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Mirrors > Home > MPE Home > Th. List > areaf | Structured version Visualization version GIF version |
Description: Area measurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.) |
Ref | Expression |
---|---|
areaf | ⊢ area:dom area⟶(0[,)+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfarea 25540 | . 2 ⊢ area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥) | |
2 | areambl 25538 | . . . . 5 ⊢ ((𝑠 ∈ dom area ∧ 𝑥 ∈ ℝ) → ((𝑠 “ {𝑥}) ∈ dom vol ∧ (vol‘(𝑠 “ {𝑥})) ∈ ℝ)) | |
3 | 2 | simprd 498 | . . . 4 ⊢ ((𝑠 ∈ dom area ∧ 𝑥 ∈ ℝ) → (vol‘(𝑠 “ {𝑥})) ∈ ℝ) |
4 | dmarea 25537 | . . . . 5 ⊢ (𝑠 ∈ dom area ↔ (𝑠 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝑠 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑠 “ {𝑥}))) ∈ 𝐿1)) | |
5 | 4 | simp3bi 1143 | . . . 4 ⊢ (𝑠 ∈ dom area → (𝑥 ∈ ℝ ↦ (vol‘(𝑠 “ {𝑥}))) ∈ 𝐿1) |
6 | 3, 5 | itgrecl 24400 | . . 3 ⊢ (𝑠 ∈ dom area → ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ ℝ) |
7 | 2 | simpld 497 | . . . . . 6 ⊢ ((𝑠 ∈ dom area ∧ 𝑥 ∈ ℝ) → (𝑠 “ {𝑥}) ∈ dom vol) |
8 | mblss 24134 | . . . . . 6 ⊢ ((𝑠 “ {𝑥}) ∈ dom vol → (𝑠 “ {𝑥}) ⊆ ℝ) | |
9 | ovolge0 24084 | . . . . . 6 ⊢ ((𝑠 “ {𝑥}) ⊆ ℝ → 0 ≤ (vol*‘(𝑠 “ {𝑥}))) | |
10 | 7, 8, 9 | 3syl 18 | . . . . 5 ⊢ ((𝑠 ∈ dom area ∧ 𝑥 ∈ ℝ) → 0 ≤ (vol*‘(𝑠 “ {𝑥}))) |
11 | mblvol 24133 | . . . . . 6 ⊢ ((𝑠 “ {𝑥}) ∈ dom vol → (vol‘(𝑠 “ {𝑥})) = (vol*‘(𝑠 “ {𝑥}))) | |
12 | 7, 11 | syl 17 | . . . . 5 ⊢ ((𝑠 ∈ dom area ∧ 𝑥 ∈ ℝ) → (vol‘(𝑠 “ {𝑥})) = (vol*‘(𝑠 “ {𝑥}))) |
13 | 10, 12 | breqtrrd 5096 | . . . 4 ⊢ ((𝑠 ∈ dom area ∧ 𝑥 ∈ ℝ) → 0 ≤ (vol‘(𝑠 “ {𝑥}))) |
14 | 5, 3, 13 | itgge0 24413 | . . 3 ⊢ (𝑠 ∈ dom area → 0 ≤ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥) |
15 | elrege0 12845 | . . 3 ⊢ (∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ (0[,)+∞) ↔ (∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ ℝ ∧ 0 ≤ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)) | |
16 | 6, 14, 15 | sylanbrc 585 | . 2 ⊢ (𝑠 ∈ dom area → ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ (0[,)+∞)) |
17 | 1, 16 | fmpti 6878 | 1 ⊢ area:dom area⟶(0[,)+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 {csn 4569 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ◡ccnv 5556 dom cdm 5557 “ cima 5560 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 +∞cpnf 10674 ≤ cle 10678 [,)cico 12743 vol*covol 24065 volcvol 24066 𝐿1cibl 24220 ∫citg 24221 areacarea 25535 |
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 ax-addf 10618 |
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-disj 5034 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-4 11705 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-mod 13241 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 df-ibl 24225 df-itg 24226 df-0p 24273 df-area 25536 |
This theorem is referenced by: areacl 25542 areage0 25543 |
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