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Theorem hsphoidmvle2 41540
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 3782 . . . . 5 (𝜑𝑍𝑋)
41, 3ffvelrnd 6587 . . . 4 (𝜑 → (𝐴𝑍) ∈ ℝ)
5 hsphoidmvle2.b . . . . . 6 (𝜑𝐵:𝑋⟶ℝ)
65, 3ffvelrnd 6587 . . . . 5 (𝜑 → (𝐵𝑍) ∈ ℝ)
7 hsphoidmvle2.c . . . . 5 (𝜑𝐶 ∈ ℝ)
86, 7ifcld 4323 . . . 4 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ)
9 volicore 41536 . . . 4 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ) → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ∈ ℝ)
104, 8, 9syl2anc 580 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ∈ ℝ)
11 hsphoidmvle2.d . . . . 5 (𝜑𝐷 ∈ ℝ)
126, 11ifcld 4323 . . . 4 (𝜑 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ)
13 volicore 41536 . . . 4 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ) → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) ∈ ℝ)
144, 12, 13syl2anc 580 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) ∈ ℝ)
15 hsphoidmvle2.x . . . . 5 (𝜑𝑋 ∈ Fin)
16 difssd 3937 . . . . 5 (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
17 ssfi 8423 . . . . 5 ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
1815, 16, 17syl2anc 580 . . . 4 (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
19 eldifi 3931 . . . . . 6 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘𝑋)
2019adantl 474 . . . . 5 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘𝑋)
211ffvelrnda 6586 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
225ffvelrnda 6586 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
23 volicore 41536 . . . . . 6 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2421, 22, 23syl2anc 580 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2520, 24syldan 586 . . . 4 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2618, 25fprodrecl 15019 . . 3 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
27 nfv 2010 . . . 4 𝑘𝜑
2820, 21syldan 586 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝑘) ∈ ℝ)
2920, 22syldan 586 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵𝑘) ∈ ℝ)
3029rexrd 10379 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵𝑘) ∈ ℝ*)
31 icombl 23671 . . . . . 6 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ*) → ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol)
3228, 30, 31syl2anc 580 . . . . 5 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol)
33 volge0 40915 . . . . 5 (((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol → 0 ≤ (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
3432, 33syl 17 . . . 4 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 0 ≤ (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
3527, 18, 25, 34fprodge0 15059 . . 3 (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
368rexrd 10379 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ*)
37 icombl 23671 . . . . 5 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ*) → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol)
384, 36, 37syl2anc 580 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol)
3912rexrd 10379 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ*)
40 icombl 23671 . . . . 5 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ*) → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) ∈ dom vol)
414, 39, 40syl2anc 580 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) ∈ dom vol)
424rexrd 10379 . . . . 5 (𝜑 → (𝐴𝑍) ∈ ℝ*)
434leidd 10887 . . . . 5 (𝜑 → (𝐴𝑍) ≤ (𝐴𝑍))
446leidd 10887 . . . . . . . 8 (𝜑 → (𝐵𝑍) ≤ (𝐵𝑍))
4544adantr 473 . . . . . . 7 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ (𝐵𝑍))
46 iftrue 4284 . . . . . . . . 9 ((𝐵𝑍) ≤ 𝐶 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = (𝐵𝑍))
4746adantl 474 . . . . . . . 8 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = (𝐵𝑍))
486adantr 473 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ∈ ℝ)
497adantr 473 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐶 ∈ ℝ)
5011adantr 473 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐷 ∈ ℝ)
51 simpr 478 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ 𝐶)
52 hsphoidmvle2.e . . . . . . . . . . 11 (𝜑𝐶𝐷)
5352adantr 473 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐶𝐷)
5448, 49, 50, 51, 53letrd 10485 . . . . . . . . 9 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ 𝐷)
5554iftrued 4286 . . . . . . . 8 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = (𝐵𝑍))
5647, 55breq12d 4857 . . . . . . 7 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ↔ (𝐵𝑍) ≤ (𝐵𝑍)))
5745, 56mpbird 249 . . . . . 6 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
58 simpl 475 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝜑)
59 simpr 478 . . . . . . . . 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 10475 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → (𝐶 < (𝐵𝑍) ↔ ¬ (𝐵𝑍) ≤ 𝐶))
6359, 62mpbird 249 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝐶 < (𝐵𝑍))
647adantr 473 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ∈ ℝ)
656adantr 473 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → (𝐵𝑍) ∈ ℝ)
66 simpr 478 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 < (𝐵𝑍))
6764, 65, 66ltled 10476 . . . . . . . . . . 11 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ≤ (𝐵𝑍))
