hsphoidmvle - Mathbox for Glauco Siliprandi

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Theorem hsphoidmvle 41276

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 3781	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
4	1, 3	ffvelrnd 6578	. . . 4 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
5		hsphoidmvle.b	. . . . . 6 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
6	5, 3	ffvelrnd 6578	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
7		hsphoidmvle.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
8	6, 7	ifcld 4324	. . . 4 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ)
9		volicore 41271	. . . 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 575	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ∈ ℝ)
11		volicore 41271	. . . 4 ⊢ (((𝐴‘𝑍) ∈ ℝ ∧ (𝐵‘𝑍) ∈ ℝ) → (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) ∈ ℝ)
12	4, 6, 11	syl2anc 575	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) ∈ ℝ)
13		hsphoidmvle.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
14		difssd 3937	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
15		ssfi 8415	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
16	13, 14, 15	syl2anc 575	. . . 4 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
17		eldifi 3931	. . . . . 6 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑋)
18	17	adantl 469	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑋)
19	1	ffvelrnda 6577	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
20	5	ffvelrnda 6577	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
21		volicore 41271	. . . . . 6 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
22	19, 20, 21	syl2anc 575	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
23	18, 22	syldan 581	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
24	16, 23	fprodrecl 14900	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
25		nfv 2005	. . . 4 ⊢ Ⅎ𝑘𝜑
26	18, 19	syldan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐴‘𝑘) ∈ ℝ)
27	18, 20	syldan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵‘𝑘) ∈ ℝ)
28	27	rexrd 10370	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵‘𝑘) ∈ ℝ^*)
29		icombl 23541	. . . . . 6 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ^*) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
30	26, 28, 29	syl2anc 575	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
31		volge0 40650	. . . . 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 14940	. . 3 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
34	8	rexrd 10370	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ^*)
35		icombl 23541	. . . . 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 575	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ∈ dom vol)
37	6	rexrd 10370	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ^*)
38		icombl 23541	. . . . 5 ⊢ (((𝐴‘𝑍) ∈ ℝ ∧ (𝐵‘𝑍) ∈ ℝ^*) → ((𝐴‘𝑍)[,)(𝐵‘𝑍)) ∈ dom vol)
39	4, 37, 38	syl2anc 575	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(𝐵‘𝑍)) ∈ dom vol)
40	4	rexrd 10370	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ^*)
41	4	leidd 10875	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐴‘𝑍))
42		min1 12234	. . . . . 6 ⊢ (((𝐵‘𝑍) ∈ ℝ ∧ 𝐶 ∈ ℝ) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ (𝐵‘𝑍))
43	6, 7, 42	syl2anc 575	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ (𝐵‘𝑍))
44		icossico 12457	. . . . 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 858	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ⊆ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
46		volss 23510	. . . 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 1483	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ≤ (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))))
48	10, 12, 24, 33, 47	lemul1ad 11244	. 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 4123	. . . . 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 41266	. . . . 5 ⊢ (𝜑 → ((𝐻‘𝐶)‘𝐵):𝑋⟶ℝ)
53	49, 13, 50, 1, 52	hoidmvn0val 41274	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))))
54	52	ffvelrnda 6577	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐻‘𝐶)‘𝐵)‘𝑘) ∈ ℝ)
55		volicore 41271	. . . . . . 7 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (((𝐻‘𝐶)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
56	19, 54, 55	syl2anc 575	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
57	56	recnd 10349	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
58		fveq2 6404	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
59		fveq2 6404	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (((𝐻‘𝐶)‘𝐵)‘𝑍))
60	58, 59	oveq12d 6888	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)))
61	60	fveq2d 6408	. . . . . . 7 ⊢ (𝑘 = 𝑍 → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))))
62	61	adantl 469	. . . . . 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 41269	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
64	2	eldifbd 3782	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
65	64	iffalsed 4290	. . . . . . . . . 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 2840	. . . . . . . . 9 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))
67	66	oveq2d 6886	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)) = ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
68	67	fveq2d 6408	. . . . . . 7 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))))
69	68	adantr 468	. . . . . 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 2840	. . . . 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 40299	. . . 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 468	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐶 ∈ ℝ)
73	13	adantr 468	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑋 ∈ Fin)
74	5	adantr 468	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐵:𝑋⟶ℝ)
75	51, 72, 73, 74, 18	hsphoival 41269	. . . . . . . . 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	syl6eleq 2895	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
78		eldifn 3932	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
79		elunnel2 39686	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ {𝑍}) → 𝑘 ∈ 𝑌)
80	77, 78, 79	syl2anc 575	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑌)
81	80	adantl 469	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑌)
82	81	iftrued 4287	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐶, (𝐵‘𝑘), 𝐶)) = (𝐵‘𝑘))
83	75, 82	eqtrd 2840	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (𝐵‘𝑘))
84	83	oveq2d 6886	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
85	84	fveq2d 6408	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
86	85	prodeq2dv 14870	. . . . 5 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
87	86	oveq2d 6886	. . . 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 2844	. . 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 41267	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
90	50	neneqd 2983	. . . . 5 ⊢ (𝜑 → ¬ 𝑋 = ∅)
91	90	iffalsed 4290	. . . 4 ⊢ (𝜑 → if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
92	22	recnd 10349	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℂ)
93		fveq2 6404	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → (𝐵‘𝑘) = (𝐵‘𝑍))
94	58, 93	oveq12d 6888	. . . . . . 7 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
95	94	fveq2d 6408	. . . . . 6 ⊢ (𝑘 = 𝑍 → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))))
96	95	adantl 469	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))))
97	13, 92, 3, 96	fprodsplit1 40299	. . . 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 2844	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = ((vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
99	88, 98	breq12d 4857	. 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 248	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) ≤ (𝐴(𝐿‘𝑋)𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1637 ∈ wcel 2156 ∖ cdif 3766 ∪ cun 3767 ⊆ wss 3769 ∅c0 4116 ifcif 4279 {csn 4370 class class class wbr 4844 ↦ cmpt 4923 dom cdm 5311 ⟶wf 6093 ‘cfv 6097 (class class class)co 6870 ↦ cmpt2 6872 ↑_𝑚 cmap 8088 Fincfn 8188 ℝcr 10216 0cc0 10217 · cmul 10222 ℝ^*cxr 10354 ≤ cle 10356 [,)cico 12391 ∏cprod 14852 volcvol 23440

