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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoimbl | Structured version Visualization version GIF version |
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, 24-Dec-2020.) |
Ref | Expression |
---|---|
hoimbl.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
hoimbl.s | ⊢ 𝑆 = dom (voln‘𝑋) |
hoimbl.a | ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
hoimbl.b | ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
Ref | Expression |
---|---|
hoimbl | ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoimbl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
2 | 1 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 ∈ Fin) |
3 | 2 | rrnmbl 46145 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑m 𝑋) ∈ dom (voln‘𝑋)) |
4 | reex 11236 | . . . . . . . . 9 ⊢ ℝ ∈ V | |
5 | mapdm0 8861 | . . . . . . . . 9 ⊢ (ℝ ∈ V → (ℝ ↑m ∅) = {∅}) | |
6 | 4, 5 | ax-mp 5 | . . . . . . . 8 ⊢ (ℝ ↑m ∅) = {∅} |
7 | 6 | eqcomi 2734 | . . . . . . 7 ⊢ {∅} = (ℝ ↑m ∅) |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝑋 = ∅ → {∅} = (ℝ ↑m ∅)) |
9 | id 22 | . . . . . . . 8 ⊢ (𝑋 = ∅ → 𝑋 = ∅) | |
10 | 9 | ixpeq1d 8928 | . . . . . . 7 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = X𝑖 ∈ ∅ ((𝐴‘𝑖)[,)(𝐵‘𝑖))) |
11 | ixp0x 8945 | . . . . . . . 8 ⊢ X𝑖 ∈ ∅ ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = {∅} | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝑋 = ∅ → X𝑖 ∈ ∅ ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = {∅}) |
13 | 10, 12 | eqtrd 2765 | . . . . . 6 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = {∅}) |
14 | oveq2 7427 | . . . . . 6 ⊢ (𝑋 = ∅ → (ℝ ↑m 𝑋) = (ℝ ↑m ∅)) | |
15 | 8, 13, 14 | 3eqtr4d 2775 | . . . . 5 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = (ℝ ↑m 𝑋)) |
16 | 15 | adantl 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = (ℝ ↑m 𝑋)) |
17 | hoimbl.s | . . . . 5 ⊢ 𝑆 = dom (voln‘𝑋) | |
18 | 17 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑆 = dom (voln‘𝑋)) |
19 | 16, 18 | eleq12d 2819 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆 ↔ (ℝ ↑m 𝑋) ∈ dom (voln‘𝑋))) |
20 | 3, 19 | mpbird 256 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
21 | 1 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin) |
22 | 9 | necon3bi 2956 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) |
23 | 22 | adantl 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
24 | hoimbl.a | . . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | |
25 | 24 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ) |
26 | hoimbl.b | . . . 4 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) | |
27 | 26 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ) |
28 | id 22 | . . . . . 6 ⊢ (𝑤 = 𝑥 → 𝑤 = 𝑥) | |
29 | eqidd 2726 | . . . . . 6 ⊢ (𝑤 = 𝑥 → ℝ = ℝ) | |
30 | 28 | ixpeq1d 8928 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑗 ∈ 𝑥 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ)) |
31 | eqeq1 2729 | . . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑗 = ℎ ↔ 𝑖 = ℎ)) | |
32 | 31 | ifbid 4553 | . . . . . . . . 9 ⊢ (𝑗 = 𝑖 → if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) |
33 | 32 | cbvixpv 8934 | . . . . . . . 8 ⊢ X𝑗 ∈ 𝑥 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ) |
34 | 33 | a1i 11 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → X𝑗 ∈ 𝑥 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) |
35 | 30, 34 | eqtrd 2765 | . . . . . 6 ⊢ (𝑤 = 𝑥 → X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) |
36 | 28, 29, 35 | mpoeq123dv 7495 | . . . . 5 ⊢ (𝑤 = 𝑥 → (ℎ ∈ 𝑤, 𝑧 ∈ ℝ ↦ X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ)) = (ℎ ∈ 𝑥, 𝑧 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ))) |
37 | eqeq2 2737 | . . . . . . . . 9 ⊢ (ℎ = 𝑙 → (𝑖 = ℎ ↔ 𝑖 = 𝑙)) | |
38 | 37 | ifbid 4553 | . . . . . . . 8 ⊢ (ℎ = 𝑙 → if(𝑖 = ℎ, (-∞(,)𝑧), ℝ) = if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ)) |
39 | 38 | ixpeq2dv 8932 | . . . . . . 7 ⊢ (ℎ = 𝑙 → X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ)) |
40 | oveq2 7427 | . . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (-∞(,)𝑧) = (-∞(,)𝑦)) | |
41 | 40 | ifeq1d 4549 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ) = if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)) |
42 | 41 | ixpeq2dv 8932 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)) |
43 | 39, 42 | cbvmpov 7515 | . . . . . 6 ⊢ (ℎ ∈ 𝑥, 𝑧 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) = (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)) |
44 | 43 | a1i 11 | . . . . 5 ⊢ (𝑤 = 𝑥 → (ℎ ∈ 𝑥, 𝑧 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) = (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ))) |
45 | 36, 44 | eqtrd 2765 | . . . 4 ⊢ (𝑤 = 𝑥 → (ℎ ∈ 𝑤, 𝑧 ∈ ℝ ↦ X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ)) = (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ))) |
46 | 45 | cbvmptv 5262 | . . 3 ⊢ (𝑤 ∈ Fin ↦ (ℎ ∈ 𝑤, 𝑧 ∈ ℝ ↦ X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ))) = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ))) |
47 | 21, 23, 17, 25, 27, 46 | hoimbllem 46161 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
48 | 20, 47 | pm2.61dan 811 | 1 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 ∅c0 4322 ifcif 4530 {csn 4630 ↦ cmpt 5232 dom cdm 5678 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 ↑m cmap 8845 Xcixp 8916 Fincfn 8964 ℝcr 11144 -∞cmnf 11283 (,)cioo 13364 [,)cico 13366 volncvoln 46069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 ax-cc 10465 ax-ac2 10493 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 ax-mulf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 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-fi 9441 df-sup 9472 df-inf 9473 df-oi 9540 df-dju 9931 df-card 9969 df-acn 9972 df-ac 10146 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13798 df-seq 14008 df-exp 14068 df-hash 14334 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-clim 15476 df-rlim 15477 df-sum 15677 df-prod 15894 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-starv 17267 df-tset 17271 df-ple 17272 df-ds 17274 df-unif 17275 df-rest 17423 df-0g 17442 df-topgen 17444 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-grp 18917 df-minusg 18918 df-subg 19103 df-cmn 19766 df-abl 19767 df-mgp 20104 df-rng 20122 df-ur 20151 df-ring 20204 df-cring 20205 df-oppr 20302 df-dvdsr 20325 df-unit 20326 df-invr 20356 df-dvr 20369 df-drng 20655 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-cnfld 21314 df-top 22857 df-topon 22874 df-bases 22910 df-cmp 23352 df-ovol 25454 df-vol 25455 df-salg 45840 df-sumge0 45894 df-mea 45981 df-ome 46021 df-caragen 46023 df-ovoln 46068 df-voln 46070 |
This theorem is referenced by: opnvonmbllem2 46164 hoimbl2 46196 vonhoi 46198 vonioolem1 46211 vonioolem2 46212 vonicclem1 46214 vonicclem2 46215 |
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