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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 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 ∈ Fin) |
3 | 2 | rrnmbl 41755 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑𝑚 𝑋) ∈ dom (voln‘𝑋)) |
4 | reex 10363 | . . . . . . . . 9 ⊢ ℝ ∈ V | |
5 | mapdm0 8155 | . . . . . . . . 9 ⊢ (ℝ ∈ V → (ℝ ↑𝑚 ∅) = {∅}) | |
6 | 4, 5 | ax-mp 5 | . . . . . . . 8 ⊢ (ℝ ↑𝑚 ∅) = {∅} |
7 | 6 | eqcomi 2787 | . . . . . . 7 ⊢ {∅} = (ℝ ↑𝑚 ∅) |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝑋 = ∅ → {∅} = (ℝ ↑𝑚 ∅)) |
9 | id 22 | . . . . . . . 8 ⊢ (𝑋 = ∅ → 𝑋 = ∅) | |
10 | 9 | ixpeq1d 8206 | . . . . . . 7 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = X𝑖 ∈ ∅ ((𝐴‘𝑖)[,)(𝐵‘𝑖))) |
11 | ixp0x 8222 | . . . . . . . 8 ⊢ X𝑖 ∈ ∅ ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = {∅} | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝑋 = ∅ → X𝑖 ∈ ∅ ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = {∅}) |
13 | 10, 12 | eqtrd 2814 | . . . . . 6 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = {∅}) |
14 | oveq2 6930 | . . . . . 6 ⊢ (𝑋 = ∅ → (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 ∅)) | |
15 | 8, 13, 14 | 3eqtr4d 2824 | . . . . 5 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = (ℝ ↑𝑚 𝑋)) |
16 | 15 | adantl 475 | . . . 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 2853 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆 ↔ (ℝ ↑𝑚 𝑋) ∈ dom (voln‘𝑋))) |
20 | 3, 19 | mpbird 249 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
21 | 1 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin) |
22 | 9 | necon3bi 2995 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) |
23 | 22 | adantl 475 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
24 | hoimbl.a | . . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | |
25 | 24 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ) |
26 | hoimbl.b | . . . 4 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) | |
27 | 26 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ) |
28 | id 22 | . . . . . 6 ⊢ (𝑤 = 𝑥 → 𝑤 = 𝑥) | |
29 | eqidd 2779 | . . . . . 6 ⊢ (𝑤 = 𝑥 → ℝ = ℝ) | |
30 | 28 | ixpeq1d 8206 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑗 ∈ 𝑥 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ)) |
31 | eqeq1 2782 | . . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑗 = ℎ ↔ 𝑖 = ℎ)) | |
32 | 31 | ifbid 4329 | . . . . . . . . 9 ⊢ (𝑗 = 𝑖 → if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) |
33 | 32 | cbvixpv 8212 | . . . . . . . 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 2814 | . . . . . 6 ⊢ (𝑤 = 𝑥 → X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) |
36 | 28, 29, 35 | mpt2eq123dv 6994 | . . . . 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 2789 | . . . . . . . . 9 ⊢ (ℎ = 𝑙 → (𝑖 = ℎ ↔ 𝑖 = 𝑙)) | |
38 | 37 | ifbid 4329 | . . . . . . . 8 ⊢ (ℎ = 𝑙 → if(𝑖 = ℎ, (-∞(,)𝑧), ℝ) = if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ)) |
39 | 38 | ixpeq2dv 8210 | . . . . . . 7 ⊢ (ℎ = 𝑙 → X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ)) |
40 | oveq2 6930 | . . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (-∞(,)𝑧) = (-∞(,)𝑦)) | |
41 | 40 | ifeq1d 4325 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ) = if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)) |
42 | 41 | ixpeq2dv 8210 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)) |
43 | 39, 42 | cbvmpt2v 7012 | . . . . . 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 2814 | . . . 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 4985 | . . 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 41771 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
48 | 20, 47 | pm2.61dan 803 | 1 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 Vcvv 3398 ∅c0 4141 ifcif 4307 {csn 4398 ↦ cmpt 4965 dom cdm 5355 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 ↑𝑚 cmap 8140 Xcixp 8194 Fincfn 8241 ℝcr 10271 -∞cmnf 10409 (,)cioo 12487 [,)cico 12489 volncvoln 41679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cc 9592 ax-ac2 9620 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-disj 4855 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-omul 7848 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-acn 9101 df-ac 9272 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-rlim 14628 df-sum 14825 df-prod 15039 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-rest 16469 df-0g 16488 df-topgen 16490 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-subg 17975 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-cnfld 20143 df-top 21106 df-topon 21123 df-bases 21158 df-cmp 21599 df-ovol 23668 df-vol 23669 df-salg 41453 df-sumge0 41504 df-mea 41591 df-ome 41631 df-caragen 41633 df-ovoln 41678 df-voln 41680 |
This theorem is referenced by: opnvonmbllem2 41774 hoimbl2 41806 vonhoi 41808 vonioolem1 41821 vonioolem2 41822 vonicclem1 41824 vonicclem2 41825 |
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