Theorem hsphoidmvle 46746

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 3910	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
4	1, 3	ffvelcdmd 7027	. . . 4 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
5		hsphoidmvle.b	. . . . . 6 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
6	5, 3	ffvelcdmd 7027	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
7		hsphoidmvle.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
8	6, 7	ifcld 4523	. . . 4 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ)
9		volicore 46741	. . . 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 46741	. . . 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 4086	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
15		ssfi 9093	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
16	13, 14, 15	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
17		eldifi 4080	. . . . . 6 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑋)
18	17	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑋)
19	1	ffvelcdmda 7026	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
20	5	ffvelcdmda 7026	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
21		volicore 46741	. . . . . 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 15867	. . 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 11173	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵‘𝑘) ∈ ℝ*)
29		icombl 25512	. . . . . 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 46121	. . . . 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 15907	. . 3 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
34	8	rexrd 11173	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ*)
35		icombl 25512	. . . . 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 11173	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ*)
38		icombl 25512	. . . . 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 11173	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ*)
41	4	leidd 11694	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐴‘𝑍))
42		min1 13095	. . . . . 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 13323	. . . . 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 25481	. . . 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 12072	. 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 4291	. . . . 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 46736	. . . . 5 ⊢ (𝜑 → ((𝐻‘𝐶)‘𝐵):𝑋⟶ℝ)
53	49, 13, 50, 1, 52	hoidmvn0val 46744	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))))
54	52	ffvelcdmda 7026	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐻‘𝐶)‘𝐵)‘𝑘) ∈ ℝ)
55		volicore 46741	. . . . . . 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 11151	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
58		fveq2 6831	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
59		fveq2 6831	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (((𝐻‘𝐶)‘𝐵)‘𝑍))
60	58, 59	oveq12d 7373	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)))
61	60	fveq2d 6835	. . . . . . 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 46739	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
64	2	eldifbd 3911	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
65	64	iffalsed 4487	. . . . . . . . . 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 2768	. . . . . . . . 9 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))
67	66	oveq2d 7371	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)) = ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
68	67	fveq2d 6835	. . . . . . 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 2768	. . . . 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 45755	. . . 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 46739	. . . . . . . . 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 2843	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
78		eldifn 4081	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
79		elunnel2 4104	. . . . . . . . . . . 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 4484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐶, (𝐵‘𝑘), 𝐶)) = (𝐵‘𝑘))
83	75, 82	eqtrd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (𝐵‘𝑘))
84	83	oveq2d 7371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
85	84	fveq2d 6835	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
86	85	prodeq2dv 15836	. . . . 5 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
87	86	oveq2d 7371	. . . 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 2772	. . 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 46737	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
90	50	neneqd 2934	. . . . 5 ⊢ (𝜑 → ¬ 𝑋 = ∅)
91	90	iffalsed 4487	. . . 4 ⊢ (𝜑 → if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
92	22	recnd 11151	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℂ)
93		fveq2 6831	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → (𝐵‘𝑘) = (𝐵‘𝑍))
94	58, 93	oveq12d 7373	. . . . . . 7 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
95	94	fveq2d 6835	. . . . . 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 45755	. . . 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 2772	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = ((vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
99	88, 98	breq12d 5108	. 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 3895   ∪ cun 3896   ⊆ wss 3898  ∅c0 4282  ifcif 4476  {csn 4577   class class class wbr 5095   ↦ cmpt 5176  dom cdm 5621  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355   ∈ cmpo 7357   ↑_m cmap 8759  Fincfn 8879  ℝcr 11016  0cc0 11017   · cmul 11022  ℝ^*cxr 11156   ≤ cle 11158  [,)cico 13254  ∏cprod 15817  volcvol 25411

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 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-inf2 9542  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094  ax-pre-sup 11095

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-of 7619  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-er 8631  df-map 8761  df-pm 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fi 9306  df-sup 9337  df-inf 9338  df-oi 9407  df-dju 9805  df-card 9843  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-div 11786  df-nn 12137  df-2 12199  df-3 12200  df-n0 12393  df-z 12480  df-uz 12743  df-q 12853  df-rp 12897  df-xneg 13017  df-xadd 13018  df-xmul 13019  df-ioo 13256  df-ico 13258  df-icc 13259  df-fz 13415  df-fzo 13562  df-fl 13703  df-seq 13916  df-exp 13976  df-hash 14245  df-cj 15013  df-re 15014  df-im 15015  df-sqrt 15149  df-abs 15150  df-clim 15402  df-rlim 15403  df-sum 15601  df-prod 15818  df-rest 17333  df-topgen 17354  df-psmet 21292  df-xmet 21293  df-met 21294  df-bl 21295  df-mopn 21296  df-top 22829  df-topon 22846  df-bases 22881  df-cmp 23322  df-ovol 25412  df-vol 25413

This theorem is referenced by:  sge0hsphoire  46749  hoidmvlelem1  46755  hoidmvlelem4  46758  hspmbllem2  46787