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Theorem hsphoidmvle 40104
Description: The dimensional volume of a half-open interval intersected with a half-space, is less than or equal to the dimensional volume of the original half-open interval. Used in the last inequality of step (e) of Lemma 115B of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
hsphoidmvle.l 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
hsphoidmvle.x (𝜑𝑋 ∈ Fin)
hsphoidmvle.z (𝜑𝑍 ∈ (𝑋𝑌))
hsphoidmvle.y 𝑋 = (𝑌 ∪ {𝑍})
hsphoidmvle.c (𝜑𝐶 ∈ ℝ)
hsphoidmvle.h 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))
hsphoidmvle.a (𝜑𝐴:𝑋⟶ℝ)
hsphoidmvle.b (𝜑𝐵:𝑋⟶ℝ)
Assertion
Ref Expression
hsphoidmvle (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)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   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 hsphoidmvle
StepHypRef Expression
1 hsphoidmvle.a . . . . 5 (𝜑𝐴:𝑋⟶ℝ)
2 hsphoidmvle.z . . . . . 6 (𝜑𝑍 ∈ (𝑋𝑌))
32eldifad 3567 . . . . 5 (𝜑𝑍𝑋)
41, 3ffvelrnd 6316 . . . 4 (𝜑 → (𝐴𝑍) ∈ ℝ)
5 hsphoidmvle.b . . . . . 6 (𝜑𝐵:𝑋⟶ℝ)
65, 3ffvelrnd 6316 . . . . 5 (𝜑 → (𝐵𝑍) ∈ ℝ)
7 hsphoidmvle.c . . . . 5 (𝜑𝐶 ∈ ℝ)
86, 7ifcld 4103 . . . 4 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ)
9 volicore 40099 . . . 4 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ) → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ∈ ℝ)
104, 8, 9syl2anc 692 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ∈ ℝ)
11 volicore 40099 . . . 4 (((𝐴𝑍) ∈ ℝ ∧ (𝐵𝑍) ∈ ℝ) → (vol‘((𝐴𝑍)[,)(𝐵𝑍))) ∈ ℝ)
124, 6, 11syl2anc 692 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)(𝐵𝑍))) ∈ ℝ)
13 hsphoidmvle.x . . . . 5 (𝜑𝑋 ∈ Fin)
14 difssd 3716 . . . . 5 (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
15 ssfi 8124 . . . . 5 ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
1613, 14, 15syl2anc 692 . . . 4 (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
17 eldifi 3710 . . . . . 6 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘𝑋)
1817adantl 482 . . . . 5 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘𝑋)
191ffvelrnda 6315 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
205ffvelrnda 6315 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
21 volicore 40099 . . . . . 6 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2219, 20, 21syl2anc 692 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2318, 22syldan 487 . . . 4 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
2416, 23fprodrecl 14608 . . 3 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
25 nfv 1840 . . . 4 𝑘𝜑
2618, 19syldan 487 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝑘) ∈ ℝ)
2718, 20syldan 487 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵𝑘) ∈ ℝ)
2827rexrd 10033 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵𝑘) ∈ ℝ*)
29 icombl 23239 . . . . . 6 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ*) → ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol)
3026, 28, 29syl2anc 692 . . . . 5 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol)
31 volge0 39481 . . . . 5 (((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom vol → 0 ≤ (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
3230, 31syl 17 . . . 4 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 0 ≤ (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
3325, 16, 23, 32fprodge0 14649 . . 3 (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
348rexrd 10033 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ*)
35 icombl 23239 . . . . 5 (((𝐴𝑍) ∈ ℝ ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ∈ ℝ*) → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol)
364, 34, 35syl2anc 692 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol)
