (𝑃𝑗)) = ((𝑄𝑆) · 0))
432174mul01d 10082 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑄𝑆) · 0) = 0)
433432ad2antrr 757 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → ((𝑄𝑆) · 0) = 0)
434431, 433eqtrd 2639 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → ((𝑄𝑆) · (𝑃𝑗)) = 0)
435434oveq2d 6539 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + 0))
436390addid1d 10083 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + 0) = ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))
437436adantr 479 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + 0) = ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))
438435, 437eqtrd 2639 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))
439424adantr 479 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
440438, 439eqbrtrd 4595 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) = 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
441427, 428, 429, 440syl21anc 1316 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) = 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
442 simpl 471 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃𝑗) = 0) → (𝜑𝑗 ∈ (1...𝑀)))
443 neqne 2785 . . . . . . . . . . . . . . 15 (¬ (𝑃𝑗) = 0 → (𝑃𝑗) ≠ 0)
444443adantl 480 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃𝑗) = 0) → (𝑃𝑗) ≠ 0)
445405adantr 479 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ∈ ℝ)
446209adantr 479 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝜑)
447211adantr 479 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑗 ∈ ℕ)
448 simpr 475 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (𝑃𝑗) ≠ 0)
4492adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ ℕ) → 𝑍 ∈ (𝑋𝑌))
450201adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ ℕ) → ¬ 𝑍𝑌)
451 eqid 2605 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘)))
452161, 219, 449, 450, 7, 112, 359, 451hoiprodp1 39278 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) = (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍)))))
453452adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) = (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍)))))
454218adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝑃𝑗) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
455219adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → 𝑌 ∈ Fin)
456218adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝑃𝑗) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
457 fveq2 6084 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑌 = ∅ → (𝐿𝑌) = (𝐿‘∅))
458457oveqd 6540 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑌 = ∅ → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) = ((𝐽𝑗)(𝐿‘∅)(𝐾𝑗)))
459458adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) = ((𝐽𝑗)(𝐿‘∅)(𝐾𝑗)))
460250adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐽𝑗):𝑌⟶ℝ)
461 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑌 = ∅ → 𝑌 = ∅)
462461eqcomd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑌 = ∅ → ∅ = 𝑌)
463462adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ∅ = 𝑌)
464463feq2d 5926 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽𝑗):∅⟶ℝ ↔ (𝐽𝑗):𝑌⟶ℝ))
465460, 464mpbird 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐽𝑗):∅⟶ℝ)
466271adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐾𝑗):𝑌⟶ℝ)
467463feq2d 5926 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐾𝑗):∅⟶ℝ ↔ (𝐾𝑗):𝑌⟶ℝ))
468466, 467mpbird 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐾𝑗):∅⟶ℝ)
469161, 465, 468hoidmv0val 39273 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽𝑗)(𝐿‘∅)(𝐾𝑗)) = 0)
470456, 459, 4693eqtrd 2643 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝑃𝑗) = 0)
471470adantlr 746 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑌 = ∅) → (𝑃𝑗) = 0)
472 neneq 2783 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃𝑗) ≠ 0 → ¬ (𝑃𝑗) = 0)
473472ad2antlr 758 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑌 = ∅) → ¬ (𝑃𝑗) = 0)
474471, 473pm2.65da 597 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ¬ 𝑌 = ∅)
475474neqned 2784 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → 𝑌 ≠ ∅)