6867adantr 473 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ (𝐵𝑍))
69 iftrue 4284 . . . . . . . . . . . 12 ((𝐵𝑍) ≤ 𝐷 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = (𝐵𝑍))
7069eqcomd 2806 . . . . . . . . . . 11 ((𝐵𝑍) ≤ 𝐷 → (𝐵𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7170adantl 474 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → (𝐵𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7268, 71breqtrd 4870 . . . . . . . . 9 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7352ad2antrr 718 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐶𝐷)
74 iffalse 4287 . . . . . . . . . . . 12 (¬ (𝐵𝑍) ≤ 𝐷 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = 𝐷)
7574eqcomd 2806 . . . . . . . . . . 11 (¬ (𝐵𝑍) ≤ 𝐷𝐷 = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7675adantl 474 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐷 = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7773, 76breqtrd 4870 . . . . . . . . 9 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7872, 77pm2.61dan 848 . . . . . . . 8 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7958, 63, 78syl2anc 580 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
80 iffalse 4287 . . . . . . . . 9 (¬ (𝐵𝑍) ≤ 𝐶 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = 𝐶)
8180adantl 474 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = 𝐶)
8281breq1d 4854 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → (if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ↔ 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
8379, 82mpbird 249 . . . . . 6 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
8457, 83pm2.61dan 848 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
85 icossico 12491 . . . . 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 868 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ⊆ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
87 volss 23640 . . . 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 1491 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ≤ (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))))
8910, 14, 26, 35, 88lemul1ad 11256 . 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 4123 . . . . 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 41531 . . . . 5 (𝜑 → ((𝐻𝐶)‘𝐵):𝑋⟶ℝ)
9490, 15, 91, 1, 93hoidmvn0val 41539 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))))
9593ffvelrnda 6586 . . . . . . 7 ((𝜑𝑘𝑋) → (((𝐻𝐶)‘𝐵)‘𝑘) ∈ ℝ)
96 volicore 41536 . . . . . . 7 (((𝐴𝑘) ∈ ℝ ∧ (((𝐻𝐶)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
9721, 95, 96syl2anc 580 . . . . . 6 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
9897recnd 10358 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
99 fveq2 6412 . . . . . . . . 9 (𝑘 = 𝑍 → (𝐴𝑘) = (𝐴𝑍))
100 fveq2 6412 . . . . . . . . 9 (𝑘 = 𝑍 → (((𝐻𝐶)‘𝐵)‘𝑘) = (((𝐻𝐶)‘𝐵)‘𝑍))
10199, 100oveq12d 6897 . . . . . . . 8 (𝑘 = 𝑍 → ((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)) = ((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍)))
102101fveq2d 6416 . . . . . . 7 (𝑘 = 𝑍 → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))))
103102adantl 474 . . . . . 6 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))))
10492, 7, 15, 5, 3hsphoival 41534 . . . . . . . . . 10 (𝜑 → (((𝐻𝐶)‘𝐵)‘𝑍) = if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)))
1052eldifbd 3783 . . . . . . . . . . 11 (𝜑 → ¬ 𝑍𝑌)
106105iffalsed 4289 . . . . . . . . . 10 (𝜑 → if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) = if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))
107104, 106eqtrd 2834 . . . . . . . . 9 (𝜑 → (((𝐻𝐶)‘𝐵)‘𝑍) = if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))
108107oveq2d 6895 . . . . . . . 8 (𝜑 → ((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍)) = ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)))
109108fveq2d 6416 . . . . . . 7 (𝜑 → (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
110109adantr 473 . . . . . 6 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
111103, 110eqtrd 2834 . . . . 5 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
11215, 98, 3, 111fprodsplit1 40564 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)))))
1137adantr 473 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐶 ∈ ℝ)
11415adantr 473 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑋 ∈ Fin)
1155adantr 473 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐵:𝑋⟶ℝ)
11692, 113, 114, 115, 20hsphoival 41534 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐶)‘𝐵)‘𝑘) = if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐶, (𝐵𝑘), 𝐶)))
117 hsphoidmvle2.y . . . . . . . . . . . . 13 𝑋 = (𝑌 ∪ {𝑍})
11819, 117syl6eleq 2889 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
119 eldifn 3932 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
120 elunnel2 39953 . . . . . . . . . . . 12 ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ {𝑍}) → 𝑘𝑌)
121118, 119, 120syl2anc 580 . . . . . . . . . . 11 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘𝑌)
122121adantl 474 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘𝑌)
123122iftrued 4286 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐶, (𝐵𝑘), 𝐶)) = (𝐵𝑘))
