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Mirrors > Home > MPE Home > Th. List > itg2lcl | Structured version Visualization version GIF version |
Description: The set of lower sums is a set of extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
itg2val.1 | ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} |
Ref | Expression |
---|---|
itg2lcl | ⊢ 𝐿 ⊆ ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg2val.1 | . 2 ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} | |
2 | itg1cl 23858 | . . . . . 6 ⊢ (𝑔 ∈ dom ∫1 → (∫1‘𝑔) ∈ ℝ) | |
3 | 2 | rexrd 10413 | . . . . 5 ⊢ (𝑔 ∈ dom ∫1 → (∫1‘𝑔) ∈ ℝ*) |
4 | simpr 479 | . . . . . 6 ⊢ ((𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) → 𝑥 = (∫1‘𝑔)) | |
5 | 4 | eleq1d 2891 | . . . . 5 ⊢ ((𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) → (𝑥 ∈ ℝ* ↔ (∫1‘𝑔) ∈ ℝ*)) |
6 | 3, 5 | syl5ibrcom 239 | . . . 4 ⊢ (𝑔 ∈ dom ∫1 → ((𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) → 𝑥 ∈ ℝ*)) |
7 | 6 | rexlimiv 3236 | . . 3 ⊢ (∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) → 𝑥 ∈ ℝ*) |
8 | 7 | abssi 3904 | . 2 ⊢ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ* |
9 | 1, 8 | eqsstri 3860 | 1 ⊢ 𝐿 ⊆ ℝ* |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1656 ∈ wcel 2164 {cab 2811 ∃wrex 3118 ⊆ wss 3798 class class class wbr 4875 dom cdm 5346 ‘cfv 6127 ∘𝑟 cofr 7161 ℝ*cxr 10397 ≤ cle 10399 ∫1citg1 23788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-cda 9312 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-q 12079 df-rp 12120 df-xadd 12240 df-ioo 12474 df-ico 12476 df-icc 12477 df-fz 12627 df-fzo 12768 df-fl 12895 df-seq 13103 df-exp 13162 df-hash 13418 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 df-sum 14801 df-xmet 20106 df-met 20107 df-ovol 23637 df-vol 23638 df-mbf 23792 df-itg1 23793 |
This theorem is referenced by: itg2cl 23905 itg2ub 23906 itg2leub 23907 |
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