476250adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝐽𝑗):𝑌⟶ℝ)
477271adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝐾𝑗):𝑌⟶ℝ)
478161, 455, 475, 476, 477hoidmvn0val 39274 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) = ∏𝑘𝑌 (vol‘(((𝐽𝑗)‘𝑘)[,)((𝐾𝑗)‘𝑘))))
479248adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝐽𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))
480218adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝑃𝑗) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
481248adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝐽𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))
482481, 232eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝐽𝑗) = 𝐹)
483269adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝐾𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))
484483, 258eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝐾𝑗) = 𝐹)
485482, 484oveq12d 6541 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) = (𝐹(𝐿𝑌)𝐹))
486161, 164, 229hoidmvval0b 39280 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → (𝐹(𝐿𝑌)𝐹) = 0)
487486ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝐹(𝐿𝑌)𝐹) = 0)
488480, 485, 4873eqtrd 2643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝑃𝑗) = 0)
489488adantlr 746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → (𝑃𝑗) = 0)
490472ad2antlr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → ¬ (𝑃𝑗) = 0)
491489, 490condan 830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)))
492491iftrued 4039 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹) = ((𝐶𝑗) ↾ 𝑌))
493479, 492eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝐽𝑗) = ((𝐶𝑗) ↾ 𝑌))
494493fveq1d 6086 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐽𝑗)‘𝑘) = (((𝐶𝑗) ↾ 𝑌)‘𝑘))
495494adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → ((𝐽𝑗)‘𝑘) = (((𝐶𝑗) ↾ 𝑌)‘𝑘))
496 fvres 6098 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘𝑌 → (((𝐶𝑗) ↾ 𝑌)‘𝑘) = ((𝐶𝑗)‘𝑘))
497496adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → (((𝐶𝑗) ↾ 𝑌)‘𝑘) = ((𝐶𝑗)‘𝑘))
498495, 497eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → ((𝐽𝑗)‘𝑘) = ((𝐶𝑗)‘𝑘))
499269adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝐾𝑗) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))
500491, 253syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹) = ((𝐷𝑗) ↾ 𝑌))
501499, 500eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝐾𝑗) = ((𝐷𝑗) ↾ 𝑌))
502501fveq1d 6086 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐾𝑗)‘𝑘) = (((𝐷𝑗) ↾ 𝑌)‘𝑘))
503502adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → ((𝐾𝑗)‘𝑘) = (((𝐷𝑗) ↾ 𝑌)‘𝑘))
504 fvres 6098 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘𝑌 → (((𝐷𝑗) ↾ 𝑌)‘𝑘) = ((𝐷𝑗)‘𝑘))
505504adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → (((𝐷𝑗) ↾ 𝑌)‘𝑘) = ((𝐷𝑗)‘𝑘))
506503, 505eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → ((𝐾𝑗)‘𝑘) = ((𝐷𝑗)‘𝑘))
507498, 506oveq12d 6541 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → (((𝐽𝑗)‘𝑘)[,)((𝐾𝑗)‘𝑘)) = (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
508507fveq2d 6088 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ 𝑘𝑌) → (vol‘(((𝐽𝑗)‘𝑘)[,)((𝐾𝑗)‘𝑘))) = (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
509508prodeq2dv 14434 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐽𝑗)‘𝑘)[,)((𝐾𝑗)‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
510478, 509eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
511358adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → 𝑆 ∈ ℝ)
512348adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → 𝑊 ∈ Fin)
51331adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (𝐷𝑗):𝑊⟶ℝ)
514 elun1 3737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘𝑌𝑘 ∈ (𝑌 ∪ {𝑍}))
515514, 7syl6eleqr 2694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘𝑌𝑘𝑊)
516515adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → 𝑘𝑊)
517357, 511, 512, 513, 516hsphoival 39269 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (((𝐻𝑆)‘(𝐷𝑗))‘𝑘) = if(𝑘𝑌, ((𝐷𝑗)‘𝑘), if(((𝐷𝑗)‘𝑘) ≤ 𝑆, ((𝐷𝑗)‘𝑘), 𝑆)))
