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Mirrors > Home > MPE Home > Th. List > itg1lea | Structured version Visualization version GIF version |
Description: Approximate version of itg1le 24317. If 𝐹 ≤ 𝐺 for almost all 𝑥, then ∫1𝐹 ≤ ∫1𝐺. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.) |
Ref | Expression |
---|---|
itg10a.1 | ⊢ (𝜑 → 𝐹 ∈ dom ∫1) |
itg10a.2 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
itg10a.3 | ⊢ (𝜑 → (vol*‘𝐴) = 0) |
itg1lea.4 | ⊢ (𝜑 → 𝐺 ∈ dom ∫1) |
itg1lea.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) |
Ref | Expression |
---|---|
itg1lea | ⊢ (𝜑 → (∫1‘𝐹) ≤ (∫1‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg1lea.4 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ dom ∫1) | |
2 | itg10a.1 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ dom ∫1) | |
3 | i1fsub 24312 | . . . . 5 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐹 ∈ dom ∫1) → (𝐺 ∘f − 𝐹) ∈ dom ∫1) | |
4 | 1, 2, 3 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐺 ∘f − 𝐹) ∈ dom ∫1) |
5 | itg10a.2 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
6 | itg10a.3 | . . . 4 ⊢ (𝜑 → (vol*‘𝐴) = 0) | |
7 | itg1lea.5 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) | |
8 | eldifi 4054 | . . . . . . 7 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ) | |
9 | i1ff 24280 | . . . . . . . . . 10 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ) | |
10 | 1, 9 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ) |
11 | 10 | ffvelrnda 6828 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℝ) |
12 | i1ff 24280 | . . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
13 | 2, 12 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
14 | 13 | ffvelrnda 6828 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
15 | 11, 14 | subge0d 11219 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥))) |
16 | 8, 15 | sylan2 595 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥))) |
17 | 7, 16 | mpbird 260 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥))) |
18 | 10 | ffnd 6488 | . . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ) |
19 | 13 | ffnd 6488 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ) |
20 | reex 10617 | . . . . . . . 8 ⊢ ℝ ∈ V | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ∈ V) |
22 | inidm 4145 | . . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ | |
23 | eqidd 2799 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
24 | eqidd 2799 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
25 | 18, 19, 21, 21, 22, 23, 24 | ofval 7398 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ∘f − 𝐹)‘𝑥) = ((𝐺‘𝑥) − (𝐹‘𝑥))) |
26 | 8, 25 | sylan2 595 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ((𝐺 ∘f − 𝐹)‘𝑥) = ((𝐺‘𝑥) − (𝐹‘𝑥))) |
27 | 17, 26 | breqtrrd 5058 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ ((𝐺 ∘f − 𝐹)‘𝑥)) |
28 | 4, 5, 6, 27 | itg1ge0a 24315 | . . 3 ⊢ (𝜑 → 0 ≤ (∫1‘(𝐺 ∘f − 𝐹))) |
29 | itg1sub 24313 | . . . 4 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐹 ∈ dom ∫1) → (∫1‘(𝐺 ∘f − 𝐹)) = ((∫1‘𝐺) − (∫1‘𝐹))) | |
30 | 1, 2, 29 | syl2anc 587 | . . 3 ⊢ (𝜑 → (∫1‘(𝐺 ∘f − 𝐹)) = ((∫1‘𝐺) − (∫1‘𝐹))) |
31 | 28, 30 | breqtrd 5056 | . 2 ⊢ (𝜑 → 0 ≤ ((∫1‘𝐺) − (∫1‘𝐹))) |
32 | itg1cl 24289 | . . . 4 ⊢ (𝐺 ∈ dom ∫1 → (∫1‘𝐺) ∈ ℝ) | |
33 | 1, 32 | syl 17 | . . 3 ⊢ (𝜑 → (∫1‘𝐺) ∈ ℝ) |
34 | itg1cl 24289 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℝ) | |
35 | 2, 34 | syl 17 | . . 3 ⊢ (𝜑 → (∫1‘𝐹) ∈ ℝ) |
36 | 33, 35 | subge0d 11219 | . 2 ⊢ (𝜑 → (0 ≤ ((∫1‘𝐺) − (∫1‘𝐹)) ↔ (∫1‘𝐹) ≤ (∫1‘𝐺))) |
37 | 31, 36 | mpbid 235 | 1 ⊢ (𝜑 → (∫1‘𝐹) ≤ (∫1‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 class class class wbr 5030 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 ℝcr 10525 0cc0 10526 ≤ cle 10665 − cmin 10859 vol*covol 24066 ∫1citg1 24219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-xmet 20084 df-met 20085 df-ovol 24068 df-vol 24069 df-mbf 24223 df-itg1 24224 |
This theorem is referenced by: itg1le 24317 itg2uba 24347 itg2splitlem 24352 |
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