376rexrd 10033 . . . . 5 (𝜑 → (𝐵𝑍) ∈ ℝ*)
38 icombl 23239 . . . . 5 (((𝐴𝑍) ∈ ℝ ∧ (𝐵𝑍) ∈ ℝ*) → ((𝐴𝑍)[,)(𝐵𝑍)) ∈ dom vol)
394, 37, 38syl2anc 692 . . . 4 (𝜑 → ((𝐴𝑍)[,)(𝐵𝑍)) ∈ dom vol)
404rexrd 10033 . . . . 5 (𝜑 → (𝐴𝑍) ∈ ℝ*)
414leidd 10538 . . . . 5 (𝜑 → (𝐴𝑍) ≤ (𝐴𝑍))
42 min1 11963 . . . . . 6 (((𝐵𝑍) ∈ ℝ ∧ 𝐶 ∈ ℝ) → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ (𝐵𝑍))
436, 7, 42syl2anc 692 . . . . 5 (𝜑 → if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ (𝐵𝑍))
44 icossico 12185 . . . . 5 ((((𝐴𝑍) ∈ ℝ* ∧ (𝐵𝑍) ∈ ℝ*) ∧ ((𝐴𝑍) ≤ (𝐴𝑍) ∧ if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶) ≤ (𝐵𝑍))) → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ⊆ ((𝐴𝑍)[,)(𝐵𝑍)))
4540, 37, 41, 43, 44syl22anc 1324 . . . 4 (𝜑 → ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ⊆ ((𝐴𝑍)[,)(𝐵𝑍)))
46 volss 23208 . . . 4 ((((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ∈ dom vol ∧ ((𝐴𝑍)[,)(𝐵𝑍)) ∈ dom vol ∧ ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) ⊆ ((𝐴𝑍)[,)(𝐵𝑍))) → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ≤ (vol‘((𝐴𝑍)[,)(𝐵𝑍))))
4736, 39, 45, 46syl3anc 1323 . . 3 (𝜑 → (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) ≤ (vol‘((𝐴𝑍)[,)(𝐵𝑍))))
4810, 12, 24, 33, 47lemul1ad 10907 . 2 (𝜑 → ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))) ≤ ((vol‘((𝐴𝑍)[,)(𝐵𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
49 hsphoidmvle.l . . . . 5 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
50 ne0i 3897 . . . . . 6 (𝑍𝑋𝑋 ≠ ∅)
513, 50syl 17 . . . . 5 (𝜑𝑋 ≠ ∅)
52 hsphoidmvle.h . . . . . 6 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))
5352, 7, 13, 5hsphoif 40094 . . . . 5 (𝜑 → ((𝐻𝐶)‘𝐵):𝑋⟶ℝ)
5449, 13, 51, 1, 53hoidmvn0val 40102 . . . 4 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))))
5553ffvelrnda 6315 . . . . . . 7 ((𝜑𝑘𝑋) → (((𝐻𝐶)‘𝐵)‘𝑘) ∈ ℝ)
56 volicore 40099 . . . . . . 7 (((𝐴𝑘) ∈ ℝ ∧ (((𝐻𝐶)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
5719, 55, 56syl2anc 692 . . . . . 6 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
5857recnd 10012 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
59 fveq2 6148 . . . . . . . . 9 (𝑘 = 𝑍 → (𝐴𝑘) = (𝐴𝑍))
60 fveq2 6148 . . . . . . . . 9 (𝑘 = 𝑍 → (((𝐻𝐶)‘𝐵)‘𝑘) = (((𝐻𝐶)‘𝐵)‘𝑍))
6159, 60oveq12d 6622 . . . . . . . 8 (𝑘 = 𝑍 → ((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)) = ((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍)))
6261fveq2d 6152 . . . . . . 7 (𝑘 = 𝑍 → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))))
6362adantl 482 . . . . . 6 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))))
6452, 7, 13, 5, 3hsphoival 40097 . . . . . . . . . 10 (𝜑 → (((𝐻𝐶)‘𝐵)‘𝑍) = if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)))
652eldifbd 3568 . . . . . . . . . . 11 (𝜑 → ¬ 𝑍𝑌)
6665iffalsed 4069 . . . . . . . . . 10 (𝜑 → if(𝑍𝑌, (𝐵𝑍), if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)) = if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))
6764, 66eqtrd 2655 . . . . . . . . 9 (𝜑 → (((𝐻𝐶)‘𝐵)‘𝑍) = if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))
6867oveq2d 6620 . . . . . . . 8 (𝜑 → ((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍)) = ((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶)))
6968fveq2d 6152 . . . . . . 7 (𝜑 → (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
7069adantr 481 . . . . . 6 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑍)[,)(((𝐻𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
7163, 70eqtrd 2655 . . . . 5 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))))
7213, 58, 3, 71fprodsplit1 39226 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)))))
737adantr 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐶 ∈ ℝ)