518 iftrue 4037 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘𝑌 → if(𝑘𝑌, ((𝐷𝑗)‘𝑘), if(((𝐷𝑗)‘𝑘) ≤ 𝑆, ((𝐷𝑗)‘𝑘), 𝑆)) = ((𝐷𝑗)‘𝑘))
519518adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → if(𝑘𝑌, ((𝐷𝑗)‘𝑘), if(((𝐷𝑗)‘𝑘) ≤ 𝑆, ((𝐷𝑗)‘𝑘), 𝑆)) = ((𝐷𝑗)‘𝑘))
520517, 519eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (((𝐻𝑆)‘(𝐷𝑗))‘𝑘) = ((𝐷𝑗)‘𝑘))
521520oveq2d 6539 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘)) = (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
522521fveq2d 6088 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) = (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
523522prodeq2dv 14434 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
524523eqcomd 2611 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))))
525524adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))))
526454, 510, 5253eqtrrd 2644 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) = (𝑃𝑗))
527357, 358, 348, 31, 32hsphoival 39269 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑗 ∈ ℕ) → (((𝐻𝑆)‘(𝐷𝑗))‘𝑍) = if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆)))
528201iffalsed 4042 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆)) = if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆))
529528adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑗 ∈ ℕ) → if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆)) = if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆))
530527, 529eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ ℕ) → (((𝐻𝑆)‘(𝐷𝑗))‘𝑍) = if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆))
531530oveq2d 6539 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ ℕ) → (((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍)) = (((𝐶𝑗)‘𝑍)[,)if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆)))
532531adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍)) = (((𝐶𝑗)‘𝑍)[,)if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆)))
533113rexrd 9941 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)‘𝑍) ∈ ℝ*)
534533adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)‘𝑍) ∈ ℝ*)
53533rexrd 9941 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑗 ∈ ℕ) → ((𝐷𝑗)‘𝑍) ∈ ℝ*)
536535adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ ℝ*)
537 icoltub 38379 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝐶𝑗)‘𝑍) ∈ ℝ* ∧ ((𝐷𝑗)‘𝑍) ∈ ℝ*𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → 𝑆 < ((𝐷𝑗)‘𝑍))
538534, 536, 491, 537syl3anc 1317 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → 𝑆 < ((𝐷𝑗)‘𝑍))
539358adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → 𝑆 ∈ ℝ)
54033adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ ℝ)
541539, 540ltnled 10031 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (𝑆 < ((𝐷𝑗)‘𝑍) ↔ ¬ ((𝐷𝑗)‘𝑍) ≤ 𝑆))
542538, 541mpbid 220 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ¬ ((𝐷𝑗)‘𝑍) ≤ 𝑆)
543542iffalsed 4042 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆) = 𝑆)
544543oveq2d 6539 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)‘𝑍)[,)if(((𝐷𝑗)‘𝑍) ≤ 𝑆, ((𝐷𝑗)‘𝑍), 𝑆)) = (((𝐶𝑗)‘𝑍)[,)𝑆))
545532, 544eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍)) = (((𝐶𝑗)‘𝑍)[,)𝑆))
546545fveq2d 6088 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍))) = (vol‘(((𝐶𝑗)‘𝑍)[,)𝑆)))
547 volico 38676 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐶𝑗)‘𝑍) ∈ ℝ ∧ 𝑆 ∈ ℝ) → (vol‘(((𝐶𝑗)‘𝑍)[,)𝑆)) = if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0))
548113, 539, 547syl2an 492 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ℕ) ∧ ((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0)) → (vol‘(((𝐶𝑗)‘𝑍)[,)𝑆)) = if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0))
549548anabss5 852 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (vol‘(((𝐶𝑗)‘𝑍)[,)𝑆)) = if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0))
550 iftrue 4037 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐶𝑗)‘𝑍) < 𝑆 → if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶𝑗)‘𝑍)))
551550adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ((𝐶𝑗)‘𝑍) < 𝑆) → if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶𝑗)‘𝑍)))
552 iffalse 4040 . . . . . . . . . . . . . . . . . . . . . . . . 25 (¬ ((𝐶𝑗)‘𝑍) < 𝑆 → if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0) = 0)
