hoidifhspdmvle - Mathbox for Glauco Siliprandi

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Theorem hoidifhspdmvle 45014

Description: The dimensional volume of the difference of a half-open interval and a half-space is less than or equal to the dimensional volume of the whole half-open interval. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypotheses**
Ref	Expression
hoidifhspdmvle.l	⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
hoidifhspdmvle.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hoidifhspdmvle.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
hoidifhspdmvle.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
hoidifhspdmvle.k	⊢ (𝜑 → 𝐾 ∈ 𝑋)
hoidifhspdmvle.d	⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ = 𝐾, if(𝑥 ≤ (𝑐‘ℎ), (𝑐‘ℎ), 𝑥), (𝑐‘ℎ)))))
hoidifhspdmvle.y	⊢ (𝜑 → 𝑌 ∈ ℝ)

**Assertion**
Ref	Expression
hoidifhspdmvle	⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)(𝐿‘𝑋)𝐵) ≤ (𝐴(𝐿‘𝑋)𝐵))

Distinct variable groups: 𝐴,𝑎,𝑏,𝑘 𝐴,𝑐,ℎ,𝑘 𝐵,𝑎,𝑏,𝑘 𝐷,𝑎,𝑏,𝑘 𝐾,𝑐,ℎ,𝑥 𝑋,𝑎,𝑏,𝑘,𝑥 𝑋,𝑐,ℎ 𝑌,𝑎,𝑏,𝑘,𝑥 𝑌,𝑐,ℎ 𝜑,𝑎,𝑏,𝑘,𝑥 𝜑,𝑐,ℎ Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥,ℎ,𝑐) 𝐷(𝑥,ℎ,𝑐) 𝐾(𝑘,𝑎,𝑏) 𝐿(𝑥,ℎ,𝑘,𝑎,𝑏,𝑐)

