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Mirrors > Home > MPE Home > Th. List > itg1cl | Structured version Visualization version GIF version |
Description: Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.) |
Ref | Expression |
---|---|
itg1cl | ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg1val 23791 | . 2 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) | |
2 | i1frn 23785 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin) | |
3 | difss 3935 | . . . 4 ⊢ (ran 𝐹 ∖ {0}) ⊆ ran 𝐹 | |
4 | ssfi 8422 | . . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (ran 𝐹 ∖ {0}) ⊆ ran 𝐹) → (ran 𝐹 ∖ {0}) ∈ Fin) | |
5 | 2, 3, 4 | sylancl 581 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → (ran 𝐹 ∖ {0}) ∈ Fin) |
6 | i1ff 23784 | . . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
7 | 6 | frnd 6263 | . . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ⊆ ℝ) |
8 | 7 | ssdifssd 3946 | . . . . 5 ⊢ (𝐹 ∈ dom ∫1 → (ran 𝐹 ∖ {0}) ⊆ ℝ) |
9 | 8 | sselda 3798 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → 𝑥 ∈ ℝ) |
10 | i1fima2sn 23788 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ) | |
11 | 9, 10 | remulcld 10359 | . . 3 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (𝑥 · (vol‘(◡𝐹 “ {𝑥}))) ∈ ℝ) |
12 | 5, 11 | fsumrecl 14806 | . 2 ⊢ (𝐹 ∈ dom ∫1 → Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥}))) ∈ ℝ) |
13 | 1, 12 | eqeltrd 2878 | 1 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ∖ cdif 3766 ⊆ wss 3769 {csn 4368 ◡ccnv 5311 dom cdm 5312 ran crn 5313 “ cima 5315 ‘cfv 6101 (class class class)co 6878 Fincfn 8195 ℝcr 10223 0cc0 10224 · cmul 10229 Σcsu 14757 volcvol 23571 ∫1citg1 23723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-z 11667 df-uz 11931 df-q 12034 df-rp 12075 df-xadd 12194 df-ioo 12428 df-ico 12430 df-icc 12431 df-fz 12581 df-fzo 12721 df-fl 12848 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-clim 14560 df-sum 14758 df-xmet 20061 df-met 20062 df-ovol 23572 df-vol 23573 df-mbf 23727 df-itg1 23728 |
This theorem is referenced by: itg1mulc 23812 itg1sub 23817 itg1lea 23820 itg2lcl 23835 itg2itg1 23844 itg2seq 23850 itg2uba 23851 itg2mulclem 23854 itg2splitlem 23856 itg2split 23857 itg2monolem1 23858 itg2monolem2 23859 itg2monolem3 23860 itg2i1fseq2 23864 itg2addlem 23866 i1fibl 23915 itg2addnclem 33949 itg2addnc 33952 ftc1anclem5 33977 ftc1anclem7 33979 ftc1anclem8 33980 |
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