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Theorem hsphoidmvle2 42744
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 3945 . . . . 5 (𝜑𝑍𝑋)
41, 3ffvelrnd 6844 . . . 4 (𝜑 → (𝐴𝑍) ∈ ℝ)
5 hsphoidmvle2.b . . . . . 6 (𝜑𝐵:𝑋⟶ℝ)
65, 3ffvelrnd 6844 . . . . 5 (𝜑 → (𝐵𝑍) ∈ ℝ)
7 hsphoidmvle2.c . . . . 5 (𝜑𝐶 ∈ ℝ)
86, 7ifcld 4508 . . . 4 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ)
9 volicore 42740 . . . 4 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ) → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ∈ ℝ)
104, 8, 9syl2anc 584 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ∈ ℝ)
11 hsphoidmvle2.d . . . . 5 (𝜑𝐷 ∈ ℝ)
126, 11ifcld 4508 . . . 4 (𝜑 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ)
13 volicore 42740 . . . 4 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ) → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) ∈ ℝ)
144, 12, 13syl2anc 584 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) ∈ ℝ)
15 hsphoidmvle2.x . . . . 5 (𝜑𝑋 ∈ Fin)
16 difssd 4106 . . . . 5 (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
17 ssfi 8726 . . . . 5 ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
1815, 16, 17syl2anc 584 . . . 4 (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
19 eldifi 4100 . . . . . 6 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘𝑋)
2019adantl 482 . . . . 5 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘𝑋)
211ffvelrnda 6843 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
225ffvelrnda 6843 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
23 volicore 42740 . . . . . 6 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2421, 22, 23syl2anc 584 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2520, 24syldan 591 . . . 4 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2618, 25fprodrecl 15295 . . 3 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
27 nfv 1906 . . . 4 𝑘𝜑
2820, 21syldan 591 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝑘) ∈ ℝ)
2920, 22syldan 591 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵𝑘) ∈ ℝ)
3029rexrd 10679 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵𝑘) ∈ ℝ*)
31 icombl 24092 . . . . . 6 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ*) → ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol)
3228, 30, 31syl2anc 584 . . . . 5 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol)
33 volge0 42122 . . . . 5 (((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol → 0 ≤ (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
3432, 33syl 17 . . . 4 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 0 ≤ (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
3527, 18, 25, 34fprodge0 15335 . . 3 (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
368rexrd 10679 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ*)
37 icombl 24092 . . . . 5 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ*) → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol)
384, 36, 37syl2anc 584 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol)
3912rexrd 10679 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ*)
40 icombl 24092 . . . . 5 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ*) → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) ∈ dom vol)
414, 39, 40syl2anc 584 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) ∈ dom vol)
424rexrd 10679 . . . . 5 (𝜑 → (𝐴𝑍) ∈ ℝ*)
434leidd 11194 . . . . 5 (𝜑 → (𝐴𝑍) ≤ (𝐴𝑍))
446leidd 11194 . . . . . . . 8 (𝜑 → (𝐵𝑍) ≤ (𝐵𝑍))
4544adantr 481 . . . . . . 7 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ (𝐵𝑍))
46 iftrue 4469 . . . . . . . . 9 ((𝐵𝑍) ≤ 𝐶 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = (𝐵𝑍))
4746adantl 482 . . . . . . . 8 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = (𝐵𝑍))
486adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ∈ ℝ)
497adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐶 ∈ ℝ)
5011adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐷 ∈ ℝ)
51 simpr 485 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ 𝐶)
52 hsphoidmvle2.e . . . . . . . . . . 11 (𝜑𝐶𝐷)
5352adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐶𝐷)
5448, 49, 50, 51, 53letrd 10785 . . . . . . . . 9 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ 𝐷)
5554iftrued 4471 . . . . . . . 8 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = (𝐵𝑍))
5647, 55breq12d 5070 . . . . . . 7 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ↔ (𝐵𝑍) ≤ (𝐵𝑍)))
5745, 56mpbird 258 . . . . . 6 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
58 simpl 483 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝜑)
59 simpr 485 . . . . . . . . 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 10775 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → (𝐶 < (𝐵𝑍) ↔ ¬ (𝐵𝑍) ≤ 𝐶))