553552adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0) = 0)
554 simpll 785 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → (𝜑𝑗 ∈ ℕ))
555 icogelb 12048 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝐶𝑗)‘𝑍) ∈ ℝ* ∧ ((𝐷𝑗)‘𝑍) ∈ ℝ*𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))) → ((𝐶𝑗)‘𝑍) ≤ 𝑆)
556534, 536, 491, 555syl3anc 1317 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)‘𝑍) ≤ 𝑆)
557556adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → ((𝐶𝑗)‘𝑍) ≤ 𝑆)
558 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → ¬ ((𝐶𝑗)‘𝑍) < 𝑆)
559557, 558jca 552 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → (((𝐶𝑗)‘𝑍) ≤ 𝑆 ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆))
560554, 113syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → ((𝐶𝑗)‘𝑍) ∈ ℝ)
561554, 358syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → 𝑆 ∈ ℝ)
562560, 561eqleltd 10028 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → (((𝐶𝑗)‘𝑍) = 𝑆 ↔ (((𝐶𝑗)‘𝑍) ≤ 𝑆 ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆)))
563559, 562mpbird 245 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → ((𝐶𝑗)‘𝑍) = 𝑆)
564 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐶𝑗)‘𝑍) = 𝑆 → ((𝐶𝑗)‘𝑍) = 𝑆)
565564eqcomd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐶𝑗)‘𝑍) = 𝑆𝑆 = ((𝐶𝑗)‘𝑍))
566565oveq1d 6538 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐶𝑗)‘𝑍) = 𝑆 → (𝑆 − ((𝐶𝑗)‘𝑍)) = (((𝐶𝑗)‘𝑍) − ((𝐶𝑗)‘𝑍)))
567566adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ ℕ) ∧ ((𝐶𝑗)‘𝑍) = 𝑆) → (𝑆 − ((𝐶𝑗)‘𝑍)) = (((𝐶𝑗)‘𝑍) − ((𝐶𝑗)‘𝑍)))
568388, 113sseldi 3561 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)‘𝑍) ∈ ℂ)
569568subidd 10227 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑗 ∈ ℕ) → (((𝐶𝑗)‘𝑍) − ((𝐶𝑗)‘𝑍)) = 0)
570569adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ ℕ) ∧ ((𝐶𝑗)‘𝑍) = 𝑆) → (((𝐶𝑗)‘𝑍) − ((𝐶𝑗)‘𝑍)) = 0)
571567, 570eqtr2d 2640 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ ℕ) ∧ ((𝐶𝑗)‘𝑍) = 𝑆) → 0 = (𝑆 − ((𝐶𝑗)‘𝑍)))
572554, 563, 571syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → 0 = (𝑆 − ((𝐶𝑗)‘𝑍)))
573553, 572eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐶𝑗)‘𝑍) < 𝑆) → if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶𝑗)‘𝑍)))
574551, 573pm2.61dan 827 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → if(((𝐶𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶𝑗)‘𝑍)))
575546, 549, 5743eqtrd 2643 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍))) = (𝑆 − ((𝐶𝑗)‘𝑍)))
576526, 575oveq12d 6541 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑆)‘(𝐷𝑗))‘𝑍)))) = ((𝑃𝑗) · (𝑆 − ((𝐶𝑗)‘𝑍))))
577389, 273sseldi 3561 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ ℕ) → (𝑃𝑗) ∈ ℂ)
578358, 113resubcld 10305 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ ℕ) → (𝑆 − ((𝐶𝑗)‘𝑍)) ∈ ℝ)
579578recnd 9920 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ ℕ) → (𝑆 − ((𝐶𝑗)‘𝑍)) ∈ ℂ)
580577, 579mulcomd 9913 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → ((𝑃𝑗) · (𝑆 − ((𝐶𝑗)‘𝑍))) = ((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
581580adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝑃𝑗) · (𝑆 − ((𝐶𝑗)‘𝑍))) = ((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
582453, 576, 5813eqtrd 2643 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) = ((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
583582oveq1d 6538 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = (((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)) + ((𝑄𝑆) · (𝑃𝑗))))
584174adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → (𝑄𝑆) ∈ ℂ)
585579, 584, 577adddird 9917 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) · (𝑃𝑗)) = (((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)) + ((𝑄𝑆) · (𝑃𝑗))))
586585eqcomd 2611 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → (((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)) + ((𝑄𝑆) · (𝑃𝑗))) = (((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) · (𝑃𝑗)))
587586adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝑆 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)) + ((𝑄𝑆) · (𝑃𝑗))) = (((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) · (𝑃𝑗)))
