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Mirrors > Home > MPE Home > Th. List > itg1lea | Structured version Visualization version GIF version |
Description: Approximate version of itg1le 24308. 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 24303 | . . . . 5 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐹 ∈ dom ∫1) → (𝐺 ∘f − 𝐹) ∈ dom ∫1) | |
4 | 1, 2, 3 | syl2anc 586 | . . . 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 4102 | . . . . . . 7 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ) | |
9 | i1ff 24271 | . . . . . . . . . 10 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ) | |
10 | 1, 9 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ) |
11 | 10 | ffvelrnda 6845 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℝ) |
12 | i1ff 24271 | . . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
13 | 2, 12 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
14 | 13 | ffvelrnda 6845 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
15 | 11, 14 | subge0d 11224 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥))) |
16 | 8, 15 | sylan2 594 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥))) |
17 | 7, 16 | mpbird 259 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥))) |
18 | 10 | ffnd 6509 | . . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ) |
19 | 13 | ffnd 6509 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ) |
20 | reex 10622 | . . . . . . . 8 ⊢ ℝ ∈ V | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ∈ V) |
22 | inidm 4194 | . . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ | |
23 | eqidd 2822 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
24 | eqidd 2822 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
25 | 18, 19, 21, 21, 22, 23, 24 | ofval 7412 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ∘f − 𝐹)‘𝑥) = ((𝐺‘𝑥) − (𝐹‘𝑥))) |
26 | 8, 25 | sylan2 594 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ((𝐺 ∘f − 𝐹)‘𝑥) = ((𝐺‘𝑥) − (𝐹‘𝑥))) |
27 | 17, 26 | breqtrrd 5086 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ ((𝐺 ∘f − 𝐹)‘𝑥)) |
28 | 4, 5, 6, 27 | itg1ge0a 24306 | . . 3 ⊢ (𝜑 → 0 ≤ (∫1‘(𝐺 ∘f − 𝐹))) |
29 | itg1sub 24304 | . . . 4 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐹 ∈ dom ∫1) → (∫1‘(𝐺 ∘f − 𝐹)) = ((∫1‘𝐺) − (∫1‘𝐹))) | |
30 | 1, 2, 29 | syl2anc 586 | . . 3 ⊢ (𝜑 → (∫1‘(𝐺 ∘f − 𝐹)) = ((∫1‘𝐺) − (∫1‘𝐹))) |
31 | 28, 30 | breqtrd 5084 | . 2 ⊢ (𝜑 → 0 ≤ ((∫1‘𝐺) − (∫1‘𝐹))) |
32 | itg1cl 24280 | . . . 4 ⊢ (𝐺 ∈ dom ∫1 → (∫1‘𝐺) ∈ ℝ) | |
33 | 1, 32 | syl 17 | . . 3 ⊢ (𝜑 → (∫1‘𝐺) ∈ ℝ) |
34 | itg1cl 24280 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℝ) | |
35 | 2, 34 | syl 17 | . . 3 ⊢ (𝜑 → (∫1‘𝐹) ∈ ℝ) |
36 | 33, 35 | subge0d 11224 | . 2 ⊢ (𝜑 → (0 ≤ ((∫1‘𝐺) − (∫1‘𝐹)) ↔ (∫1‘𝐹) ≤ (∫1‘𝐺))) |
37 | 31, 36 | mpbid 234 | 1 ⊢ (𝜑 → (∫1‘𝐹) ≤ (∫1‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 ⊆ wss 3935 class class class wbr 5058 dom cdm 5549 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ∘f cof 7401 ℝcr 10530 0cc0 10531 ≤ cle 10670 − cmin 10864 vol*covol 24057 ∫1citg1 24210 |
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 ax-addf 10610 |
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-disj 5024 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 |
This theorem is referenced by: itg1le 24308 itg2uba 24338 itg2splitlem 24343 |
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