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Mirrors > Home > MPE Home > Th. List > itg1le | Structured version Visualization version GIF version |
Description: If one simple function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.) |
Ref | Expression |
---|---|
itg1le | ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ∧ 𝐹 ∘r ≤ 𝐺) → (∫1‘𝐹) ≤ (∫1‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1128 | . 2 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ∧ 𝐹 ∘r ≤ 𝐺) → 𝐹 ∈ dom ∫1) | |
2 | 0ss 4347 | . . 3 ⊢ ∅ ⊆ ℝ | |
3 | 2 | a1i 11 | . 2 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ∧ 𝐹 ∘r ≤ 𝐺) → ∅ ⊆ ℝ) |
4 | ovol0 24021 | . . 3 ⊢ (vol*‘∅) = 0 | |
5 | 4 | a1i 11 | . 2 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ∧ 𝐹 ∘r ≤ 𝐺) → (vol*‘∅) = 0) |
6 | simp2 1129 | . 2 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ∧ 𝐹 ∘r ≤ 𝐺) → 𝐺 ∈ dom ∫1) | |
7 | simpl 483 | . . . . . . 7 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → 𝐹 ∈ dom ∫1) | |
8 | i1ff 24204 | . . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
9 | ffn 6507 | . . . . . . 7 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ) | |
10 | 7, 8, 9 | 3syl 18 | . . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → 𝐹 Fn ℝ) |
11 | simpr 485 | . . . . . . 7 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → 𝐺 ∈ dom ∫1) | |
12 | i1ff 24204 | . . . . . . 7 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ) | |
13 | ffn 6507 | . . . . . . 7 ⊢ (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ) | |
14 | 11, 12, 13 | 3syl 18 | . . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → 𝐺 Fn ℝ) |
15 | reex 10616 | . . . . . . 7 ⊢ ℝ ∈ V | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → ℝ ∈ V) |
17 | inidm 4192 | . . . . . 6 ⊢ (ℝ ∩ ℝ) = ℝ | |
18 | eqidd 2819 | . . . . . 6 ⊢ (((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
19 | eqidd 2819 | . . . . . 6 ⊢ (((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
20 | 10, 14, 16, 16, 17, 18, 19 | ofrval 7408 | . . . . 5 ⊢ (((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) ∧ 𝐹 ∘r ≤ 𝐺 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) |
21 | 20 | 3exp 1111 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (𝐹 ∘r ≤ 𝐺 → (𝑥 ∈ ℝ → (𝐹‘𝑥) ≤ (𝐺‘𝑥)))) |
22 | 21 | 3impia 1109 | . . 3 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ∧ 𝐹 ∘r ≤ 𝐺) → (𝑥 ∈ ℝ → (𝐹‘𝑥) ≤ (𝐺‘𝑥))) |
23 | eldifi 4100 | . . 3 ⊢ (𝑥 ∈ (ℝ ∖ ∅) → 𝑥 ∈ ℝ) | |
24 | 22, 23 | impel 506 | . 2 ⊢ (((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ∧ 𝐹 ∘r ≤ 𝐺) ∧ 𝑥 ∈ (ℝ ∖ ∅)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) |
25 | 1, 3, 5, 6, 24 | itg1lea 24240 | 1 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ∧ 𝐹 ∘r ≤ 𝐺) → (∫1‘𝐹) ≤ (∫1‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 ∅c0 4288 class class class wbr 5057 dom cdm 5548 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 ∘r cofr 7397 ℝcr 10524 0cc0 10525 ≤ cle 10664 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-ofr 7399 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: itg2itg1 24264 itg2i1fseq2 24284 itg2addnclem 34824 ftc1anclem5 34852 |
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