588579, 584addcomd 10085 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → ((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) = ((𝑄𝑆) + (𝑆 − ((𝐶𝑗)‘𝑍))))
589153adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ ℕ) → 𝑄 ∈ ℂ)
590154adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ ℕ) → 𝑆 ∈ ℂ)
591589, 590, 568npncand 10263 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → ((𝑄𝑆) + (𝑆 − ((𝐶𝑗)‘𝑍))) = (𝑄 − ((𝐶𝑗)‘𝑍)))
592588, 591eqtrd 2639 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → ((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) = (𝑄 − ((𝐶𝑗)‘𝑍)))
593592oveq1d 6538 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → (((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) · (𝑃𝑗)) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
594593adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝑆 − ((𝐶𝑗)‘𝑍)) + (𝑄𝑆)) · (𝑃𝑗)) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
595583, 587, 5943eqtrd 2643 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
596446, 447, 448, 595syl21anc 1316 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
597 eqid 2605 . . . . . . . . . . . . . . . . . . . 20 𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘)))
598161, 219, 32, 450, 7, 112, 413, 597hoiprodp1 39278 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) = (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍)))))
599209, 211, 598syl2anc 690 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) = (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍)))))
600599adantr 479 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) = (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍)))))
601510eqcomd 2611 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))) = ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))
602412adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → 𝑄 ∈ ℝ)
603357, 602, 512, 513, 516hsphoival 39269 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (((𝐻𝑄)‘(𝐷𝑗))‘𝑘) = if(𝑘𝑌, ((𝐷𝑗)‘𝑘), if(((𝐷𝑗)‘𝑘) ≤ 𝑄, ((𝐷𝑗)‘𝑘), 𝑄)))
604 iftrue 4037 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘𝑌 → if(𝑘𝑌, ((𝐷𝑗)‘𝑘), if(((𝐷𝑗)‘𝑘) ≤ 𝑄, ((𝐷𝑗)‘𝑘), 𝑄)) = ((𝐷𝑗)‘𝑘))
605604adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → if(𝑘𝑌, ((𝐷𝑗)‘𝑘), if(((𝐷𝑗)‘𝑘) ≤ 𝑄, ((𝐷𝑗)‘𝑘), 𝑄)) = ((𝐷𝑗)‘𝑘))
606603, 605eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (((𝐻𝑄)‘(𝐷𝑗))‘𝑘) = ((𝐷𝑗)‘𝑘))
607606oveq2d 6539 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘)) = (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
608607fveq2d 6088 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑌) → (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) = (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
609608prodeq2dv 14434 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
610609adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) = ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
611601, 610, 4543eqtr4d 2649 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) = (𝑃𝑗))
612446, 447, 448, 611syl21anc 1316 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) = (𝑃𝑗))
613357, 412, 348, 31, 32hsphoival 39269 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → (((𝐻𝑄)‘(𝐷𝑗))‘𝑍) = if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄)))
614211, 613syldan 485 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (1...𝑀)) → (((𝐻𝑄)‘(𝐷𝑗))‘𝑍) = if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄)))
615614adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐻𝑄)‘(𝐷𝑗))‘𝑍) = if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄)))
616201iffalsed 4042 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄)) = if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄))
617616ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → if(𝑍𝑌, ((𝐷𝑗)‘𝑍), if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄)) = if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄))
618211, 33syldan 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐷𝑗)‘𝑍) ∈ ℝ)
619618adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ((𝐷𝑗)‘𝑍) = 𝑄) → ((𝐷𝑗)‘𝑍) ∈ ℝ)
