Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hsphoidmvle2 Structured version   Visualization version   GIF version

Theorem hsphoidmvle2 39299
Description: The dimensional volume of a half-open interval intersected with a two half-spaces. Used in the last inequality of step (c) of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
hsphoidmvle2.l 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
hsphoidmvle2.x (𝜑𝑋 ∈ Fin)
hsphoidmvle2.z (𝜑𝑍 ∈ (𝑋𝑌))
hsphoidmvle2.y 𝑋 = (𝑌 ∪ {𝑍})
hsphoidmvle2.c (𝜑𝐶 ∈ ℝ)
hsphoidmvle2.d (𝜑𝐷 ∈ ℝ)
hsphoidmvle2.e (𝜑𝐶𝐷)
hsphoidmvle2.h 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))
hsphoidmvle2.a (𝜑𝐴:𝑋⟶ℝ)
hsphoidmvle2.b (𝜑𝐵:𝑋⟶ℝ)
Assertion
Ref Expression
hsphoidmvle2 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑘   𝐵,𝑎,𝑏,𝑘   𝐵,𝑐,𝑗,𝑘   𝐶,𝑎,𝑏,𝑘,𝑥   𝐶,𝑐,𝑗,𝑥   𝐷,𝑎,𝑏,𝑘,𝑥   𝐷,𝑐,𝑗   𝐻,𝑎,𝑏,𝑘   𝑋,𝑎,𝑏,𝑘,𝑥   𝑋,𝑐,𝑗   𝑌,𝑐,𝑗,𝑥   𝑍,𝑐,𝑗,𝑘,𝑥   𝜑,𝑎,𝑏,𝑘,𝑥   𝜑,𝑐,𝑗
Allowed substitution hints:   𝐴(𝑥,𝑗,𝑐)   𝐵(𝑥)   𝐻(𝑥,𝑗,𝑐)   𝐿(𝑥,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑌(𝑘,𝑎,𝑏)   𝑍(𝑎,𝑏)

Proof of Theorem hsphoidmvle2
StepHypRef Expression
1 hsphoidmvle2.a . . . . 5 (𝜑𝐴:𝑋⟶ℝ)
2 hsphoidmvle2.z . . . . . 6 (𝜑𝑍 ∈ (𝑋𝑌))
32eldifad 3551 . . . . 5 (𝜑𝑍𝑋)
41, 3ffvelrnd 6253 . . . 4 (𝜑 → (𝐴𝑍) ∈ ℝ)
5 hsphoidmvle2.b . . . . . 6 (𝜑𝐵:𝑋⟶ℝ)
65, 3ffvelrnd 6253 . . . . 5 (𝜑 → (𝐵𝑍) ∈ ℝ)
7 hsphoidmvle2.c . . . . 5 (𝜑𝐶 ∈ ℝ)
86, 7ifcld 4080 . . . 4 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ)
9 volicore 39295 . . . 4 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ) → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ∈ ℝ)
104, 8, 9syl2anc 690 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ∈ ℝ)
11 hsphoidmvle2.d . . . . 5 (𝜑𝐷 ∈ ℝ)
126, 11ifcld 4080 . . . 4 (𝜑 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ)
13 volicore 39295 . . . 4 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ) → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) ∈ ℝ)
144, 12, 13syl2anc 690 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) ∈ ℝ)
15 hsphoidmvle2.x . . . . 5 (𝜑𝑋 ∈ Fin)
16 difssd 3699 . . . . 5 (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
17 ssfi 8043 . . . . 5 ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
1815, 16, 17syl2anc 690 . . . 4 (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
19 eldifi 3693 . . . . . 6 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘𝑋)
2019adantl 480 . . . . 5 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘𝑋)
211ffvelrnda 6252 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
225ffvelrnda 6252 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
23 volicore 39295 . . . . . 6 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2421, 22, 23syl2anc 690 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2520, 24syldan 485 . . . 4 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2618, 25fprodrecl 14471 . . 3 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
27 nfv 1829 . . . 4 𝑘𝜑
2820, 21syldan 485 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝑘) ∈ ℝ)
2920, 22syldan 485 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵𝑘) ∈ ℝ)
3029rexrd 9946 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵𝑘) ∈ ℝ*)
31 icombl 23084 . . . . . 6 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ*) → ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol)
3228, 30, 31syl2anc 690 . . . . 5 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol)
33 volge0 38677 . . . . 5 (((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol → 0 ≤ (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
3432, 33syl 17 . . . 4 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 0 ≤ (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
3527, 18, 25, 34fprodge0 14512 . . 3 (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
368rexrd 9946 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ*)
37 icombl 23084 . . . . 5 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ*) → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol)
384, 36, 37syl2anc 690 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol)
3912rexrd 9946 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ*)
40 icombl 23084 . . . . 5 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ*) → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) ∈ dom vol)
