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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoimbl2 | 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, 8-Apr-2021.) |
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) |
Ref | Expression |
---|---|
hoimbl2 | ⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ∈ 𝑆) |
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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