![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 25732 | . 2 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) | |
2 | i1frn 25726 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin) | |
3 | difss 4146 | . . . 4 ⊢ (ran 𝐹 ∖ {0}) ⊆ ran 𝐹 | |
4 | ssfi 9212 | . . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (ran 𝐹 ∖ {0}) ⊆ ran 𝐹) → (ran 𝐹 ∖ {0}) ∈ Fin) | |
5 | 2, 3, 4 | sylancl 586 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → (ran 𝐹 ∖ {0}) ∈ Fin) |
6 | i1ff 25725 | . . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
7 | 6 | frnd 6745 | . . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ⊆ ℝ) |
8 | 7 | ssdifssd 4157 | . . . . 5 ⊢ (𝐹 ∈ dom ∫1 → (ran 𝐹 ∖ {0}) ⊆ ℝ) |
9 | 8 | sselda 3995 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → 𝑥 ∈ ℝ) |
10 | i1fima2sn 25729 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ) | |
11 | 9, 10 | remulcld 11289 | . . 3 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (𝑥 · (vol‘(◡𝐹 “ {𝑥}))) ∈ ℝ) |
12 | 5, 11 | fsumrecl 15767 | . 2 ⊢ (𝐹 ∈ dom ∫1 → Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥}))) ∈ ℝ) |
13 | 1, 12 | eqeltrd 2839 | 1 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 ◡ccnv 5688 dom cdm 5689 ran crn 5690 “ cima 5692 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℝcr 11152 0cc0 11153 · cmul 11158 Σcsu 15719 volcvol 25512 ∫1citg1 25664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xadd 13153 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-xmet 21375 df-met 21376 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 |
This theorem is referenced by: itg1mulc 25754 itg1sub 25759 itg1lea 25762 itg2lcl 25777 itg2itg1 25786 itg2seq 25792 itg2uba 25793 itg2mulclem 25796 itg2splitlem 25798 itg2split 25799 itg2monolem1 25800 itg2monolem2 25801 itg2monolem3 25802 itg2i1fseq2 25806 itg2addlem 25808 i1fibl 25858 itg2addnclem 37658 itg2addnc 37661 ftc1anclem5 37684 ftc1anclem7 37686 ftc1anclem8 37687 |
Copyright terms: Public domain | W3C validator |