hoimbl2 - Mathbox for Glauco Siliprandi

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Theorem hoimbl2 44203

Description: Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
hoimbl2.k	⊢ Ⅎ𝑘𝜑
hoimbl2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hoimbl2.s	⊢ 𝑆 = dom (voln‘𝑋)
hoimbl2.a	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
hoimbl2.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)

**Assertion**
Ref	Expression
hoimbl2	⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ∈ 𝑆)

Distinct variable group: 𝑘,𝑋 Allowed substitution hints: 𝜑(𝑘) 𝐴(𝑘) 𝐵(𝑘) 𝑆(𝑘)

Proof of Theorem hoimbl2 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋)
2		hoimbl2.k	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
3		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑋
4	2, 3	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑋)
5		nfcsb1v 3857	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴
6		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑘ℝ
7	5, 6	nfel 2921	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ
8	4, 7	nfim 1899	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)
9		eleq1w 2821	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑋 ↔ 𝑗 ∈ 𝑋))
10	9	anbi2d 629	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑋) ↔ (𝜑 ∧ 𝑗 ∈ 𝑋)))
11		csbeq1a 3846	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴)
12	11	eleq1d 2823	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ))
13	10, 12	imbi12d 345	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)))
14		hoimbl2.a	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
15	8, 13, 14	chvarfv 2233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)
16		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑘𝑗
17	16	nfcsb1 3856	. . . . . . 7 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴
18		eqid 2738	. . . . . . 7 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐴) = (𝑘 ∈ 𝑋 ↦ 𝐴)
19	16, 17, 11, 18	fvmptf 6896	. . . . . 6 ⊢ ((𝑗 ∈ 𝑋 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ) → ((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
20	1, 15, 19	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
21	16	nfcsb1 3856	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵
22	21, 6	nfel 2921	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ
23	4, 22	nfim 1899	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)
24		csbeq1a 3846	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
25	24	eleq1d 2823	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ))
26	10, 25	imbi12d 345	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)))
27		hoimbl2.b	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)
28	23, 26, 27	chvarfv 2233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)
29		eqid 2738	. . . . . . 7 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐵) = (𝑘 ∈ 𝑋 ↦ 𝐵)
30	16, 21, 24, 29	fvmptf 6896	. . . . . 6 ⊢ ((𝑗 ∈ 𝑋 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ) → ((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵)
31	1, 28, 30	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵)
32	20, 31	oveq12d 7293	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = (⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵))
33	32	ixpeq2dva 8700	. . 3 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵))
34		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑗(𝐴[,)𝐵)
35		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑘[,)
36	5, 35, 21	nfov 7305	. . . . . 6 ⊢ Ⅎ𝑘(⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵)
37	11, 24	oveq12d 7293	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴[,)𝐵) = (⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵))
38	34, 36, 37	cbvixp 8702	. . . . 5 ⊢ X𝑘 ∈ 𝑋 (𝐴[,)𝐵) = X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵)
39	38	eqcomi 2747	. . . 4 ⊢ X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵) = X𝑘 ∈ 𝑋 (𝐴[,)𝐵)
40	39	a1i 11	. . 3 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵) = X𝑘 ∈ 𝑋 (𝐴[,)𝐵))
41	33, 40	eqtr2d 2779	. 2 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) = X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)))
42		hoimbl2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
43		hoimbl2.s	. . 3 ⊢ 𝑆 = dom (voln‘𝑋)
44	2, 14, 18	fmptdf 6991	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℝ)
45	2, 27, 29	fmptdf 6991	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℝ)
46	42, 43, 44, 45	hoimbl 44169	. 2 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) ∈ 𝑆)
47	41, 46	eqeltrd 2839	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2106 ⦋csb 3832 ↦ cmpt 5157 dom cdm 5589 ‘cfv 6433 (class class class)co 7275 Xcixp 8685 Fincfn 8733 ℝcr 10870 [,)cico 13081 volncvoln 44076

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-ac2 10219 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-acn 9700 df-ac 9872 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-prod 15616 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-rest 17133 df-0g 17152 df-topgen 17154 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-subg 18752 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-top 22043 df-topon 22060 df-bases 22096 df-cmp 22538 df-ovol 24628 df-vol 24629 df-salg 43850 df-sumge0 43901 df-mea 43988 df-ome 44028 df-caragen 44030 df-ovoln 44075 df-voln 44077

This theorem is referenced by: vonhoire 44210

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