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Mirrors > Home > MPE Home > Th. List > itg1lea | Structured version Visualization version GIF version |
Description: Approximate version of itg1le 24241. 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 24236 | . . . . 5 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐹 ∈ dom ∫1) → (𝐺 ∘f − 𝐹) ∈ dom ∫1) | |
4 | 1, 2, 3 | syl2anc 584 | . . . 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 4100 | . . . . . . 7 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ) | |
9 | i1ff 24204 | . . . . . . . . . 10 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ) | |
10 | 1, 9 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ) |
11 | 10 | ffvelrnda 6843 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℝ) |
12 | i1ff 24204 | . . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
13 | 2, 12 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
14 | 13 | ffvelrnda 6843 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
15 | 11, 14 | subge0d 11218 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥))) |
16 | 8, 15 | sylan2 592 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥))) |
17 | 7, 16 | mpbird 258 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥))) |
18 | 10 | ffnd 6508 | . . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ) |
19 | 13 | ffnd 6508 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ) |
20 | reex 10616 | . . . . . . . 8 ⊢ ℝ ∈ V | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ∈ V) |
22 | inidm 4192 | . . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ | |
23 | eqidd 2819 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
24 | eqidd 2819 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
25 | 18, 19, 21, 21, 22, 23, 24 | ofval 7407 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ∘f − 𝐹)‘𝑥) = ((𝐺‘𝑥) − (𝐹‘𝑥))) |
26 | 8, 25 | sylan2 592 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ((𝐺 ∘f − 𝐹)‘𝑥) = ((𝐺‘𝑥) − (𝐹‘𝑥))) |
27 | 17, 26 | breqtrrd 5085 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ ((𝐺 ∘f − 𝐹)‘𝑥)) |
28 | 4, 5, 6, 27 | itg1ge0a 24239 | . . 3 ⊢ (𝜑 → 0 ≤ (∫1‘(𝐺 ∘f − 𝐹))) |
29 | itg1sub 24237 | . . . 4 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐹 ∈ dom ∫1) → (∫1‘(𝐺 ∘f − 𝐹)) = ((∫1‘𝐺) − (∫1‘𝐹))) | |
30 | 1, 2, 29 | syl2anc 584 | . . 3 ⊢ (𝜑 → (∫1‘(𝐺 ∘f − 𝐹)) = ((∫1‘𝐺) − (∫1‘𝐹))) |
31 | 28, 30 | breqtrd 5083 | . 2 ⊢ (𝜑 → 0 ≤ ((∫1‘𝐺) − (∫1‘𝐹))) |
32 | itg1cl 24213 | . . . 4 ⊢ (𝐺 ∈ dom ∫1 → (∫1‘𝐺) ∈ ℝ) | |
33 | 1, 32 | syl 17 | . . 3 ⊢ (𝜑 → (∫1‘𝐺) ∈ ℝ) |
34 | itg1cl 24213 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℝ) | |
35 | 2, 34 | syl 17 | . . 3 ⊢ (𝜑 → (∫1‘𝐹) ∈ ℝ) |
36 | 33, 35 | subge0d 11218 | . 2 ⊢ (𝜑 → (0 ≤ ((∫1‘𝐺) − (∫1‘𝐹)) ↔ (∫1‘𝐹) ≤ (∫1‘𝐺))) |
37 | 31, 36 | mpbid 233 | 1 ⊢ (𝜑 → (∫1‘𝐹) ≤ (∫1‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 class class class wbr 5057 dom cdm 5548 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 ℝcr 10524 0cc0 10525 ≤ cle 10664 − cmin 10858 vol*covol 23990 ∫1citg1 24143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 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 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 df-xmet 20466 df-met 20467 df-ovol 23992 df-vol 23993 df-mbf 24147 df-itg1 24148 |
This theorem is referenced by: itg1le 24241 itg2uba 24271 itg2splitlem 24276 |
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