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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccvonmbllem | Structured version Visualization version GIF version |
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.) |
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)))) |
Ref | Expression |
---|---|
iccvonmbllem | ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) ∈ 𝑆) |
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 7236 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))) ∈ V) |
6 | 2, 5 | fvmpt2d 7018 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛)))) |
7 | iccvonmbllem.a | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | |
8 | 7 | ffvelcdmda 7094 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ) |
9 | 8 | adantlr 714 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ) |
10 | nnrecre 12285 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ) | |
11 | 10 | ad2antlr 726 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ) |
12 | 9, 11 | resubcld 11673 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → ((𝐴‘𝑖) − (1 / 𝑛)) ∈ ℝ) |
13 | 6, 12 | fvmpt2d 7018 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑖) = ((𝐴‘𝑖) − (1 / 𝑛))) |
14 | 13 | an32s 651 | . . . . . . . 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 7236 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))) ∈ V) |
18 | 16, 17 | fvmpt2d 7018 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛)))) |
19 | iccvonmbllem.b | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) | |
20 | 19 | ffvelcdmda 7094 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ) |
21 | 20 | adantlr 714 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ) |
22 | 21, 11 | readdcld 11274 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) + (1 / 𝑛)) ∈ ℝ) |
23 | 18, 22 | fvmpt2d 7018 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑛)‘𝑖) = ((𝐵‘𝑖) + (1 / 𝑛))) |
24 | 23 | an32s 651 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)‘𝑖) = ((𝐵‘𝑖) + (1 / 𝑛))) |
25 | 14, 24 | oveq12d 7438 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = (((𝐴‘𝑖) − (1 / 𝑛))(,)((𝐵‘𝑖) + (1 / 𝑛)))) |
26 | 25 | iineq2dv 5021 | . . . . . 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 44927 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ∩ 𝑛 ∈ ℕ (((𝐴‘𝑖) − (1 / 𝑛))(,)((𝐵‘𝑖) + (1 / 𝑛))) = ((𝐴‘𝑖)[,](𝐵‘𝑖))) |
28 | 26, 27 | eqtrd 2768 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = ((𝐴‘𝑖)[,](𝐵‘𝑖))) |
29 | 28 | ixpeq2dva 8931 | . . . 4 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) = X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖))) |
30 | 29 | eqcomd 2734 | . . 3 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) = X𝑖 ∈ 𝑋 ∩ 𝑛 ∈ ℕ (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖))) |
31 | eqidd 2729 | . . 3 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) = X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖))) | |
32 | nnn0 44760 | . . . . 5 ⊢ ℕ ≠ ∅ | |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ ≠ ∅) |
34 | ixpiin 8943 | . . . 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 2776 | . 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 46000 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
39 | nnct 13979 | . . . 4 ⊢ ℕ ≼ ω | |
40 | 39 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≼ ω) |
41 | 12 | fmpttd 7125 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))):𝑋⟶ℝ) |
42 | ressxr 11289 | . . . . . . 7 ⊢ ℝ ⊆ ℝ* | |
43 | 42 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℝ ⊆ ℝ*) |
44 | 41, 43 | fssd 6740 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))):𝑋⟶ℝ*) |
45 | 6 | feq1d 6707 | . . . . 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 7125 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))):𝑋⟶ℝ) |
48 | 47, 43 | fssd 6740 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))):𝑋⟶ℝ*) |
49 | 18 | feq1d 6707 | . . . . 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 46065 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑖 ∈ 𝑋 (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) ∈ 𝑆) |
52 | 38, 40, 33, 51 | saliincl 45715 | . 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ X𝑖 ∈ 𝑋 (((𝐶‘𝑛)‘𝑖)(,)((𝐷‘𝑛)‘𝑖)) ∈ 𝑆) |
53 | 36, 52 | eqeltrd 2829 | 1 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ⊆ wss 3947 ∅c0 4323 ∩ ciin 4997 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5678 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ωcom 7870 Xcixp 8916 ≼ cdom 8962 Fincfn 8964 ℝcr 11138 1c1 11140 + caddc 11142 ℝ*cxr 11278 − cmin 11475 / cdiv 11902 ℕcn 12243 (,)cioo 13357 [,]cicc 13360 volncvoln 45926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cc 10459 ax-ac2 10487 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-dju 9925 df-card 9963 df-acn 9966 df-ac 10140 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-rlim 15466 df-sum 15666 df-prod 15883 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-prds 17429 df-pws 17431 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-ghm 19168 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-rhm 20411 df-subrng 20483 df-subrg 20508 df-drng 20626 df-field 20627 df-abv 20697 df-staf 20725 df-srng 20726 df-lmod 20745 df-lss 20816 df-lmhm 20907 df-lvec 20988 df-sra 21058 df-rgmod 21059 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-cnfld 21280 df-refld 21537 df-phl 21558 df-dsmm 21666 df-frlm 21681 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cmp 23304 df-xms 24239 df-ms 24240 df-nm 24504 df-ngp 24505 df-tng 24506 df-nrg 24507 df-nlm 24508 df-clm 25003 df-cph 25109 df-tcph 25110 df-rrx 25326 df-ovol 25406 df-vol 25407 df-salg 45697 df-sumge0 45751 df-mea 45838 df-ome 45878 df-caragen 45880 df-ovoln 45925 df-voln 45927 |
This theorem is referenced by: iccvonmbl 46067 |
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