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Mirrors > Home > MPE Home > Th. List > itg2leub | Structured version Visualization version GIF version |
Description: Any upper bound on the integrals of all simple functions 𝐺 dominated by 𝐹 is greater than (∫2‘𝐹), the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
itg2leub | ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → ((∫2‘𝐹) ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} | |
2 | 1 | itg2val 24323 | . . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )) |
3 | 2 | adantr 483 | . . 3 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → (∫2‘𝐹) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )) |
4 | 3 | breq1d 5068 | . 2 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → ((∫2‘𝐹) ≤ 𝐴 ↔ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ≤ 𝐴)) |
5 | 1 | itg2lcl 24322 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ* |
6 | supxrleub 12713 | . . . . 5 ⊢ (({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ* ∧ 𝐴 ∈ ℝ*) → (sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑧 ≤ 𝐴)) | |
7 | 5, 6 | mpan 688 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑧 ≤ 𝐴)) |
8 | 7 | adantl 484 | . . 3 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → (sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑧 ≤ 𝐴)) |
9 | eqeq1 2825 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = (∫1‘𝑔) ↔ 𝑧 = (∫1‘𝑔))) | |
10 | 9 | anbi2d 630 | . . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ (𝑔 ∘r ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)))) |
11 | 10 | rexbidv 3297 | . . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)))) |
12 | 11 | ralab 3683 | . . . 4 ⊢ (∀𝑧 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑧 ≤ 𝐴 ↔ ∀𝑧(∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴)) |
13 | r19.23v 3279 | . . . . . . 7 ⊢ (∀𝑔 ∈ dom ∫1((𝑔 ∘r ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ (∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴)) | |
14 | ancomst 467 | . . . . . . . . 9 ⊢ (((𝑔 ∘r ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ ((𝑧 = (∫1‘𝑔) ∧ 𝑔 ∘r ≤ 𝐹) → 𝑧 ≤ 𝐴)) | |
15 | impexp 453 | . . . . . . . . 9 ⊢ (((𝑧 = (∫1‘𝑔) ∧ 𝑔 ∘r ≤ 𝐹) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (∫1‘𝑔) → (𝑔 ∘r ≤ 𝐹 → 𝑧 ≤ 𝐴))) | |
16 | 14, 15 | bitri 277 | . . . . . . . 8 ⊢ (((𝑔 ∘r ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (∫1‘𝑔) → (𝑔 ∘r ≤ 𝐹 → 𝑧 ≤ 𝐴))) |
17 | 16 | ralbii 3165 | . . . . . . 7 ⊢ (∀𝑔 ∈ dom ∫1((𝑔 ∘r ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ ∀𝑔 ∈ dom ∫1(𝑧 = (∫1‘𝑔) → (𝑔 ∘r ≤ 𝐹 → 𝑧 ≤ 𝐴))) |
18 | 13, 17 | bitr3i 279 | . . . . . 6 ⊢ ((∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ ∀𝑔 ∈ dom ∫1(𝑧 = (∫1‘𝑔) → (𝑔 ∘r ≤ 𝐹 → 𝑧 ≤ 𝐴))) |
19 | 18 | albii 1816 | . . . . 5 ⊢ (∀𝑧(∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ ∀𝑧∀𝑔 ∈ dom ∫1(𝑧 = (∫1‘𝑔) → (𝑔 ∘r ≤ 𝐹 → 𝑧 ≤ 𝐴))) |
20 | ralcom4 3235 | . . . . . 6 ⊢ (∀𝑔 ∈ dom ∫1∀𝑧(𝑧 = (∫1‘𝑔) → (𝑔 ∘r ≤ 𝐹 → 𝑧 ≤ 𝐴)) ↔ ∀𝑧∀𝑔 ∈ dom ∫1(𝑧 = (∫1‘𝑔) → (𝑔 ∘r ≤ 𝐹 → 𝑧 ≤ 𝐴))) | |
21 | fvex 6677 | . . . . . . . 8 ⊢ (∫1‘𝑔) ∈ V | |
22 | breq1 5061 | . . . . . . . . 9 ⊢ (𝑧 = (∫1‘𝑔) → (𝑧 ≤ 𝐴 ↔ (∫1‘𝑔) ≤ 𝐴)) | |
23 | 22 | imbi2d 343 | . . . . . . . 8 ⊢ (𝑧 = (∫1‘𝑔) → ((𝑔 ∘r ≤ 𝐹 → 𝑧 ≤ 𝐴) ↔ (𝑔 ∘r ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴))) |
24 | 21, 23 | ceqsalv 3532 | . . . . . . 7 ⊢ (∀𝑧(𝑧 = (∫1‘𝑔) → (𝑔 ∘r ≤ 𝐹 → 𝑧 ≤ 𝐴)) ↔ (𝑔 ∘r ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴)) |
25 | 24 | ralbii 3165 | . . . . . 6 ⊢ (∀𝑔 ∈ dom ∫1∀𝑧(𝑧 = (∫1‘𝑔) → (𝑔 ∘r ≤ 𝐹 → 𝑧 ≤ 𝐴)) ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴)) |
26 | 20, 25 | bitr3i 279 | . . . . 5 ⊢ (∀𝑧∀𝑔 ∈ dom ∫1(𝑧 = (∫1‘𝑔) → (𝑔 ∘r ≤ 𝐹 → 𝑧 ≤ 𝐴)) ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴)) |
27 | 19, 26 | bitri 277 | . . . 4 ⊢ (∀𝑧(∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑧 = (∫1‘𝑔)) → 𝑧 ≤ 𝐴) ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴)) |
28 | 12, 27 | bitri 277 | . . 3 ⊢ (∀𝑧 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑧 ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴)) |
29 | 8, 28 | syl6bb 289 | . 2 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → (sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴))) |
30 | 4, 29 | bitrd 281 | 1 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → ((∫2‘𝐹) ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 ∃wrex 3139 ⊆ wss 3935 class class class wbr 5058 dom cdm 5549 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ∘r cofr 7402 supcsup 8898 ℝcr 10530 0cc0 10531 +∞cpnf 10666 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 [,]cicc 12735 ∫1citg1 24210 ∫2citg2 24211 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xadd 12502 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-xmet 20532 df-met 20533 df-ovol 24059 df-vol 24060 df-mbf 24214 df-itg1 24215 df-itg2 24216 |
This theorem is referenced by: itg2itg1 24331 itg2le 24334 itg2seq 24337 itg2lea 24339 itg2mulclem 24341 itg2splitlem 24343 itg2split 24344 itg2mono 24348 ftc1anclem5 34965 |
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