**Proof of Theorem hoidifhspdmvle**
Step	Hyp	Ref	Expression
1		nfv 1917	. . 3 ⊢ Ⅎ𝑘𝜑
2		hoidifhspdmvle.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
3		hoidifhspdmvle.d	. . . . . 6 ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ = 𝐾, if(𝑥 ≤ (𝑐‘ℎ), (𝑐‘ℎ), 𝑥), (𝑐‘ℎ)))))
4		hoidifhspdmvle.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℝ)
5		hoidifhspdmvle.a	. . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
6	3, 4, 2, 5	hoidifhspf 45012	. . . . 5 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴):𝑋⟶ℝ)
7	6	ffvelcdmda 7055	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐷‘𝑌)‘𝐴)‘𝑘) ∈ ℝ)
8		hoidifhspdmvle.b	. . . . 5 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
9	8	ffvelcdmda 7055	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
10		volicore 44975	. . . 4 ⊢ (((((𝐷‘𝑌)‘𝐴)‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
11	7, 9, 10	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
12	9	rexrd 11229	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
13		icombl 24980	. . . . 5 ⊢ (((((𝐷‘𝑌)‘𝐴)‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ^*) → ((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
14	7, 12, 13	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
15		volge0 44355	. . . 4 ⊢ (((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol → 0 ≤ (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))))
16	14, 15	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 0 ≤ (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))))
17	5	ffvelcdmda 7055	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
18		volicore 44975	. . . 4 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
19	17, 9, 18	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
20		icombl 24980	. . . . 5 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ^*) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
21	17, 12, 20	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
22	17	rexrd 11229	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
23	4	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑌 ∈ ℝ)
24	23	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → 𝑌 ∈ ℝ)
25	17	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (𝐴‘𝑘) ∈ ℝ)
26		max2 13131	. . . . . . . 8 ⊢ ((𝑌 ∈ ℝ ∧ (𝐴‘𝑘) ∈ ℝ) → (𝐴‘𝑘) ≤ if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌))
27	24, 25, 26	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (𝐴‘𝑘) ≤ if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌))
28	2	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑋 ∈ Fin)
29	5	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴:𝑋⟶ℝ)
30		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
31	3, 23, 28, 29, 30	hoidifhspval3 45013	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐷‘𝑌)‘𝐴)‘𝑘) = if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))
32	31	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (((𝐷‘𝑌)‘𝐴)‘𝑘) = if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))
33		iftrue 4512	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)) = if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌))
34	33	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)) = if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌))
35	32, 34	eqtr2d 2772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌) = (((𝐷‘𝑌)‘𝐴)‘𝑘))
36	27, 35	breqtrd 5151	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (𝐴‘𝑘) ≤ (((𝐷‘𝑌)‘𝐴)‘𝑘))
37	17	leidd 11745	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐴‘𝑘))
38	37	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (𝐴‘𝑘) ≤ (𝐴‘𝑘))
39	31	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (((𝐷‘𝑌)‘𝐴)‘𝑘) = if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))
40		iffalse 4515	. . . . . . . . 9 ⊢ (¬ 𝑘 = 𝐾 → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)) = (𝐴‘𝑘))
41	40	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)) = (𝐴‘𝑘))
42	39, 41	eqtr2d 2772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (𝐴‘𝑘) = (((𝐷‘𝑌)‘𝐴)‘𝑘))
43	38, 42	breqtrd 5151	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (𝐴‘𝑘) ≤ (((𝐷‘𝑌)‘𝐴)‘𝑘))
44	36, 43	pm2.61dan 811	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (((𝐷‘𝑌)‘𝐴)‘𝑘))
45	9	leidd 11745	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ≤ (𝐵‘𝑘))
46		icossico 13359	. . . . 5 ⊢ ((((𝐴‘𝑘) ∈ ℝ^* ∧ (𝐵‘𝑘) ∈ ℝ^*) ∧ ((𝐴‘𝑘) ≤ (((𝐷‘𝑌)‘𝐴)‘𝑘) ∧ (𝐵‘𝑘) ≤ (𝐵‘𝑘))) → ((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
47	22, 12, 44, 45, 46	syl22anc 837	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
48		volss 24949	. . . 4 ⊢ ((((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol ∧ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol ∧ ((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))) ≤ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
49	14, 21, 47, 48	syl3anc 1371	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))) ≤ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
50	1, 2, 11, 16, 19, 49	fprodle 15905	. 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
51		hoidifhspdmvle.l	. . . 4 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
52		hoidifhspdmvle.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
53	52	ne0d 4315	. . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅)
54	51, 2, 53, 6, 8	hoidmvn0val 44978	. . 3 ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))))
55	51, 2, 53, 5, 8	hoidmvn0val 44978	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
56	54, 55	breq12d 5138	. 2 ⊢ (𝜑 → ((((𝐷‘𝑌)‘𝐴)(𝐿‘𝑋)𝐵) ≤ (𝐴(𝐿‘𝑋)𝐵) ↔ ∏𝑘 ∈ 𝑋 (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
57	50, 56	mpbird 256	1 ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)(𝐿‘𝑋)𝐵) ≤ (𝐴(𝐿‘𝑋)𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3928 ∅c0 4302 ifcif 4506 class class class wbr 5125 ↦ cmpt 5208 dom cdm 5653 ⟶wf 6512 ‘cfv 6516 (class class class)co 7377 ∈ cmpo 7379 ↑_m cmap 8787 Fincfn 8905 ℝcr 11074 0cc0 11075 ℝ^*cxr 11212 ≤ cle 11214 [,)cico 13291 ∏cprod 15814 volcvol 24879

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-se 5609 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-of 7637 df-om 7823 df-1st 7941 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8670 df-map 8789 df-pm 8790 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9470 df-dju 9861 df-card 9899 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-div 11837 df-nn 12178 df-2 12240 df-3 12241 df-n0 12438 df-z 12524 df-uz 12788 df-q 12898 df-rp 12940 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13450 df-fzo 13593 df-fl 13722 df-seq 13932 df-exp 13993 df-hash 14256 df-cj 15011 df-re 15012 df-im 15013 df-sqrt 15147 df-abs 15148 df-clim 15397 df-rlim 15398 df-sum 15598 df-prod 15815 df-rest 17333 df-topgen 17354 df-psmet 20840 df-xmet 20841 df-met 20842 df-bl 20843 df-mopn 20844 df-top 22295 df-topon 22312 df-bases 22348 df-cmp 22790 df-ovol 24880 df-vol 24881

This theorem is referenced by: hspmbllem2 45021

W3C validator