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Theorem hsphoidmvle2 46600
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 3963 . . . . 5 (𝜑𝑍𝑋)
41, 3ffvelcdmd 7105 . . . 4 (𝜑 → (𝐴𝑍) ∈ ℝ)
5 hsphoidmvle2.b . . . . . 6 (𝜑𝐵:𝑋⟶ℝ)
65, 3ffvelcdmd 7105 . . . . 5 (𝜑 → (𝐵𝑍) ∈ ℝ)
7 hsphoidmvle2.c . . . . 5 (𝜑𝐶 ∈ ℝ)
86, 7ifcld 4572 . . . 4 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ)
9 volicore 46596 . . . 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 4572 . . . 4 (𝜑 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ)
13 volicore 46596 . . . 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 4137 . . . . 5 (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
17 ssfi 9213 . . . . 5 ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
1815, 16, 17syl2anc 584 . . . 4 (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
19 eldifi 4131 . . . . . 6 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘𝑋)
2019adantl 481 . . . . 5 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘𝑋)
211ffvelcdmda 7104 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
225ffvelcdmda 7104 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
23 volicore 46596 . . . . . 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 15989 . . 3 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
27 nfv 1914 . . . 4 𝑘𝜑
2820, 21syldan 591 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝑘) ∈ ℝ)
2920, 22syldan 591 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵𝑘) ∈ ℝ)
3029rexrd 11311 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵𝑘) ∈ ℝ*)
31 icombl 25599 . . . . . 6 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ*) → ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol)
3228, 30, 31syl2anc 584 . . . . 5 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol)
33 volge0 45976 . . . . 5 (((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol → 0 ≤ (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
3432, 33syl 17 . . . 4 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 0 ≤ (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
3527, 18, 25, 34fprodge0 16029 . . 3 (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
368rexrd 11311 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ*)
37 icombl 25599 . . . . 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 11311 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ∈ ℝ*)
40 icombl 25599 . . . . 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 11311 . . . . 5 (𝜑 → (𝐴𝑍) ∈ ℝ*)
434leidd 11829 . . . . 5 (𝜑 → (𝐴𝑍) ≤ (𝐴𝑍))
446leidd 11829 . . . . . . . 8 (𝜑 → (𝐵𝑍) ≤ (𝐵𝑍))
4544adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ (𝐵𝑍))
46 iftrue 4531 . . . . . . . . 9 ((𝐵𝑍) ≤ 𝐶 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = (𝐵𝑍))
4746adantl 481 . . . . . . . 8 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = (𝐵𝑍))
486adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ∈ ℝ)
497adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐶 ∈ ℝ)
5011adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐷 ∈ ℝ)
51 simpr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ 𝐶)
52 hsphoidmvle2.e . . . . . . . . . . 11 (𝜑𝐶𝐷)
5352adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → 𝐶𝐷)
5448, 49, 50, 51, 53letrd 11418 . . . . . . . . 9 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (𝐵𝑍) ≤ 𝐷)
5554iftrued 4533 . . . . . . . 8 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = (𝐵𝑍))
5647, 55breq12d 5156 . . . . . . 7 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → (if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ↔ (𝐵𝑍) ≤ (𝐵𝑍)))
5745, 56mpbird 257 . . . . . 6 ((𝜑 ∧ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
58 simpl 482 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝜑)
59 simpr 484 . . . . . . . . 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 11408 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → (𝐶 < (𝐵𝑍) ↔ ¬ (𝐵𝑍) ≤ 𝐶))
6359, 62mpbird 257 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → 𝐶 < (𝐵𝑍))
647adantr 480 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ∈ ℝ)
656adantr 480 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → (𝐵𝑍) ∈ ℝ)
66 simpr 484 . . . . . . . . . . . 12 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 < (𝐵𝑍))
6764, 65, 66ltled 11409 . . . . . . . . . . 11 ((𝜑𝐶 < (𝐵𝑍)) → 𝐶 ≤ (𝐵𝑍))
6867adantr 480 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ (𝐵𝑍))
69 iftrue 4531 . . . . . . . . . . . 12 ((𝐵𝑍) ≤ 𝐷 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = (𝐵𝑍))
7069eqcomd 2743 . . . . . . . . . . 11 ((𝐵𝑍) ≤ 𝐷 → (𝐵𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7170adantl 481 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → (𝐵𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7268, 71breqtrd 5169 . . . . . . . . 9 (((𝜑𝐶 < (𝐵𝑍)) ∧ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7352ad2antrr 726 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐶𝐷)
74 iffalse 4534 . . . . . . . . . . . 12 (¬ (𝐵𝑍) ≤ 𝐷 → if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) = 𝐷)
7574eqcomd 2743 . . . . . . . . . . 11 (¬ (𝐵𝑍) ≤ 𝐷𝐷 = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7675adantl 481 . . . . . . . . . 10 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐷 = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7773, 76breqtrd 5169 . . . . . . . . 9 (((𝜑𝐶 < (𝐵𝑍)) ∧ ¬ (𝐵𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
7872, 77pm2.61dan 813 . . . . . . . 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 4534 . . . . . . . . 9 (¬ (𝐵𝑍) ≤ 𝐶 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = 𝐶)
8180adantl 481 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) = 𝐶)
8281breq1d 5153 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → (if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷) ↔ 𝐶 ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
8379, 82mpbird 257 . . . . . 6 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝐶) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
8457, 83pm2.61dan 813 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
85 icossico 13457 . . . . 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 839 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ⊆ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
87 volss 25568 . . . 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 1373 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ≤ (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))))