6359, 62mpbird 258 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝐶 < (𝐵𝑍))
647adantr 481 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ∈ ℝ)
656adantr 481 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → (𝐵𝑍) ∈ ℝ)
66 simpr 485 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 < (𝐵𝑍))
6764, 65, 66ltled 10776 . . . . . . . . . . 11 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ≤ (𝐵𝑍))
6867adantr 481 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ (𝐵𝑍))
69 iftrue 4469 . . . . . . . . . . . 12 ((𝐵𝑍) ≤ 𝐷 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = (𝐵𝑍))
7069eqcomd 2824 . . . . . . . . . . 11 ((𝐵𝑍) ≤ 𝐷 → (𝐵𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7170adantl 482 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → (𝐵𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7268, 71breqtrd 5083 . . . . . . . . 9 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7352ad2antrr 722 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐶𝐷)
74 iffalse 4472 . . . . . . . . . . . 12 (¬ (𝐵𝑍) ≤ 𝐷 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = 𝐷)
7574eqcomd 2824 . . . . . . . . . . 11 (¬ (𝐵𝑍) ≤ 𝐷𝐷 = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7675adantl 482 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐷 = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7773, 76breqtrd 5083 . . . . . . . . 9 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7872, 77pm2.61dan 809 . . . . . . . 8 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7958, 63, 78syl2anc 584 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
80 iffalse 4472 . . . . . . . . 9 (¬ (𝐵𝑍) ≤ 𝐶 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = 𝐶)
8180adantl 482 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = 𝐶)
8281breq1d 5067 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → (if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ↔ 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
8379, 82mpbird 258 . . . . . 6 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
8457, 83pm2.61dan 809 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
85 icossico 12794 . . . . 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 834 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ⊆ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
87 volss 24061 . . . 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 1363 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ≤ (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))))
8910, 14, 26, 35, 88lemul1ad 11567 . 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))))))
913ne0d 4298 . . . . 5 (𝜑𝑋 ≠ ∅)
92 hsphoidmvle2.h . . . . . 6 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))
9392, 7, 15, 5hsphoif 42735 . . . . 5 (𝜑 → ((𝐻𝐶)‘𝐵):𝑋⟶ℝ)
9490, 15, 91, 1, 93hoidmvn0val 42743 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))))
9593ffvelrnda 6843 . . . . . . 7 ((𝜑𝑘𝑋) → (((𝐻𝐶)‘𝐵)‘𝑘) ∈ ℝ)
96 volicore 42740 . . . . . . 7 (((𝐴𝑘) ∈ ℝ ∧ (((𝐻𝐶)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
9721, 95, 96syl2anc 584 . . . . . 6 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
9897recnd 10657 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
99 fveq2 6663 . . . . . . . . 9 (𝑘 = 𝑍 → (𝐴𝑘) = (𝐴𝑍))
100 fveq2 6663 . . . . . . . . 9 (𝑘 = 𝑍 → (((𝐻𝐶)‘𝐵)‘𝑘) = (((𝐻𝐶)‘𝐵)‘𝑍))
10199, 100oveq12d 7163 . . . . . . . 8 (𝑘 = 𝑍 → ((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)) = ((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍)))
102101fveq2d 6667 . . . . . . 7 (𝑘 = 𝑍 → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))))
103102adantl 482 . . . . . 6 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))))
10492, 7, 15, 5, 3hsphoival 42738 . . . . . . . . . 10 (𝜑 → (((𝐻𝐶)‘𝐵)‘𝑍) = if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)))
1052eldifbd 3946 . . . . . . . . . . 11 (𝜑 → ¬ 𝑍𝑌)
106105iffalsed 4474 . . . . . . . . . 10 (𝜑 → if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) = if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))
107104, 106eqtrd 2853 . . . . . . . . 9 (𝜑 → (((𝐻𝐶)‘𝐵)‘𝑍) = if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))
108107oveq2d 7161 . . . . . . . 8 (𝜑 → ((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍)) = ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)))
109108fveq2d 6667 . . . . . . 7 (𝜑 → (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
110109adantr 481 . . . . . 6 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
111103, 110eqtrd 2853 . . . . 5 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
11215, 98, 3, 111fprodsplit1 41750 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)))))
1137adantr 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐶 ∈ ℝ)