620 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ((𝐷𝑗)‘𝑍) = 𝑄) → ((𝐷𝑗)‘𝑍) = 𝑄)
621619, 620eqled 9987 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ((𝐷𝑗)‘𝑍) = 𝑄) → ((𝐷𝑗)‘𝑍) ≤ 𝑄)
622621iftrued 4039 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ((𝐷𝑗)‘𝑍) = 𝑄) → if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄) = ((𝐷𝑗)‘𝑍))
623622, 620eqtrd 2639 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ((𝐷𝑗)‘𝑍) = 𝑄) → if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄) = 𝑄)
624623adantlr 746 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ((𝐷𝑗)‘𝑍) = 𝑄) → if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄) = 𝑄)
62567adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑄 ∈ ℝ)
626625adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → 𝑄 ∈ ℝ)
627626adantlr 746 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → 𝑄 ∈ ℝ)
628618adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → ((𝐷𝑗)‘𝑍) ∈ ℝ)
629628adantlr 746 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → ((𝐷𝑗)‘𝑍) ∈ ℝ)
63040a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑄 = inf(𝑉, ℝ, < ))
631446, 39syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑉 ⊆ ℝ)
632148ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ∃𝑥𝑉𝑦𝑉 𝑥𝑦)
633 simplr 787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑗 ∈ (1...𝑀))
634210, 491sylanl2 680 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)))
635633, 634jca 552 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (𝑗 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))))
636 rabid 3090 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑗 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↔ (𝑗 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))))
637635, 636sylibr 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑗 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))})
638 eqidd 2606 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) = ((𝐷𝑗)‘𝑍))
639 fveq2 6084 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑖 = 𝑗 → (𝐷𝑖) = (𝐷𝑗))
640639fveq1d 6086 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑖 = 𝑗 → ((𝐷𝑖)‘𝑍) = ((𝐷𝑗)‘𝑍))
641640eqeq2d 2615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑖 = 𝑗 → (((𝐷𝑗)‘𝑍) = ((𝐷𝑖)‘𝑍) ↔ ((𝐷𝑗)‘𝑍) = ((𝐷𝑗)‘𝑍)))
642641rspcev 3277 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
643637, 638, 642syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ((𝐷𝑗)‘𝑍) = ((𝐷𝑖)‘𝑍))
644 fvex 6094 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐷𝑗)‘𝑍) ∈ V
645644a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ V)
64616, 643, 645elrnmptd 38160 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍)))
647646, 14syl6eleqr 2694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ 𝑂)
648 elun2 3738 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝐷𝑗)‘𝑍) ∈ 𝑂 → ((𝐷𝑗)‘𝑍) ∈ ({(𝐵𝑍)} ∪ 𝑂))
649647, 648syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ ({(𝐵𝑍)} ∪ 𝑂))
65059a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ({(𝐵𝑍)} ∪ 𝑂) = 𝑉)
651649, 650eleqtrd 2685 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐷𝑗)‘𝑍) ∈ 𝑉)
652 lbinfle 10823 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑉 ⊆ ℝ ∧ ∃𝑥𝑉𝑦𝑉 𝑥𝑦 ∧ ((𝐷𝑗)‘𝑍) ∈ 𝑉) → inf(𝑉, ℝ, < ) ≤ ((𝐷𝑗)‘𝑍))
653631, 632, 651, 652syl3anc 1317 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → inf(𝑉, ℝ, < ) ≤ ((𝐷𝑗)‘𝑍))
654630, 653eqbrtrd 4595 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑄 ≤ ((𝐷𝑗)‘𝑍))
655654adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → 𝑄 ≤ ((𝐷𝑗)‘𝑍))
656 neqne 2785 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (¬ ((𝐷𝑗)‘𝑍) = 𝑄 → ((𝐷𝑗)‘𝑍) ≠ 𝑄)
657656adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → ((𝐷𝑗)‘𝑍) ≠ 𝑄)
658627, 629, 655, 657leneltd 10038 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → 𝑄 < ((𝐷𝑗)‘𝑍))
659627, 629ltnled 10031 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → (𝑄 < ((𝐷𝑗)‘𝑍) ↔ ¬ ((𝐷𝑗)‘𝑍) ≤ 𝑄))
660658, 659mpbid 220 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → ¬ ((𝐷𝑗)‘𝑍) ≤ 𝑄)