8910, 14, 26, 35, 88lemul1ad 12207 . 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 4342 . . . . 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 46591 . . . . 5 (𝜑 → ((𝐻𝐶)‘𝐵):𝑋⟶ℝ)
9490, 15, 91, 1, 93hoidmvn0val 46599 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))))
9593ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑘𝑋) → (((𝐻𝐶)‘𝐵)‘𝑘) ∈ ℝ)
96 volicore 46596 . . . . . . 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 11289 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
99 fveq2 6906 . . . . . . . . 9 (𝑘 = 𝑍 → (𝐴𝑘) = (𝐴𝑍))
100 fveq2 6906 . . . . . . . . 9 (𝑘 = 𝑍 → (((𝐻𝐶)‘𝐵)‘𝑘) = (((𝐻𝐶)‘𝐵)‘𝑍))
10199, 100oveq12d 7449 . . . . . . . 8 (𝑘 = 𝑍 → ((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)) = ((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍)))
102101fveq2d 6910 . . . . . . 7 (𝑘 = 𝑍 → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))))
103102adantl 481 . . . . . 6 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))))
10492, 7, 15, 5, 3hsphoival 46594 . . . . . . . . . 10 (𝜑 → (((𝐻𝐶)‘𝐵)‘𝑍) = if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)))
1052eldifbd 3964 . . . . . . . . . . 11 (𝜑 → ¬ 𝑍𝑌)
106105iffalsed 4536 . . . . . . . . . 10 (𝜑 → if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) = if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))
107104, 106eqtrd 2777 . . . . . . . . 9 (𝜑 → (((𝐻𝐶)‘𝐵)‘𝑍) = if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))
108107oveq2d 7447 . . . . . . . 8 (𝜑 → ((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍)) = ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)))
109108fveq2d 6910 . . . . . . 7 (𝜑 → (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
110109adantr 480 . . . . . 6 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
111103, 110eqtrd 2777 . . . . 5 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
11215, 98, 3, 111fprodsplit1 45608 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)))))
1137adantr 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐶 ∈ ℝ)
11415adantr 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑋 ∈ Fin)
1155adantr 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐵:𝑋⟶ℝ)
11692, 113, 114, 115, 20hsphoival 46594 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐶)‘𝐵)‘𝑘) = if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐶, (𝐵𝑘), 𝐶)))
117 hsphoidmvle2.y . . . . . . . . . . . . 13 𝑋 = (𝑌 ∪ {𝑍})
11819, 117eleqtrdi 2851 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
119 eldifn 4132 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
120 elunnel2 4155 . . . . . . . . . . . 12 ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ {𝑍}) → 𝑘𝑌)
121118, 119, 120syl2anc 584 . . . . . . . . . . 11 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘𝑌)
122121adantl 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘𝑌)
123122iftrued 4533 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐶, (𝐵𝑘), 𝐶)) = (𝐵𝑘))
124116, 123eqtrd 2777 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐶)‘𝐵)‘𝑘) = (𝐵𝑘))
125124oveq2d 7447 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
126125fveq2d 6910 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
127126prodeq2dv 15958 . . . . 5 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
128127oveq2d 7447 . . . 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 2781 . . 3 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
13092, 11, 15, 5hsphoif 46591 . . . . 5 (𝜑 → ((𝐻𝐷)‘𝐵):𝑋⟶ℝ)
13190, 15, 91, 1, 130hoidmvn0val 46599 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))))
132130ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑘𝑋) → (((𝐻𝐷)‘𝐵)‘𝑘) ∈ ℝ)
133 volicore 46596 . . . . . . 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 11289 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) ∈ ℂ)
136 fveq2 6906 . . . . . . . 8 (𝑘 = 𝑍 → (((𝐻𝐷)‘𝐵)‘𝑘) = (((𝐻𝐷)‘𝐵)‘𝑍))
13799, 136oveq12d 7449 . . . . . . 7 (𝑘 = 𝑍 → ((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)) = ((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍)))
138137fveq2d 6910 . . . . . 6 (𝑘 = 𝑍 → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))))
139138adantl 481 . . . . 5 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))))
14015, 135, 3, 139fprodsplit1 45608 . . . 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 46594 . . . . . . . 8 (𝜑 → (((𝐻𝐷)‘𝐵)‘𝑍) = if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
142105iffalsed 4536 . . . . . . . 8 (𝜑 → if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
143141, 142eqtrd 2777 . . . . . . 7 (𝜑 → (((𝐻𝐷)‘𝐵)‘𝑍) = if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))
144143oveq2d 7447 . . . . . 6 (𝜑 → ((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍)) = ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷)))
145144fveq2d 6910 . . . . 5 (𝜑 → (vol‘((𝐴𝑍)[,)(((𝐻𝐷)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))))
14611adantr 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐷 ∈ ℝ)
14792, 146, 114, 115, 20hsphoival 46594 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐷)‘𝐵)‘𝑘) = if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐷, (𝐵𝑘), 𝐷)))
148122iftrued 4533 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐷, (𝐵𝑘), 𝐷)) = (𝐵𝑘))
149147, 148eqtrd 2777 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐷)‘𝐵)‘𝑘) = (𝐵𝑘))
150149oveq2d 7447 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
151150fveq2d 6910 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
152151prodeq2dv 15958 . . . . 5 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐷)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
153145, 152oveq12d 7449 . . . 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 2781 . . 3 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐷, (𝐵𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
155129, 154breq12d 5156 . 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 257 1 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  cdif 3948  cun 3949  wss 3951  c0 4333  ifcif 4525  {csn 4626   class class class wbr 5143  cmpt 5225  dom cdm 5685  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  m cmap 8866  Fincfn 8985  cr 11154  0cc0 11155   · cmul 11160  *cxr 11294   < clt 11295  cle 11296  [,)cico 13389  cprod 15939  volcvol 25498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-prod 15940  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500
This theorem is referenced by:  hoidmvlelem1  46610  hoidmvlelem2  46611
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