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2068 ax-7 2104 ax-8 2158 ax-9 2165 ax-10 2185 ax-11 2201 ax-12 2214 ax-13 2420 ax-ext 2784 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5096 ax-un 7175 ax-inf2 8781 ax-cnex 10273 ax-resscn 10274 ax-1cn 10275 ax-icn 10276 ax-addcl 10277 ax-addrcl 10278 ax-mulcl 10279 ax-mulrcl 10280 ax-mulcom 10281 ax-addass 10282 ax-mulass 10283 ax-distr 10284 ax-i2m1 10285 ax-1ne0 10286 ax-1rid 10287 ax-rnegex 10288 ax-rrecex 10289 ax-cnre 10290 ax-pre-lttri 10291 ax-pre-lttrn 10292 ax-pre-ltadd 10293 ax-pre-mulgt0 10294 ax-pre-sup 10295

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3or 1101 df-3an 1102 df-tru 1641 df-fal 1651 df-ex 1860 df-nf 1864 df-sb 2061 df-eu 2634 df-mo 2635 df-clab 2793 df-cleq 2799 df-clel 2802 df-nfc 2937 df-ne 2979 df-nel 3082 df-ral 3101 df-rex 3102 df-reu 3103 df-rmo 3104 df-rab 3105 df-v 3393 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4117 df-if 4280 df-pw 4353 df-sn 4371 df-pr 4373 df-tp 4375 df-op 4377 df-uni 4631 df-int 4670 df-iun 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-se 5271 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5893 df-ord 5939 df-on 5940 df-lim 5941 df-suc 5942 df-iota 6060 df-fun 6099 df-fn 6100 df-f 6101 df-f1 6102 df-fo 6103 df-f1o 6104 df-fv 6105 df-isom 6106 df-riota 6831 df-ov 6873 df-oprab 6874 df-mpt2 6875 df-of 7123 df-om 7292 df-1st 7394 df-2nd 7395 df-wrecs 7638 df-recs 7700 df-rdg 7738 df-1o 7792 df-2o 7793 df-oadd 7796 df-er 7975 df-map 8090 df-pm 8091 df-en 8189 df-dom 8190 df-sdom 8191 df-fin 8192 df-fi 8552 df-sup 8583 df-inf 8584 df-oi 8650 df-card 9044 df-cda 9271 df-pnf 10357 df-mnf 10358 df-xr 10359 df-ltxr 10360 df-le 10361 df-sub 10549 df-neg 10550 df-div 10966 df-nn 11302 df-2 11360 df-3 11361 df-n0 11556 df-z 11640 df-uz 11901 df-q 12004 df-rp 12043 df-xneg 12158 df-xadd 12159 df-xmul 12160 df-ioo 12393 df-ico 12395 df-icc 12396 df-fz 12546 df-fzo 12686 df-fl 12813 df-seq 13021 df-exp 13080 df-hash 13334 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-clim 14438 df-rlim 14439 df-sum 14636 df-prod 14853 df-rest 16284 df-topgen 16305 df-psmet 19942 df-xmet 19943 df-met 19944 df-bl 19945 df-mopn 19946 df-top 20908 df-topon 20925 df-bases 20960 df-cmp 21400 df-ovol 23441 df-vol 23442

This theorem is referenced by: sge0hsphoire 41279 hoidmvlelem1 41285 hoidmvlelem4 41288 hspmbllem2 41317

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