661660iffalsed 4042 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) ∧ ¬ ((𝐷𝑗)‘𝑍) = 𝑄) → if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄) = 𝑄)
662624, 661pm2.61dan 827 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → if(((𝐷𝑗)‘𝑍) ≤ 𝑄, ((𝐷𝑗)‘𝑍), 𝑄) = 𝑄)
663615, 617, 6623eqtrd 2643 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐻𝑄)‘(𝐷𝑗))‘𝑍) = 𝑄)
664663oveq2d 6539 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍)) = (((𝐶𝑗)‘𝑍)[,)𝑄))
665664fveq2d 6088 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍))) = (vol‘(((𝐶𝑗)‘𝑍)[,)𝑄)))
666209, 211, 113syl2anc 690 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)‘𝑍) ∈ ℝ)
667666adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)‘𝑍) ∈ ℝ)
668446, 67syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑄 ∈ ℝ)
669 volico 38676 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶𝑗)‘𝑍) ∈ ℝ ∧ 𝑄 ∈ ℝ) → (vol‘(((𝐶𝑗)‘𝑍)[,)𝑄)) = if(((𝐶𝑗)‘𝑍) < 𝑄, (𝑄 − ((𝐶𝑗)‘𝑍)), 0))
670667, 668, 669syl2anc 690 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (vol‘(((𝐶𝑗)‘𝑍)[,)𝑄)) = if(((𝐶𝑗)‘𝑍) < 𝑄, (𝑄 − ((𝐶𝑗)‘𝑍)), 0))
671446, 75syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑆 ∈ ℝ)
672446, 447, 448, 556syl21anc 1316 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)‘𝑍) ≤ 𝑆)
673446, 144syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → 𝑆 < 𝑄)
674667, 671, 668, 672, 673lelttrd 10042 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)‘𝑍) < 𝑄)
675674iftrued 4039 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → if(((𝐶𝑗)‘𝑍) < 𝑄, (𝑄 − ((𝐶𝑗)‘𝑍)), 0) = (𝑄 − ((𝐶𝑗)‘𝑍)))
676665, 670, 6753eqtrd 2643 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍))) = (𝑄 − ((𝐶𝑗)‘𝑍)))
677612, 676oveq12d 6541 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (∏𝑘𝑌 (vol‘(((𝐶𝑗)‘𝑘)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑘))) · (vol‘(((𝐶𝑗)‘𝑍)[,)(((𝐻𝑄)‘(𝐷𝑗))‘𝑍)))) = ((𝑃𝑗) · (𝑄 − ((𝐶𝑗)‘𝑍))))
678209, 153syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑄 ∈ ℂ)
679388, 666sseldi 3561 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)‘𝑍) ∈ ℂ)
680678, 679subcld 10239 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (1...𝑀)) → (𝑄 − ((𝐶𝑗)‘𝑍)) ∈ ℂ)
681306, 680mulcomd 9913 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝑃𝑗) · (𝑄 − ((𝐶𝑗)‘𝑍))) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
682681adantr 479 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝑃𝑗) · (𝑄 − ((𝐶𝑗)‘𝑍))) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
683600, 677, 6823eqtrd 2643 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) = ((𝑄 − ((𝐶𝑗)‘𝑍)) · (𝑃𝑗)))
684596, 683eqtr4d 2642 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) = ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
685445, 684eqled 9987 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ (𝑃𝑗) ≠ 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
686442, 444, 685syl2anc 690 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃𝑗) = 0) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
687441, 686pm2.61dan 827 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
688207, 405, 418, 687fsumle 14314 . . . . . . . . . . 11 (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))) + ((𝑄𝑆) · (𝑃𝑗))) ≤ Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
689370, 406, 416, 419, 426, 688leadd12dd 38273 . . . . . . . . . 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)))))
690322mpteq1d 4656 . . . . . . . . . . . . 13 (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))) = (𝑗 ∈ ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))
691690fveq2d 6088 . . . . . . . . . . . 12 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) = (Σ^‘(𝑗 ∈ ((ℤ‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))))
692211, 415syldan 485 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ (0[,]+∞))
693325, 327, 329, 333, 420, 692sge0splitmpt 39104 . . . . . . . . . . . 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)))))))
694691, 693eqtrd 2639 . . . . . . . . . . 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)))))))
695209, 211, 414syl2anc 690 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))) ∈ (0[,)+∞))