414, 39, 40syl2anc 690 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) ∈ dom vol)
424rexrd 9946 . . . . 5 (𝜑 → (𝐴𝑍) ∈ ℝ*)
434leidd 10446 . . . . 5 (𝜑 → (𝐴𝑍) ≤ (𝐴𝑍))
446leidd 10446 . . . . . . . 8 (𝜑 → (𝐵𝑍) ≤ (𝐵𝑍))
4544adantr 479 . . . . . . 7 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ (𝐵𝑍))
46 iftrue 4041 . . . . . . . . 9 ((𝐵𝑍) ≤ 𝐶 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = (𝐵𝑍))
4746adantl 480 . . . . . . . 8 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = (𝐵𝑍))
486adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ∈ ℝ)
497adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐶 ∈ ℝ)
5011adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐷 ∈ ℝ)
51 simpr 475 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ 𝐶)
52 hsphoidmvle2.e . . . . . . . . . . 11 (𝜑𝐶𝐷)
5352adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐶𝐷)
5448, 49, 50, 51, 53letrd 10046 . . . . . . . . 9 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ 𝐷)
5554iftrued 4043 . . . . . . . 8 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = (𝐵𝑍))
5647, 55breq12d 4590 . . . . . . 7 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ↔ (𝐵𝑍) ≤ (𝐵𝑍)))
5745, 56mpbird 245 . . . . . 6 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
58 simpl 471 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝜑)
59 simpr 475 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → ¬ (𝐵𝑍) ≤ 𝐶)
6058, 7syl 17 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝐶 ∈ ℝ)
6158, 6syl 17 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ∈ ℝ)
6260, 61ltnled 10036 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → (𝐶 < (𝐵𝑍) ↔ ¬ (𝐵𝑍) ≤ 𝐶))
6359, 62mpbird 245 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝐶 < (𝐵𝑍))
647adantr 479 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ∈ ℝ)
656adantr 479 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → (𝐵𝑍) ∈ ℝ)
66 simpr 475 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 < (𝐵𝑍))
6764, 65, 66ltled 10037 . . . . . . . . . . 11 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ≤ (𝐵𝑍))
6867adantr 479 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ (𝐵𝑍))
69 iftrue 4041 . . . . . . . . . . . 12 ((𝐵𝑍) ≤ 𝐷 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = (𝐵𝑍))
7069eqcomd 2615 . . . . . . . . . . 11 ((𝐵𝑍) ≤ 𝐷 → (𝐵𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7170adantl 480 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → (𝐵𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7268, 71breqtrd 4603 . . . . . . . . 9 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7352ad2antrr 757 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐶𝐷)
74 iffalse 4044 . . . . . . . . . . . 12 (¬ (𝐵𝑍) ≤ 𝐷 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = 𝐷)
7574eqcomd 2615 . . . . . . . . . . 11 (¬ (𝐵𝑍) ≤ 𝐷𝐷 = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7675adantl 480 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐷 = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7773, 76breqtrd 4603 . . . . . . . . 9 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7872, 77pm2.61dan 827 . . . . . . . 8 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7958, 63, 78syl2anc 690 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
80 iffalse 4044 . . . . . . . . 9 (¬ (𝐵𝑍) ≤ 𝐶 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = 𝐶)
8180adantl 480 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = 𝐶)
8281breq1d 4587 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → (if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ↔ 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
8379, 82mpbird 245 . . . . . 6 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
8457, 83pm2.61dan 827 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
85 icossico 12073 . . . . 5 ((((𝐴𝑍) ∈ ℝ* ∧ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ*) ∧ ((𝐴𝑍) ≤ (𝐴𝑍) ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ⊆ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
8642, 39, 43, 84, 85syl22anc 1318 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ⊆ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
