hoidifhspdmvle - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hoidifhspdmvle		Structured version Visualization version GIF version

Theorem hoidifhspdmvle 45635

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 1916	. . 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 45633	. . . . 5 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴):𝑋⟶ℝ)
7	6	ffvelcdmda 7086	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐷‘𝑌)‘𝐴)‘𝑘) ∈ ℝ)
8		hoidifhspdmvle.b	. . . . 5 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
9	8	ffvelcdmda 7086	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
10		volicore 45596	. . . 4 ⊢ (((((𝐷‘𝑌)‘𝐴)‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
11	7, 9, 10	syl2anc 583	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
12	9	rexrd 11269	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
13		icombl 25314	. . . . 5 ⊢ (((((𝐷‘𝑌)‘𝐴)‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ^*) → ((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
14	7, 12, 13	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
15		volge0 44976	. . . 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 7086	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
18		volicore 45596	. . . 4 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
19	17, 9, 18	syl2anc 583	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
20		icombl 25314	. . . . 5 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ^*) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
21	17, 12, 20	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
22	17	rexrd 11269	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
23	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑌 ∈ ℝ)
24	23	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → 𝑌 ∈ ℝ)
25	17	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (𝐴‘𝑘) ∈ ℝ)
26		max2 13171	. . . . . . . 8 ⊢ ((𝑌 ∈ ℝ ∧ (𝐴‘𝑘) ∈ ℝ) → (𝐴‘𝑘) ≤ if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌))
27	24, 25, 26	syl2anc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (𝐴‘𝑘) ≤ if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌))
28	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑋 ∈ Fin)
29	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴:𝑋⟶ℝ)
30		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
31	3, 23, 28, 29, 30	hoidifhspval3 45634	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐷‘𝑌)‘𝐴)‘𝑘) = if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))
32	31	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (((𝐷‘𝑌)‘𝐴)‘𝑘) = if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))
33		iftrue 4534	. . . . . . . . 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 481	. . . . . . . 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 5174	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (𝐴‘𝑘) ≤ (((𝐷‘𝑌)‘𝐴)‘𝑘))
37	17	leidd 11785	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐴‘𝑘))
38	37	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (𝐴‘𝑘) ≤ (𝐴‘𝑘))
39	31	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (((𝐷‘𝑌)‘𝐴)‘𝑘) = if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))
40		iffalse 4537	. . . . . . . . 9 ⊢ (¬ 𝑘 = 𝐾 → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)) = (𝐴‘𝑘))
41	40	adantl 481	. . . . . . . 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 5174	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (𝐴‘𝑘) ≤ (((𝐷‘𝑌)‘𝐴)‘𝑘))
44	36, 43	pm2.61dan 810	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (((𝐷‘𝑌)‘𝐴)‘𝑘))
45	9	leidd 11785	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ≤ (𝐵‘𝑘))
46		icossico 13399	. . . . 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 836	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
48		volss 25283	. . . 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 1370	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))) ≤ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
50	1, 2, 11, 16, 19, 49	fprodle 15945	. 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 4335	. . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅)
54	51, 2, 53, 6, 8	hoidmvn0val 45599	. . 3 ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((((𝐷‘𝑌)‘𝐴)‘𝑘)[,)(𝐵‘𝑘))))
55	51, 2, 53, 5, 8	hoidmvn0val 45599	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
56	54, 55	breq12d 5161	. 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 257	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 395 = wceq 1540 ∈ wcel 2105 ⊆ wss 3948 ∅c0 4322 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 ↑_m cmap 8824 Fincfn 8943 ℝcr 11113 0cc0 11114 ℝ^*cxr 11252 ≤ cle 11254 [,)cico 13331 ∏cprod 15854 volcvol 25213

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-er 8707 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-dju 9900 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-rlim 15438 df-sum 15638 df-prod 15855 df-rest 17373 df-topgen 17394 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-top 22617 df-topon 22634 df-bases 22670 df-cmp 23112 df-ovol 25214 df-vol 25215

This theorem is referenced by: hspmbllem2 45642

W3C validator