124116, 123eqtrd 2834 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐶)‘𝐵)‘𝑘) = (𝐵𝑘))
125124oveq2d 6895 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
126125fveq2d 6416 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
127126prodeq2dv 14989 . . . . 5 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
128127oveq2d 6895 . . . 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 2838 . . 3 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
13092, 11, 15, 5hsphoif 41531 . . . . 5 (𝜑 → ((𝐻𝐷)‘𝐵):𝑋⟶ℝ)
13190, 15, 91, 1, 130hoidmvn0val 41539 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))))
132130ffvelrnda 6586 . . . . . . 7 ((𝜑𝑘𝑋) → (((𝐻𝐷)‘𝐵)‘𝑘) ∈ ℝ)
133 volicore 41536 . . . . . . 7 (((𝐴𝑘) ∈ ℝ ∧ (((𝐻𝐷)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℝ)
13421, 132, 133syl2anc 580 . . . . . 6 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℝ)
135134recnd 10358 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℂ)
136 fveq2 6412 . . . . . . . 8 (𝑘 = 𝑍 → (((𝐻𝐷)‘𝐵)‘𝑘) = (((𝐻𝐷)‘𝐵)‘𝑍))
13799, 136oveq12d 6897 . . . . . . 7 (𝑘 = 𝑍 → ((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)) = ((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍)))
138137fveq2d 6416 . . . . . 6 (𝑘 = 𝑍 → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))))
139138adantl 474 . . . . 5 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))))
14015, 135, 3, 139fprodsplit1 40564 . . . 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 41534 . . . . . . . 8 (𝜑 → (((𝐻𝐷)‘𝐵)‘𝑍) = if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
142105iffalsed 4289 . . . . . . . 8 (𝜑 → if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
143141, 142eqtrd 2834 . . . . . . 7 (𝜑 → (((𝐻𝐷)‘𝐵)‘𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
144143oveq2d 6895 . . . . . 6 (𝜑 → ((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍)) = ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
145144fveq2d 6416 . . . . 5 (𝜑 → (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))))
14611adantr 473 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐷 ∈ ℝ)
14792, 146, 114, 115, 20hsphoival 41534 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐷)‘𝐵)‘𝑘) = if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐷, (𝐵𝑘), 𝐷)))
148122iftrued 4286 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐷, (𝐵𝑘), 𝐷)) = (𝐵𝑘))
149147, 148eqtrd 2834 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐷)‘𝐵)‘𝑘) = (𝐵𝑘))
150149oveq2d 6895 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
151150fveq2d 6416 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
152151prodeq2dv 14989 . . . . 5 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
153145, 152oveq12d 6897 . . . 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 2838 . . 3 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
155129, 154breq12d 4857 . 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 249 1 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wcel 2157  cdif 3767  cun 3768  wss 3770  c0 4116  ifcif 4278  {csn 4369   class class class wbr 4844  cmpt 4923  dom cdm 5313  wf 6098  cfv 6102  (class class class)co 6879  cmpt2 6881  𝑚 cmap 8096  Fincfn 8196  cr 10224  0cc0 10225   · cmul 10230  *cxr 10363   < clt 10364  cle 10365  [,)cico 12425  cprod 14971  volcvol 23570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184  ax-inf2 8789  ax-cnex 10281  ax-resscn 10282  ax-1cn 10283  ax-icn 10284  ax-addcl 10285  ax-addrcl 10286  ax-mulcl 10287  ax-mulrcl 10288  ax-mulcom 10289  ax-addass 10290  ax-mulass 10291  ax-distr 10292  ax-i2m1 10293  ax-1ne0 10294  ax-1rid 10295  ax-rnegex 10296  ax-rrecex 10297  ax-cnre 10298  ax-pre-lttri 10299  ax-pre-lttrn 10300  ax-pre-ltadd 10301  ax-pre-mulgt0 10302  ax-pre-sup 10303
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-se 5273  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-pred 5899  df-ord 5945  df-on 5946  df-lim 5947  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-isom 6111  df-riota 6840  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-of 7132  df-om 7301  df-1st 7402  df-2nd 7403  df-wrecs 7646  df-recs 7708  df-rdg 7746  df-1o 7800  df-2o 7801  df-oadd 7804  df-er 7983  df-map 8098  df-pm 8099  df-en 8197  df-dom 8198  df-sdom 8199  df-fin 8200  df-fi 8560  df-sup 8591  df-inf 8592  df-oi 8658  df-card 9052  df-cda 9279  df-pnf 10366  df-mnf 10367  df-xr 10368  df-ltxr 10369  df-le 10370  df-sub 10559  df-neg 10560  df-div 10978  df-nn 11314  df-2 11375  df-3 11376  df-n0 11580  df-z 11666  df-uz 11930  df-q 12033  df-rp 12074  df-xneg 12192  df-xadd 12193  df-xmul 12194  df-ioo 12427  df-ico 12429  df-icc 12430  df-fz 12580  df-fzo 12720  df-fl 12847  df-seq 13055  df-exp 13114  df-hash 13370  df-cj 14179  df-re 14180  df-im 14181  df-sqrt 14315  df-abs 14316  df-clim 14559  df-rlim 14560  df-sum 14757  df-prod 14972  df-rest 16397  df-topgen 16418  df-psmet 20059  df-xmet 20060  df-met 20061  df-bl 20062  df-mopn 20063  df-top 21026  df-topon 21043  df-bases 21078  df-cmp 21518  df-ovol 23571  df-vol 23572
This theorem is referenced by:  hoidmvlelem1  41550  hoidmvlelem2  41551
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