87 volss 23053 . . . 4 ((((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol ∧ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) ∈ dom vol ∧ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ⊆ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ≤ (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))))
8838, 41, 86, 87syl3anc 1317 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ≤ (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))))
8910, 14, 26, 35, 88lemul1ad 10815 . 2 (𝜑 → ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))) ≤ ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
90 hsphoidmvle2.l . . . . 5 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
91 ne0i 3879 . . . . . 6 (𝑍𝑋𝑋 ≠ ∅)
923, 91syl 17 . . . . 5 (𝜑𝑋 ≠ ∅)
93 hsphoidmvle2.h . . . . . 6 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))
9493, 7, 15, 5hsphoif 39290 . . . . 5 (𝜑 → ((𝐻𝐶)‘𝐵):𝑋⟶ℝ)
9590, 15, 92, 1, 94hoidmvn0val 39298 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))))
9694ffvelrnda 6252 . . . . . . 7 ((𝜑𝑘𝑋) → (((𝐻𝐶)‘𝐵)‘𝑘) ∈ ℝ)
97 volicore 39295 . . . . . . 7 (((𝐴𝑘) ∈ ℝ ∧ (((𝐻𝐶)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
9821, 96, 97syl2anc 690 . . . . . 6 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
9998recnd 9925 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
100 fveq2 6088 . . . . . . . . 9 (𝑘 = 𝑍 → (𝐴𝑘) = (𝐴𝑍))
101 fveq2 6088 . . . . . . . . 9 (𝑘 = 𝑍 → (((𝐻𝐶)‘𝐵)‘𝑘) = (((𝐻𝐶)‘𝐵)‘𝑍))
102100, 101oveq12d 6545 . . . . . . . 8 (𝑘 = 𝑍 → ((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)) = ((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍)))
103102fveq2d 6092 . . . . . . 7 (𝑘 = 𝑍 → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))))
104103adantl 480 . . . . . 6 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))))
10593, 7, 15, 5, 3hsphoival 39293 . . . . . . . . . 10 (𝜑 → (((𝐻𝐶)‘𝐵)‘𝑍) = if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)))
1062eldifbd 3552 . . . . . . . . . . 11 (𝜑 → ¬ 𝑍𝑌)
107106iffalsed 4046 . . . . . . . . . 10 (𝜑 → if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) = if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))
108105, 107eqtrd 2643 . . . . . . . . 9 (𝜑 → (((𝐻𝐶)‘𝐵)‘𝑍) = if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))
109108oveq2d 6543 . . . . . . . 8 (𝜑 → ((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍)) = ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)))
110109fveq2d 6092 . . . . . . 7 (𝜑 → (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
111110adantr 479 . . . . . 6 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
112104, 111eqtrd 2643 . . . . 5 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
11315, 99, 3, 112fprodsplit1 38484 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)))))
1147adantr 479 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐶 ∈ ℝ)
11515adantr 479 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑋 ∈ Fin)
1165adantr 479 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐵:𝑋⟶ℝ)
11793, 114, 115, 116, 20hsphoival 39293 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐶)‘𝐵)‘𝑘) = if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐶, (𝐵𝑘), 𝐶)))
118 hsphoidmvle2.y . . . . . . . . . . . . 13 𝑋 = (𝑌 ∪ {𝑍})
11919, 118syl6eleq 2697 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
120 eldifn 3694 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
121 elunnel2 38045 . . . . . . . . . . . 12 ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ {𝑍}) → 𝑘𝑌)
122119, 120, 121syl2anc 690 . . . . . . . . . . 11 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘𝑌)
123122adantl 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘𝑌)
124123iftrued 4043 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐶, (𝐵𝑘), 𝐶)) = (𝐵𝑘))
125117, 124eqtrd 2643 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐶)‘𝐵)‘𝑘) = (𝐵𝑘))
126125oveq2d 6543 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
127126fveq2d 6092 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
128127prodeq2dv 14441 . . . . 5 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
129128oveq2d 6543 . . . 4 (𝜑 → ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)))) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
13095, 113, 1293eqtrd 2647 . . 3 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