11415adantr 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑋 ∈ Fin)
1155adantr 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐵:𝑋⟶ℝ)
11692, 113, 114, 115, 20hsphoival 42738 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐶)‘𝐵)‘𝑘) = if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐶, (𝐵𝑘), 𝐶)))
117 hsphoidmvle2.y . . . . . . . . . . . . 13 𝑋 = (𝑌 ∪ {𝑍})
11819, 117eleqtrdi 2920 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
119 eldifn 4101 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
120 elunnel2 41173 . . . . . . . . . . . 12 ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ {𝑍}) → 𝑘𝑌)
121118, 119, 120syl2anc 584 . . . . . . . . . . 11 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘𝑌)
122121adantl 482 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘𝑌)
123122iftrued 4471 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐶, (𝐵𝑘), 𝐶)) = (𝐵𝑘))
124116, 123eqtrd 2853 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐶)‘𝐵)‘𝑘) = (𝐵𝑘))
125124oveq2d 7161 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
126125fveq2d 6667 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
127126prodeq2dv 15265 . . . . 5 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
128127oveq2d 7161 . . . 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)))))
12994, 112, 1283eqtrd 2857 . . 3 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
13092, 11, 15, 5hsphoif 42735 . . . . 5 (𝜑 → ((𝐻𝐷)‘𝐵):𝑋⟶ℝ)
13190, 15, 91, 1, 130hoidmvn0val 42743 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))))
132130ffvelrnda 6843 . . . . . . 7 ((𝜑𝑘𝑋) → (((𝐻𝐷)‘𝐵)‘𝑘) ∈ ℝ)
133 volicore 42740 . . . . . . 7 (((𝐴𝑘) ∈ ℝ ∧ (((𝐻𝐷)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℝ)
13421, 132, 133syl2anc 584 . . . . . 6 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℝ)
135134recnd 10657 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℂ)
136 fveq2 6663 . . . . . . . 8 (𝑘 = 𝑍 → (((𝐻𝐷)‘𝐵)‘𝑘) = (((𝐻𝐷)‘𝐵)‘𝑍))
13799, 136oveq12d 7163 . . . . . . 7 (𝑘 = 𝑍 → ((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)) = ((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍)))
138137fveq2d 6667 . . . . . 6 (𝑘 = 𝑍 → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))))
139138adantl 482 . . . . 5 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))))
14015, 135, 3, 139fprodsplit1 41750 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = ((vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)))))
14192, 11, 15, 5, 3hsphoival 42738 . . . . . . . 8 (𝜑 → (((𝐻𝐷)‘𝐵)‘𝑍) = if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
142105iffalsed 4474 . . . . . . . 8 (𝜑 → if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
143141, 142eqtrd 2853 . . . . . . 7 (𝜑 → (((𝐻𝐷)‘𝐵)‘𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
144143oveq2d 7161 . . . . . 6 (𝜑 → ((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍)) = ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
145144fveq2d 6667 . . . . 5 (𝜑 → (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))))
14611adantr 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐷 ∈ ℝ)
14792, 146, 114, 115, 20hsphoival 42738 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐷)‘𝐵)‘𝑘) = if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐷, (𝐵𝑘), 𝐷)))
148122iftrued 4471 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐷, (𝐵𝑘), 𝐷)) = (𝐵𝑘))
149147, 148eqtrd 2853 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐷)‘𝐵)‘𝑘) = (𝐵𝑘))
150149oveq2d 7161 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
151150fveq2d 6667 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
152151prodeq2dv 15265 . . . . 5 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
153145, 152oveq12d 7163 . . . 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)))))
154131, 140, 1533eqtrd 2857 . . 3 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
155129, 154breq12d 5070 . 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))))))
15689, 155mpbird 258 1 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  cdif 3930  cun 3931  wss 3933  c0 4288  ifcif 4463  {csn 4557   class class class wbr 5057  cmpt 5137  dom cdm 5548  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  m cmap 8395  Fincfn 8497  cr 10524  0cc0 10525   · cmul 10530  *cxr 10662   < clt 10663  cle 10664  [,)cico 12728  cprod 15247  volcvol 23991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-rlim 14834  df-sum 15031  df-prod 15248  df-rest 16684  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-top 21430  df-topon 21447  df-bases 21482  df-cmp 21923  df-ovol 23992  df-vol 23993
This theorem is referenced by:  hoidmvlelem1  42754  hoidmvlelem2  42755
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