iccvonmbllem - Mathbox for Glauco Siliprandi

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Theorem iccvonmbllem 47106

Description: Any n-dimensional closed interval is Lebesgue measurable. This is the second statement in Proposition 115G (c) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
iccvonmbllem.x	⊢ (𝜑 → 𝑋 ∈ Fin)
iccvonmbllem.s	⊢ 𝑆 = dom (voln‘𝑋)
iccvonmbllem.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
iccvonmbllem.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
iccvonmbllem.c	⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))))
iccvonmbllem.d	⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))))

**Assertion**
Ref	Expression
iccvonmbllem	⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) ∈ 𝑆)

Distinct variable groups: 𝐴,𝑛 𝐵,𝑛 𝐶,𝑖 𝐷,𝑖 𝑆,𝑛 𝑖,𝑋,𝑛 𝜑,𝑖,𝑛 Allowed substitution hints: 𝐴(𝑖) 𝐵(𝑖) 𝐶(𝑛) 𝐷(𝑛) 𝑆(𝑖)

**Proof of Theorem iccvonmbllem**
Step	Hyp	Ref	Expression
1		iccvonmbllem.c	. . . . . . . . . . . 12 ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))))
2	1	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛)))))
3		iccvonmbllem.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ Fin)
4	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
5	4	mptexd 7179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))) ∈ V)
6	2, 5	fvmpt2d 6961	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))))
7		iccvonmbllem.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
8	7	ffvelcdmda 7036	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ)
9	8	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ)
10		nnrecre 12219	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
11	10	ad2antlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
12	9, 11	resubcld 11578	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → ((𝐴‘𝑖) − (1 / 𝑛)) ∈ ℝ)
13	6, 12	fvmpt2d 6961	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑖) = ((𝐴‘𝑖) − (1 / 𝑛)))
14	13	an32s 653	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛)‘𝑖) = ((𝐴‘𝑖) − (1 / 𝑛)))
15		iccvonmbllem.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))))
16	15	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛)))))
17	4	mptexd 7179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))) ∈ V)
18	16, 17	fvmpt2d 6961	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))))
19		iccvonmbllem.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
20	19	ffvelcdmda 7036	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ)
21	20	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ)
22	21, 11	readdcld 11174	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) + (1 / 𝑛)) ∈ ℝ)
23	18, 22	fvmpt2d 6961	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑛)‘𝑖) = ((𝐵‘𝑖) + (1 / 𝑛)))
24	23	an32s 653	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)‘𝑖) = ((𝐵‘𝑖) + (1 / 𝑛)))
25	14, 24	oveq12d 7385	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = (((𝐴‘𝑖) − (1 / 𝑛))(,)((𝐵‘𝑖) + (1 / 𝑛))))
26	25	iineq2dv 4959	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = ∩ 𝑛 ∈ ℕ (((𝐴‘𝑖) − (1 / 𝑛))(,)((𝐵‘𝑖) + (1 / 𝑛))))
27	8, 20	iooiinicc 45972	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ∩ 𝑛 ∈ ℕ (((𝐴‘𝑖) − (1 / 𝑛))(,)((𝐵‘𝑖) + (1 / 𝑛))) = ((𝐴‘𝑖)[,](𝐵‘𝑖)))
28	26, 27	eqtrd 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = ((𝐴‘𝑖)[,](𝐵‘𝑖)))
29	28	ixpeq2dva 8860	. . . 4 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)))
30	29	eqcomd 2742	. . 3 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) = X𝑖 ∈ 𝑋 ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)))
31		eqidd 2737	. . 3 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) = X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)))
32		nnn0 45807	. . . . 5 ⊢ ℕ ≠ ∅
33	32	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ≠ ∅)
34		ixpiin 8872	. . . 4 ⊢ (ℕ ≠ ∅ → X𝑖 ∈ 𝑋 ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = ∩ 𝑛 ∈ ℕ X𝑖 ∈ 𝑋 (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)))
35	33, 34	syl 17	. . 3 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = ∩ 𝑛 ∈ ℕ X𝑖 ∈ 𝑋 (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)))
36	30, 31, 35	3eqtr3d 2779	. 2 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) = ∩ 𝑛 ∈ ℕ X𝑖 ∈ 𝑋 (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)))
37		iccvonmbllem.s	. . . 4 ⊢ 𝑆 = dom (voln‘𝑋)
38	3, 37	dmovnsal 47040	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
39		nnct 13943	. . . 4 ⊢ ℕ ≼ ω
40	39	a1i 11	. . 3 ⊢ (𝜑 → ℕ ≼ ω)
41	12	fmpttd 7067	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))):𝑋⟶ℝ)
42		ressxr 11189	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
43	42	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℝ ⊆ ℝ^*)
44	41, 43	fssd 6685	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))):𝑋⟶ℝ^*)
45	6	feq1d 6650	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ^* ↔ (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))):𝑋⟶ℝ^*))
46	44, 45	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ^*)
47	22	fmpttd 7067	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))):𝑋⟶ℝ)
48	47, 43	fssd 6685	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))):𝑋⟶ℝ^*)
49	18	feq1d 6650	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛):𝑋⟶ℝ^* ↔ (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))):𝑋⟶ℝ^*))
50	48, 49	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛):𝑋⟶ℝ^*)
51	4, 37, 46, 50	ioovonmbl 47105	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑖 ∈ 𝑋 (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) ∈ 𝑆)
52	38, 40, 33, 51	saliincl 46755	. 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ X𝑖 ∈ 𝑋 (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) ∈ 𝑆)
53	36, 52	eqeltrd 2836	1 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 Vcvv 3429 ⊆ wss 3889 ∅c0 4273 ∩ ciin 4934 class class class wbr 5085 ↦ cmpt 5166 dom cdm 5631 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ωcom 7817 Xcixp 8845 ≼ cdom 8891 Fincfn 8893 ℝcr 11037 1c1 11039 + caddc 11041 ℝ^*cxr 11178 − cmin 11377 / cdiv 11807 ℕcn 12174 (,)cioo 13298 [,]cicc 13301 volncvoln 46966

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cc 10357 ax-ac2 10385 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-prod 15869 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-prds 17410 df-pws 17412 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-subg 19099 df-ghm 19188 df-cntz 19292 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-rhm 20452 df-subrng 20523 df-subrg 20547 df-drng 20708 df-field 20709 df-abv 20786 df-staf 20816 df-srng 20817 df-lmod 20857 df-lss 20927 df-lmhm 21017 df-lvec 21098 df-sra 21168 df-rgmod 21169 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-refld 21585 df-phl 21606 df-dsmm 21712 df-frlm 21727 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cmp 23352 df-xms 24285 df-ms 24286 df-nm 24547 df-ngp 24548 df-tng 24549 df-nrg 24550 df-nlm 24551 df-clm 25030 df-cph 25135 df-tcph 25136 df-rrx 25352 df-ovol 25431 df-vol 25432 df-salg 46737 df-sumge0 46791 df-mea 46878 df-ome 46918 df-caragen 46920 df-ovoln 46965 df-voln 46967

This theorem is referenced by: iccvonmbl 47107

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