13193, 11, 15, 5hsphoif 39290 . . . . 5 (𝜑 → ((𝐻𝐷)‘𝐵):𝑋⟶ℝ)
13290, 15, 92, 1, 131hoidmvn0val 39298 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))))
133131ffvelrnda 6252 . . . . . . 7 ((𝜑𝑘𝑋) → (((𝐻𝐷)‘𝐵)‘𝑘) ∈ ℝ)
134 volicore 39295 . . . . . . 7 (((𝐴𝑘) ∈ ℝ ∧ (((𝐻𝐷)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℝ)
13521, 133, 134syl2anc 690 . . . . . 6 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℝ)
136135recnd 9925 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℂ)
137 fveq2 6088 . . . . . . . 8 (𝑘 = 𝑍 → (((𝐻𝐷)‘𝐵)‘𝑘) = (((𝐻𝐷)‘𝐵)‘𝑍))
138100, 137oveq12d 6545 . . . . . . 7 (𝑘 = 𝑍 → ((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)) = ((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍)))
139138fveq2d 6092 . . . . . 6 (𝑘 = 𝑍 → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))))
140139adantl 480 . . . . 5 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))))
14115, 136, 3, 140fprodsplit1 38484 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = ((vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)))))
14293, 11, 15, 5, 3hsphoival 39293 . . . . . . . 8 (𝜑 → (((𝐻𝐷)‘𝐵)‘𝑍) = if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
143106iffalsed 4046 . . . . . . . 8 (𝜑 → if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
144142, 143eqtrd 2643 . . . . . . 7 (𝜑 → (((𝐻𝐷)‘𝐵)‘𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
145144oveq2d 6543 . . . . . 6 (𝜑 → ((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍)) = ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
146145fveq2d 6092 . . . . 5 (𝜑 → (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))))
14711adantr 479 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐷 ∈ ℝ)
14893, 147, 115, 116, 20hsphoival 39293 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐷)‘𝐵)‘𝑘) = if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐷, (𝐵𝑘), 𝐷)))
149123iftrued 4043 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐷, (𝐵𝑘), 𝐷)) = (𝐵𝑘))
150148, 149eqtrd 2643 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐷)‘𝐵)‘𝑘) = (𝐵𝑘))
151150oveq2d 6543 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
152151fveq2d 6092 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
153152prodeq2dv 14441 . . . . 5 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
154146, 153oveq12d 6545 . . . 4 (𝜑 → ((vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)))) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
155132, 141, 1543eqtrd 2647 . . 3 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
156130, 155breq12d 4590 . 2 (𝜑 → ((𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)) ↔ ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))) ≤ ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))))
15789, 156mpbird 245 1 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  wne 2779  cdif 3536  cun 3537  wss 3539  c0 3873  ifcif 4035  {csn 4124   class class class wbr 4577  cmpt 4637  dom cdm 5028  wf 5786  cfv 5790  (class class class)co 6527  cmpt2 6529  𝑚 cmap 7722  Fincfn 7819  cr 9792  0cc0 9793   · cmul 9798  *cxr 9930   < clt 9931  cle 9932  [,)cico 12007  cprod 14423  volcvol 22984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871
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 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6773  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-2o 7426  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-fi 8178  df-sup 8209  df-inf 8210  df-oi 8276  df-card 8626  df-cda 8851  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-n0 11143  df-z 11214  df-uz 11523  df-q 11624  df-rp 11668  df-xneg 11781  df-xadd 11782  df-xmul 11783  df-ioo 12009  df-ico 12011  df-icc 12012  df-fz 12156  df-fzo 12293  df-fl 12413  df-seq 12622  df-exp 12681  df-hash 12938  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-clim 14016  df-rlim 14017  df-sum 14214  df-prod 14424  df-rest 15855  df-topgen 15876  df-psmet 19508  df-xmet 19509  df-met 19510  df-bl 19511  df-mopn 19512  df-top 20469  df-bases 20470  df-topon 20471  df-cmp 20948  df-ovol 22985  df-vol 22986
This theorem is referenced by:  hoidmvlelem1  39309  hoidmvlelem2  39310
  Copyright terms: Public domain W3C validator