hoimbl2 - Mathbox for Glauco Siliprandi

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Theorem hoimbl2 41396

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 471	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋)
2		hoimbl2.k	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
3		nfv 1995	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑋
4	2, 3	nfan 1980	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑋)
5		nfcsb1v 3698	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴
6		nfcv 2913	. . . . . . . . 9 ⊢ Ⅎ𝑘ℝ
7	5, 6	nfel 2926	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ
8	4, 7	nfim 1977	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)
9		eleq1w 2833	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑋 ↔ 𝑗 ∈ 𝑋))
10	9	anbi2d 614	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑋) ↔ (𝜑 ∧ 𝑗 ∈ 𝑋)))
11		csbeq1a 3691	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴)
12	11	eleq1d 2835	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ))
13	10, 12	imbi12d 333	. . . . . . 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	chvar 2424	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)
16		nfcv 2913	. . . . . . 7 ⊢ Ⅎ𝑘𝑗
17	16	nfcsb1 3697	. . . . . . 7 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴
18		eqid 2771	. . . . . . 7 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐴) = (𝑘 ∈ 𝑋 ↦ 𝐴)
19	16, 17, 11, 18	fvmptf 6442	. . . . . 6 ⊢ ((𝑗 ∈ 𝑋 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ) → ((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
20	1, 15, 19	syl2anc 573	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
21	16	nfcsb1 3697	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵
22	21, 6	nfel 2926	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ
23	4, 22	nfim 1977	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)
24		csbeq1a 3691	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
25	24	eleq1d 2835	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ))
26	10, 25	imbi12d 333	. . . . . . 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	chvar 2424	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)
29		eqid 2771	. . . . . . 7 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐵) = (𝑘 ∈ 𝑋 ↦ 𝐵)
30	16, 21, 24, 29	fvmptf 6442	. . . . . 6 ⊢ ((𝑗 ∈ 𝑋 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ) → ((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵)
31	1, 28, 30	syl2anc 573	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵)
32	20, 31	oveq12d 6810	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = (⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵))
33	32	ixpeq2dva 8077	. . 3 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵))
34		nfcv 2913	. . . . . 6 ⊢ Ⅎ𝑗(𝐴[,)𝐵)
35		nfcv 2913	. . . . . . 7 ⊢ Ⅎ𝑘[,)
36	5, 35, 21	nfov 6821	. . . . . 6 ⊢ Ⅎ𝑘(⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵)
37	11, 24	oveq12d 6810	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴[,)𝐵) = (⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵))
38	34, 36, 37	cbvixp 8079	. . . . 5 ⊢ X𝑘 ∈ 𝑋 (𝐴[,)𝐵) = X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴[,)⦋𝑗 / 𝑘⦌𝐵)
39	38	eqcomi 2780	. . . 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 2806	. 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 6528	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℝ)
45	2, 27, 29	fmptdf 6528	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℝ)
46	42, 43, 44, 45	hoimbl 41362	. 2 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,)((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) ∈ 𝑆)
47	41, 46	eqeltrd 2850	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 Ⅎwnf 1856 ∈ wcel 2145 ⦋csb 3682 ↦ cmpt 4863 dom cdm 5249 ‘cfv 6029 (class class class)co 6792 Xcixp 8062 Fincfn 8109 ℝcr 10137 [,)cico 12378 volncvoln 41269

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cc 9459 ax-ac2 9487 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 ax-addf 10217 ax-mulf 10218

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-disj 4755 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-tpos 7504 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-omul 7718 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fi 8473 df-sup 8504 df-inf 8505 df-oi 8571 df-card 8965 df-acn 8968 df-ac 9139 df-cda 9192 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11496 df-z 11581 df-dec 11697 df-uz 11890 df-q 11993 df-rp 12032 df-xneg 12147 df-xadd 12148 df-xmul 12149 df-ioo 12380 df-ico 12382 df-icc 12383 df-fz 12530 df-fzo 12670 df-fl 12797 df-seq 13005 df-exp 13064 df-hash 13318 df-cj 14043 df-re 14044 df-im 14045 df-sqrt 14179 df-abs 14180 df-clim 14423 df-rlim 14424 df-sum 14621 df-prod 14839 df-struct 16062 df-ndx 16063 df-slot 16064 df-base 16066 df-sets 16067 df-ress 16068 df-plusg 16158 df-mulr 16159 df-starv 16160 df-tset 16164 df-ple 16165 df-ds 16168 df-unif 16169 df-rest 16287 df-0g 16306 df-topgen 16308 df-mgm 17446 df-sgrp 17488 df-mnd 17499 df-grp 17629 df-minusg 17630 df-subg 17795 df-cmn 18398 df-abl 18399 df-mgp 18694 df-ur 18706 df-ring 18753 df-cring 18754 df-oppr 18827 df-dvdsr 18845 df-unit 18846 df-invr 18876 df-dvr 18887 df-drng 18955 df-psmet 19949 df-xmet 19950 df-met 19951 df-bl 19952 df-mopn 19953 df-cnfld 19958 df-top 20915 df-topon 20932 df-bases 20967 df-cmp 21407 df-ovol 23448 df-vol 23449 df-salg 41043 df-sumge0 41094 df-mea 41181 df-ome 41221 df-caragen 41223 df-ovoln 41268 df-voln 41270

This theorem is referenced by: vonhoire 41403

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