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Mirrors > Home > MPE Home > Th. List > itg2lecl | Structured version Visualization version GIF version |
Description: If an ∫2 integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
itg2lecl | ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ ∧ (∫2‘𝐹) ≤ 𝐴) → (∫2‘𝐹) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg2cl 24335 | . . 3 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) ∈ ℝ*) | |
2 | 1 | 3ad2ant1 1129 | . 2 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ ∧ (∫2‘𝐹) ≤ 𝐴) → (∫2‘𝐹) ∈ ℝ*) |
3 | simp2 1133 | . 2 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ ∧ (∫2‘𝐹) ≤ 𝐴) → 𝐴 ∈ ℝ) | |
4 | itg2ge0 24338 | . . 3 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘𝐹)) | |
5 | 4 | 3ad2ant1 1129 | . 2 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ ∧ (∫2‘𝐹) ≤ 𝐴) → 0 ≤ (∫2‘𝐹)) |
6 | simp3 1134 | . 2 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ ∧ (∫2‘𝐹) ≤ 𝐴) → (∫2‘𝐹) ≤ 𝐴) | |
7 | xrrege0 12570 | . 2 ⊢ ((((∫2‘𝐹) ∈ ℝ* ∧ 𝐴 ∈ ℝ) ∧ (0 ≤ (∫2‘𝐹) ∧ (∫2‘𝐹) ≤ 𝐴)) → (∫2‘𝐹) ∈ ℝ) | |
8 | 2, 3, 5, 6, 7 | syl22anc 836 | 1 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ ∧ (∫2‘𝐹) ≤ 𝐴) → (∫2‘𝐹) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 ∈ wcel 2114 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 +∞cpnf 10674 ℝ*cxr 10676 ≤ cle 10678 [,]cicc 12744 ∫2citg2 24219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xadd 12511 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-xmet 20540 df-met 20541 df-ovol 24067 df-vol 24068 df-mbf 24222 df-itg1 24223 df-itg2 24224 |
This theorem is referenced by: itg2mulc 24350 itg2split 24352 itg2monolem1 24353 itg2cnlem2 24365 iblss 24407 ibladdlem 24422 iblabs 24431 iblabsr 24432 iblmulc2 24433 bddmulibl 24441 ibladdnclem 34950 iblabsnc 34958 iblmulc2nc 34959 bddiblnc 34964 ftc1anclem4 34972 ftc1anclem7 34975 ftc1anclem8 34976 |
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