696207, 695sge0fsummpt 39083 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) = Σ𝑗 ∈ (1...𝑀)((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
697696, 419eqeltrd 2683 . . . . . . . . . . . 12 (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))) ∈ ℝ)
698 rexadd 11892 . . . . . . . . . . . 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)))))))
699416, 697, 698syl2anc 690 . . . . . . . . . . 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)))))))
700696oveq2d 6539 . . . . . . . . . . 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)))))
701694, 699, 7003eqtrrd 2644 . . . . . . . . . 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))))))
702689, 701breqtrd 4599 . . . . . . . . 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))))))
703407, 280, 200, 411, 702lemul2ad 10809 . . . . . . . 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)))))))
704402, 703eqbrtrd 4595 . . . . . . 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)))))))
705196, 279, 281, 315, 704letrd 10041 . . . . . 6 (𝜑 → ((𝐺 · (𝑆 − (𝐴𝑍))) + (𝐺 · (𝑄𝑆))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))))
706180, 705eqbrtrd 4595 . . . . 5 (𝜑 → (𝐺 · (𝑄 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))))
707152, 706jca 552 . . . 4 (𝜑 → (𝑄 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∧ (𝐺 · (𝑄 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))))))
708 oveq1 6530 . . . . . . 7 (𝑧 = 𝑄 → (𝑧 − (𝐴𝑍)) = (𝑄 − (𝐴𝑍)))
709708oveq2d 6539 . . . . . 6 (𝑧 = 𝑄 → (𝐺 · (𝑧 − (𝐴𝑍))) = (𝐺 · (𝑄 − (𝐴𝑍))))
710 fveq2 6084 . . . . . . . . . . 11 (𝑧 = 𝑄 → (𝐻𝑧) = (𝐻𝑄))
711710fveq1d 6086 . . . . . . . . . 10 (𝑧 = 𝑄 → ((𝐻𝑧)‘(𝐷𝑗)) = ((𝐻𝑄)‘(𝐷𝑗)))
712711oveq2d 6539 . . . . . . . . 9 (𝑧 = 𝑄 → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))) = ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))
713712mpteq2dv 4663 . . . . . . . 8 (𝑧 = 𝑄 → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))
714713fveq2d 6088 . . . . . . 7 (𝑧 = 𝑄 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗))))))
715714oveq2d 6539 . . . . . 6 (𝑧 = 𝑄 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))))) = ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑄)‘(𝐷𝑗)))))))
716709, 715breq12d 4586 . . . . 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))))))))
717716elrab 3326 . . . 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))))))))
718707, 717sylibr 222 . . 3 (𝜑𝑄 ∈ {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))})
719718, 68syl6eleqr 2694 . 2 (𝜑𝑄𝑈)
720 breq2 4577 . . 3 (𝑢 = 𝑄 → (𝑆 < 𝑢𝑆 < 𝑄))
721720rspcev 3277 . 2 ((𝑄𝑈𝑆 < 𝑄) → ∃𝑢𝑈 𝑆 < 𝑢)
722719, 144, 721syl2anc 690 1 (𝜑 → ∃𝑢𝑈 𝑆 < 𝑢)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775  wral 2891  wrex 2892  {crab 2895  Vcvv 3168  cdif 3532  cun 3533  cin 3534  wss 3535  c0 3869  ifcif 4031  {csn 4120   class class class wbr 4573  cmpt 4633   Or wor 4944  ran crn 5025  cres 5026  wf 5782  cfv 5786  (class class class)co 6523  cmpt2 6525  𝑚 cmap 7717  Fincfn 7814  infcinf 8203  cc 9786  cr 9787  0cc0 9788  1c1 9789   + caddc 9791   · cmul 9793  +∞cpnf 9923  *cxr 9925   < clt 9926  cle 9927  cmin 10113  cn 10863  cuz 11515  +crp 11660   +𝑒 cxad 11772  [,)cico 12000  [,]cicc 12001  ...cfz 12148  Σcsu 14206  cprod 14416  volcvol 22952  Σ^csumge0 39055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-of 6768  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-fi 8173  df-sup 8204  df-inf 8205  df-oi 8271  df-card 8621  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-3 10923  df-n0 11136  df-z 11207  df-uz 11516  df-q 11617  df-rp 11661  df-xneg 11774  df-xadd 11775  df-xmul 11776  df-ioo 12002  df-ico 12004  df-icc 12005  df-fz 12149  df-fzo 12286  df-fl 12406  df-seq 12615  df-exp 12674  df-hash 12931  df-cj 13629  df-re 13630  df-im 13631  df-sqrt 13765  df-abs 13766  df-clim 14009  df-rlim 14010  df-sum 14207  df-prod 14417  df-rest 15848  df-topgen 15869  df-psmet 19501  df-xmet 19502  df-met 19503  df-bl 19504  df-mopn 19505  df-top 20459  df-bases 20460  df-topon 20461  df-cmp 20938  df-ovol 22953  df-vol 22954  df-sumge0 39056
This theorem is referenced by:  hoidmvlelem3  39287
  Copyright terms: Public domain W3C validator