Theorem hsphoidmvle 46830

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, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
hsphoidmvle.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hsphoidmvle.z	⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
hsphoidmvle.y	⊢ 𝑋 = (𝑌 ∪ {𝑍})
hsphoidmvle.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
hsphoidmvle.h	⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
hsphoidmvle.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
hsphoidmvle.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)

Assertion

Ref	Expression
hsphoidmvle	⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) ≤ (𝐴(𝐿‘𝑋)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

Step	Hyp	Ref	Expression
1		hsphoidmvle.a	. . . . 5 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
2		hsphoidmvle.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
3	2	eldifad 3913	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
4	1, 3	ffvelcdmd 7030	. . . 4 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
5		hsphoidmvle.b	. . . . . 6 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
6	5, 3	ffvelcdmd 7030	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
7		hsphoidmvle.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
8	6, 7	ifcld 4526	. . . 4 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ)
9		volicore 46825	. . . 4 ⊢ (((𝐴‘𝑍) ∈ ℝ ∧ if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ) → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ∈ ℝ)
10	4, 8, 9	syl2anc 584	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ∈ ℝ)
11		volicore 46825	. . . 4 ⊢ (((𝐴‘𝑍) ∈ ℝ ∧ (𝐵‘𝑍) ∈ ℝ) → (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) ∈ ℝ)
12	4, 6, 11	syl2anc 584	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) ∈ ℝ)
13		hsphoidmvle.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
14		difssd 4089	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
15		ssfi 9097	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
16	13, 14, 15	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
17		eldifi 4083	. . . . . 6 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑋)
18	17	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑋)
19	1	ffvelcdmda 7029	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
20	5	ffvelcdmda 7029	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
21		volicore 46825	. . . . . 6 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
22	19, 20, 21	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
23	18, 22	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
24	16, 23	fprodrecl 15876	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
25		nfv 1915	. . . 4 ⊢ Ⅎ𝑘𝜑
26	18, 19	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐴‘𝑘) ∈ ℝ)
27	18, 20	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵‘𝑘) ∈ ℝ)
28	27	rexrd 11182	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵‘𝑘) ∈ ℝ*)
29		icombl 25521	. . . . . 6 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ*) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
30	26, 28, 29	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
31		volge0 46205	. . . . 5 ⊢ (((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol → 0 ≤ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
32	30, 31	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 0 ≤ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
33	25, 16, 23, 32	fprodge0 15916	. . 3 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
34	8	rexrd 11182	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ*)
35		icombl 25521	. . . . 5 ⊢ (((𝐴‘𝑍) ∈ ℝ ∧ if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ*) → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ∈ dom vol)
36	4, 34, 35	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ∈ dom vol)
37	6	rexrd 11182	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ*)
38		icombl 25521	. . . . 5 ⊢ (((𝐴‘𝑍) ∈ ℝ ∧ (𝐵‘𝑍) ∈ ℝ*) → ((𝐴‘𝑍)[,)(𝐵‘𝑍)) ∈ dom vol)
39	4, 37, 38	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(𝐵‘𝑍)) ∈ dom vol)
40	4	rexrd 11182	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ*)
41	4	leidd 11703	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐴‘𝑍))
42		min1 13104	. . . . . 6 ⊢ (((𝐵‘𝑍) ∈ ℝ ∧ 𝐶 ∈ ℝ) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ (𝐵‘𝑍))
43	6, 7, 42	syl2anc 584	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ (𝐵‘𝑍))
44		icossico 13332	. . . . 5 ⊢ ((((𝐴‘𝑍) ∈ ℝ* ∧ (𝐵‘𝑍) ∈ ℝ*) ∧ ((𝐴‘𝑍) ≤ (𝐴‘𝑍) ∧ if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ (𝐵‘𝑍))) → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ⊆ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
45	40, 37, 41, 43, 44	syl22anc 838	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ⊆ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
46		volss 25490	. . . 4 ⊢ ((((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ∈ dom vol ∧ ((𝐴‘𝑍)[,)(𝐵‘𝑍)) ∈ dom vol ∧ ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ⊆ ((𝐴‘𝑍)[,)(𝐵‘𝑍))) → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ≤ (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))))
47	36, 39, 45, 46	syl3anc 1373	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ≤ (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))))
48	10, 12, 24, 33, 47	lemul1ad 12081	. 2 ⊢ (𝜑 → ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) ≤ ((vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
49		hsphoidmvle.l	. . . . 5 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
50	3	ne0d 4294	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ ∅)
51		hsphoidmvle.h	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