7413adantr 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑋 ∈ Fin)
755adantr 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐵:𝑋⟶ℝ)
7652, 73, 74, 75, 18hsphoival 40097 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐶)‘𝐵)‘𝑘) = if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐶, (𝐵𝑘), 𝐶)))
77 hsphoidmvle.y . . . . . . . . . . . . 13 𝑋 = (𝑌 ∪ {𝑍})
7817, 77syl6eleq 2708 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
79 eldifn 3711 . . . . . . . . . . . 12 (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
80 elunnel2 38678 . . . . . . . . . . . 12 ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ {𝑍}) → 𝑘𝑌)
8178, 79, 80syl2anc 692 . . . . . . . . . . 11 (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘𝑌)
8281adantl 482 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘𝑌)
8382iftrued 4066 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘𝑌, (𝐵𝑘), if((𝐵𝑘) ≤ 𝐶, (𝐵𝑘), 𝐶)) = (𝐵𝑘))
8476, 83eqtrd 2655 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻𝐶)‘𝐵)‘𝑘) = (𝐵𝑘))
8584oveq2d 6620 . . . . . . 7 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
8685fveq2d 6152 . . . . . 6 ((𝜑𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
8786prodeq2dv 14578 . . . . 5 (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(((𝐻𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))
8887oveq2d 6620 . . . 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)))))
8954, 72, 883eqtrd 2659 . . 3 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) = ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
9049, 1, 5, 13hoidmvval 40095 . . . 4 (𝜑 → (𝐴(𝐿𝑋)𝐵) = if(𝑋 = ∅, 0, ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
9151neneqd 2795 . . . . 5 (𝜑 → ¬ 𝑋 = ∅)
9291iffalsed 4069 . . . 4 (𝜑 → if(𝑋 = ∅, 0, ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘)))) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
9322recnd 10012 . . . . 5 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℂ)
94 fveq2 6148 . . . . . . . 8 (𝑘 = 𝑍 → (𝐵𝑘) = (𝐵𝑍))
9559, 94oveq12d 6622 . . . . . . 7 (𝑘 = 𝑍 → ((𝐴𝑘)[,)(𝐵𝑘)) = ((𝐴𝑍)[,)(𝐵𝑍)))
9695fveq2d 6152 . . . . . 6 (𝑘 = 𝑍 → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (vol‘((𝐴𝑍)[,)(𝐵𝑍))))
9796adantl 482 . . . . 5 ((𝜑𝑘 = 𝑍) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (vol‘((𝐴𝑍)[,)(𝐵𝑍))))
9813, 93, 3, 97fprodsplit1 39226 . . . 4 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ((vol‘((𝐴𝑍)[,)(𝐵𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
9990, 92, 983eqtrd 2659 . . 3 (𝜑 → (𝐴(𝐿𝑋)𝐵) = ((vol‘((𝐴𝑍)[,)(𝐵𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
10089, 99breq12d 4626 . 2 (𝜑 → ((𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)𝐵) ↔ ((vol‘((𝐴𝑍)[,)if((𝐵𝑍) ≤ 𝐶, (𝐵𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘)))) ≤ ((vol‘((𝐴𝑍)[,)(𝐵𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴𝑘)[,)(𝐵𝑘))))))
10148, 100mpbird 247 1 (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  cdif 3552  cun 3553  wss 3555  c0 3891  ifcif 4058  {csn 4148   class class class wbr 4613  cmpt 4673  dom cdm 5074  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802  Fincfn 7899  cr 9879  0cc0 9880   · cmul 9885  *cxr 10017  cle 10019  [,)cico 12119  cprod 14560  volcvol 23139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-rlim 14154  df-sum 14351  df-prod 14561  df-rest 16004  df-topgen 16025  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-top 20621  df-bases 20622  df-topon 20623  df-cmp 21100  df-ovol 23140  df-vol 23141
This theorem is referenced by:  sge0hsphoire  40107  hoidmvlelem1  40113  hoidmvlelem4  40116  hspmbllem2  40145
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