Theorem hsphoidmvle 40800

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:   𝑥,𝑘   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 3586	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
4	1, 3	ffvelrnd 6360	. . . 4 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
5		hsphoidmvle.b	. . . . . 6 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
6	5, 3	ffvelrnd 6360	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
7		hsphoidmvle.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
8	6, 7	ifcld 4131	. . . 4 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ)
9		volicore 40795	. . . 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 693	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ∈ ℝ)
11		volicore 40795	. . . 4 ⊢ (((𝐴‘𝑍) ∈ ℝ ∧ (𝐵‘𝑍) ∈ ℝ) → (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) ∈ ℝ)
12	4, 6, 11	syl2anc 693	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) ∈ ℝ)
13		hsphoidmvle.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
14		difssd 3738	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
15		ssfi 8180	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
16	13, 14, 15	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
17		eldifi 3732	. . . . . 6 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑋)
18	17	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑋)
19	1	ffvelrnda 6359	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
20	5	ffvelrnda 6359	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
21		volicore 40795	. . . . . 6 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
22	19, 20, 21	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
23	18, 22	syldan 487	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
24	16, 23	fprodrecl 14683	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
25		nfv 1843	. . . 4 ⊢ Ⅎ𝑘𝜑
26	18, 19	syldan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐴‘𝑘) ∈ ℝ)
27	18, 20	syldan 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵‘𝑘) ∈ ℝ)
28	27	rexrd 10089	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵‘𝑘) ∈ ℝ*)
29		icombl 23332	. . . . . 6 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ*) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
30	26, 28, 29	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
31		volge0 40177	. . . . 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 14724	. . 3 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
34	8	rexrd 10089	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ*)
35		icombl 23332	. . . . 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 693	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ∈ dom vol)
37	6	rexrd 10089	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ*)
38		icombl 23332	. . . . 5 ⊢ (((𝐴‘𝑍) ∈ ℝ ∧ (𝐵‘𝑍) ∈ ℝ*) → ((𝐴‘𝑍)[,)(𝐵‘𝑍)) ∈ dom vol)
39	4, 37, 38	syl2anc 693	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(𝐵‘𝑍)) ∈ dom vol)
40	4	rexrd 10089	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ*)
41	4	leidd 10594	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐴‘𝑍))
42		min1 12020	. . . . . 6 ⊢ (((𝐵‘𝑍) ∈ ℝ ∧ 𝐶 ∈ ℝ) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ (𝐵‘𝑍))
43	6, 7, 42	syl2anc 693	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ (𝐵‘𝑍))
44		icossico 12243	. . . . 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 1327	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ⊆ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
46		volss 23301	. . . 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 1326	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ≤ (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))))
48	10, 12, 24, 33, 47	lemul1ad 10963	. 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		ne0i 3921	. . . . . 6 ⊢ (𝑍 ∈ 𝑋 → 𝑋 ≠ ∅)
51	3, 50	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ ∅)
52		hsphoidmvle.h	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
53	52, 7, 13, 5	hsphoif 40790	. . . . 5 ⊢ (𝜑 → ((𝐻‘𝐶)‘𝐵):𝑋⟶ℝ)
54	49, 13, 51, 1, 53	hoidmvn0val 40798	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))))
55	53	ffvelrnda 6359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐻‘𝐶)‘𝐵)‘𝑘) ∈ ℝ)
56		volicore 40795	. . . . . . 7 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (((𝐻‘𝐶)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
57	19, 55, 56	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
58	57	recnd 10068	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
59		fveq2 6191	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
60		fveq2 6191	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (((𝐻‘𝐶)‘𝐵)‘𝑍))
61	59, 60	oveq12d 6668	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)))
62	61	fveq2d 6195	. . . . . . 7 ⊢ (𝑘 = 𝑍 → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))))
63	62	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))))
64	52, 7, 13, 5, 3	hsphoival 40793	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
65	2	eldifbd 3587	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
66	65	iffalsed 4097	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) = if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))
67	64, 66	eqtrd 2656	. . . . . . . . 9 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))
68	67	oveq2d 6666	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)) = ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
69	68	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))))
70	69	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))))
71	63, 70	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))))
72	13, 58, 3, 71	fprodsplit1 39825	. . . 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)))))
73	7	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐶 ∈ ℝ)
74	13	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑋 ∈ Fin)
75	5	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐵:𝑋⟶ℝ)
76	52, 73, 74, 75, 18	hsphoival 40793	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐶)‘𝐵)‘𝑘) = if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐶, (𝐵‘𝑘), 𝐶)))
77		hsphoidmvle.y	. . . . . . . . . . . . 13 ⊢ 𝑋 = (𝑌 ∪ {𝑍})
78	17, 77	syl6eleq 2711	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
79		eldifn 3733	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
80		elunnel2 39198	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ {𝑍}) → 𝑘 ∈ 𝑌)
81	78, 79, 80	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑌)
82	81	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑌)
83	82	iftrued 4094	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐶, (𝐵‘𝑘), 𝐶)) = (𝐵‘𝑘))
84	76, 83	eqtrd 2656	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (𝐵‘𝑘))
85	84	oveq2d 6666	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
86	85	fveq2d 6195	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
87	86	prodeq2dv 14653	. . . . 5 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
88	87	oveq2d 6666	. . . 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)))))
89	54, 72, 88	3eqtrd 2660	. . 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)))))
90	49, 1, 5, 13	hoidmvval 40791	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
91	51	neneqd 2799	. . . . 5 ⊢ (𝜑 → ¬ 𝑋 = ∅)
92	91	iffalsed 4097	. . . 4 ⊢ (𝜑 → if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
93	22	recnd 10068	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℂ)
94		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → (𝐵‘𝑘) = (𝐵‘𝑍))
95	59, 94	oveq12d 6668	. . . . . . 7 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
96	95	fveq2d 6195	. . . . . 6 ⊢ (𝑘 = 𝑍 → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))))
97	96	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))))
98	13, 93, 3, 97	fprodsplit1 39825	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
99	90, 92, 98	3eqtrd 2660	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = ((vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
100	89, 99	breq12d 4666	. 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))))))
101	48, 100	mpbird 247	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) ≤ (𝐴(𝐿‘𝑋)𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  ifcif 4086  {csn 4177   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ↑_𝑚 cmap 7857  Fincfn 7955  ℝcr 9935  0cc0 9936   · cmul 9941  ℝ^*cxr 10073   ≤ cle 10075  [,)cico 12177  ∏cprod 14635  volcvol 23232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-prod 14636  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234

This theorem is referenced by:  sge0hsphoire  40803  hoidmvlelem1  40809  hoidmvlelem4  40812  hspmbllem2  40841