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Mirrors > Home > MPE Home > Th. List > Mathboxes > omscl | Structured version Visualization version GIF version |
Description: A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019.) |
Ref | Expression |
---|---|
omscl | ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)) ⊆ (0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3401 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | simp2 1128 | . . . . . . 7 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) → 𝑅:𝑄⟶(0[,]+∞)) | |
3 | 2 | ad2antrr 716 | . . . . . 6 ⊢ ((((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) ∧ 𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}) ∧ 𝑦 ∈ 𝑥) → 𝑅:𝑄⟶(0[,]+∞)) |
4 | ssrab2 3908 | . . . . . . . . . 10 ⊢ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ⊆ 𝒫 dom 𝑅 | |
5 | simpr 479 | . . . . . . . . . 10 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) ∧ 𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}) → 𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}) | |
6 | 4, 5 | sseldi 3819 | . . . . . . . . 9 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) ∧ 𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}) → 𝑥 ∈ 𝒫 dom 𝑅) |
7 | fdm 6299 | . . . . . . . . . . . 12 ⊢ (𝑅:𝑄⟶(0[,]+∞) → dom 𝑅 = 𝑄) | |
8 | 7 | pweqd 4384 | . . . . . . . . . . 11 ⊢ (𝑅:𝑄⟶(0[,]+∞) → 𝒫 dom 𝑅 = 𝒫 𝑄) |
9 | 2, 8 | syl 17 | . . . . . . . . . 10 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) → 𝒫 dom 𝑅 = 𝒫 𝑄) |
10 | 9 | adantr 474 | . . . . . . . . 9 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) ∧ 𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}) → 𝒫 dom 𝑅 = 𝒫 𝑄) |
11 | 6, 10 | eleqtrd 2861 | . . . . . . . 8 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) ∧ 𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}) → 𝑥 ∈ 𝒫 𝑄) |
12 | elpwi 4389 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝑄 → 𝑥 ⊆ 𝑄) | |
13 | 11, 12 | syl 17 | . . . . . . 7 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) ∧ 𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}) → 𝑥 ⊆ 𝑄) |
14 | 13 | sselda 3821 | . . . . . 6 ⊢ ((((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) ∧ 𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑄) |
15 | 3, 14 | ffvelrnd 6624 | . . . . 5 ⊢ ((((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) ∧ 𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}) ∧ 𝑦 ∈ 𝑥) → (𝑅‘𝑦) ∈ (0[,]+∞)) |
16 | 15 | ralrimiva 3148 | . . . 4 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) ∧ 𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}) → ∀𝑦 ∈ 𝑥 (𝑅‘𝑦) ∈ (0[,]+∞)) |
17 | nfcv 2934 | . . . . 5 ⊢ Ⅎ𝑦𝑥 | |
18 | 17 | esumcl 30690 | . . . 4 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝑅‘𝑦) ∈ (0[,]+∞)) → Σ*𝑦 ∈ 𝑥(𝑅‘𝑦) ∈ (0[,]+∞)) |
19 | 1, 16, 18 | sylancr 581 | . . 3 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) ∧ 𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}) → Σ*𝑦 ∈ 𝑥(𝑅‘𝑦) ∈ (0[,]+∞)) |
20 | 19 | ralrimiva 3148 | . 2 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) → ∀𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}Σ*𝑦 ∈ 𝑥(𝑅‘𝑦) ∈ (0[,]+∞)) |
21 | eqid 2778 | . . 3 ⊢ (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)) = (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)) | |
22 | 21 | rnmptss 6656 | . 2 ⊢ (∀𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}Σ*𝑦 ∈ 𝑥(𝑅‘𝑦) ∈ (0[,]+∞) → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)) ⊆ (0[,]+∞)) |
23 | 20, 22 | syl 17 | 1 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)) ⊆ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∀wral 3090 {crab 3094 Vcvv 3398 ⊆ wss 3792 𝒫 cpw 4379 ∪ cuni 4671 class class class wbr 4886 ↦ cmpt 4965 dom cdm 5355 ran crn 5356 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ωcom 7343 ≼ cdom 8239 0cc0 10272 +∞cpnf 10408 [,]cicc 12490 Σ*cesum 30687 |
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-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 |
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-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-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 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-xadd 12258 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 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-tset 16357 df-ple 16358 df-ds 16360 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-ordt 16547 df-xrs 16548 df-mre 16632 df-mrc 16633 df-acs 16635 df-ps 17586 df-tsr 17587 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-cntz 18133 df-cmn 18581 df-fbas 20139 df-fg 20140 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-ntr 21232 df-nei 21310 df-cn 21439 df-haus 21527 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-tsms 22338 df-esum 30688 |
This theorem is referenced by: omsf 30956 omssubaddlem 30959 omssubadd 30960 |
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