52	51, 7, 13, 5	hsphoif 46820	. . . . 5 ⊢ (𝜑 → ((𝐻‘𝐶)‘𝐵):𝑋⟶ℝ)
53	49, 13, 50, 1, 52	hoidmvn0val 46828	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))))
54	52	ffvelcdmda 7029	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐻‘𝐶)‘𝐵)‘𝑘) ∈ ℝ)
55		volicore 46825	. . . . . . 7 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (((𝐻‘𝐶)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
56	19, 54, 55	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
57	56	recnd 11160	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
58		fveq2 6834	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
59		fveq2 6834	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (((𝐻‘𝐶)‘𝐵)‘𝑍))
60	58, 59	oveq12d 7376	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)))
61	60	fveq2d 6838	. . . . . . 7 ⊢ (𝑘 = 𝑍 → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))))
62	61	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))))
63	51, 7, 13, 5, 3	hsphoival 46823	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
64	2	eldifbd 3914	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
65	64	iffalsed 4490	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) = if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))
66	63, 65	eqtrd 2771	. . . . . . . . 9 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))
67	66	oveq2d 7374	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)) = ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
68	67	fveq2d 6838	. . . . . . 7 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))))
69	68	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))))
70	62, 69	eqtrd 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))))
71	13, 57, 3, 70	fprodsplit1 45839	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)))))
72	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐶 ∈ ℝ)
73	13	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑋 ∈ Fin)
74	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐵:𝑋⟶ℝ)
75	51, 72, 73, 74, 18	hsphoival 46823	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐶)‘𝐵)‘𝑘) = if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐶, (𝐵‘𝑘), 𝐶)))
76		hsphoidmvle.y	. . . . . . . . . . . . 13 ⊢ 𝑋 = (𝑌 ∪ {𝑍})
77	17, 76	eleqtrdi 2846	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
78		eldifn 4084	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
79		elunnel2 4107	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ {𝑍}) → 𝑘 ∈ 𝑌)
80	77, 78, 79	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑌)
81	80	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑌)
82	81	iftrued 4487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐶, (𝐵‘𝑘), 𝐶)) = (𝐵‘𝑘))
83	75, 82	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (𝐵‘𝑘))
84	83	oveq2d 7374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
85	84	fveq2d 6838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
86	85	prodeq2dv 15845	. . . . 5 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
87	86	oveq2d 7374	. . . 4 ⊢ (𝜑 → ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)))) = ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
88	53, 71, 87	3eqtrd 2775	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) = ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
89	49, 1, 5, 13	hoidmvval 46821	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
90	50	neneqd 2937	. . . . 5 ⊢ (𝜑 → ¬ 𝑋 = ∅)
91	90	iffalsed 4490	. . . 4 ⊢ (𝜑 → if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
92	22	recnd 11160	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℂ)
93		fveq2 6834	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → (𝐵‘𝑘) = (𝐵‘𝑍))
94	58, 93	oveq12d 7376	. . . . . . 7 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
95	94	fveq2d 6838	. . . . . 6 ⊢ (𝑘 = 𝑍 → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))))
96	95	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))))
97	13, 92, 3, 96	fprodsplit1 45839	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
98	89, 91, 97	3eqtrd 2775	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = ((vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
99	88, 98	breq12d 5111	. 2 ⊢ (𝜑 → ((𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) ≤ (𝐴(𝐿‘𝑋)𝐵) ↔ ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) ≤ ((vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))))
100	48, 99	mpbird 257	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) ≤ (𝐴(𝐿‘𝑋)𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3898   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285  ifcif 4479  {csn 4580   class class class wbr 5098   ↦ cmpt 5179  dom cdm 5624  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360   ↑_m cmap 8763  Fincfn 8883  ℝcr 11025  0cc0 11026   · cmul 11031  ℝ^*cxr 11165   ≤ cle 11167  [,)cico 13263  ∏cprod 15826  volcvol 25420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9314  df-sup 9345  df-inf 9346  df-oi 9415  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-z 12489  df-uz 12752  df-q 12862  df-rp 12906  df-xneg 13026  df-xadd 13027  df-xmul 13028  df-ioo 13265  df-ico 13267  df-icc 13268  df-fz 13424  df-fzo 13571  df-fl 13712  df-seq 13925  df-exp 13985  df-hash 14254  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-clim 15411  df-rlim 15412  df-sum 15610  df-prod 15827  df-rest 17342  df-topgen 17363  df-psmet 21301  df-xmet 21302  df-met 21303  df-bl 21304  df-mopn 21305  df-top 22838  df-topon 22855  df-bases 22890  df-cmp 23331  df-ovol 25421  df-vol 25422

This theorem is referenced by:  sge0hsphoire  46833  hoidmvlelem1  46839  hoidmvlelem4  46842  hspmbllem2  46871