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Mirrors > Home > MPE Home > Th. List > itg2lea | Structured version Visualization version GIF version |
Description: Approximate version of itg2le 23506. If 𝐹 ≤ 𝐺 for almost all 𝑥, then ∫2𝐹 ≤ ∫2𝐺. (Contributed by Mario Carneiro, 11-Aug-2014.) |
Ref | Expression |
---|---|
itg2lea.1 | ⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞)) |
itg2lea.2 | ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞)) |
itg2lea.3 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
itg2lea.4 | ⊢ (𝜑 → (vol*‘𝐴) = 0) |
itg2lea.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) |
Ref | Expression |
---|---|
itg2lea | ⊢ (𝜑 → (∫2‘𝐹) ≤ (∫2‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg2lea.2 | . . . . . 6 ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞)) | |
2 | 1 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → 𝐺:ℝ⟶(0[,]+∞)) |
3 | simprl 794 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → 𝑓 ∈ dom ∫1) | |
4 | itg2lea.3 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
5 | 4 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → 𝐴 ⊆ ℝ) |
6 | itg2lea.4 | . . . . . 6 ⊢ (𝜑 → (vol*‘𝐴) = 0) | |
7 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → (vol*‘𝐴) = 0) |
8 | i1ff 23443 | . . . . . . . . 9 ⊢ (𝑓 ∈ dom ∫1 → 𝑓:ℝ⟶ℝ) | |
9 | 8 | ad2antrl 764 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → 𝑓:ℝ⟶ℝ) |
10 | eldifi 3732 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ) | |
11 | ffvelrn 6357 | . . . . . . . 8 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝑓‘𝑥) ∈ ℝ) | |
12 | 9, 10, 11 | syl2an 494 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝑓‘𝑥) ∈ ℝ) |
13 | 12 | rexrd 10089 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝑓‘𝑥) ∈ ℝ*) |
14 | iccssxr 12256 | . . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ* | |
15 | itg2lea.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞)) | |
16 | 15 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → 𝐹:ℝ⟶(0[,]+∞)) |
17 | ffvelrn 6357 | . . . . . . . 8 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,]+∞)) | |
18 | 16, 10, 17 | syl2an 494 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ∈ (0[,]+∞)) |
19 | 14, 18 | sseldi 3601 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ∈ ℝ*) |
20 | ffvelrn 6357 | . . . . . . . 8 ⊢ ((𝐺:ℝ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ (0[,]+∞)) | |
21 | 2, 10, 20 | syl2an 494 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐺‘𝑥) ∈ (0[,]+∞)) |
22 | 14, 21 | sseldi 3601 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐺‘𝑥) ∈ ℝ*) |
23 | simprr 796 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → 𝑓 ∘𝑟 ≤ 𝐹) | |
24 | ffn 6045 | . . . . . . . . . . 11 ⊢ (𝑓:ℝ⟶ℝ → 𝑓 Fn ℝ) | |
25 | 9, 24 | syl 17 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → 𝑓 Fn ℝ) |
26 | ffn 6045 | . . . . . . . . . . 11 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹 Fn ℝ) | |
27 | 16, 26 | syl 17 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → 𝐹 Fn ℝ) |
28 | reex 10027 | . . . . . . . . . . 11 ⊢ ℝ ∈ V | |
29 | 28 | a1i 11 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → ℝ ∈ V) |
30 | inidm 3822 | . . . . . . . . . 10 ⊢ (ℝ ∩ ℝ) = ℝ | |
31 | eqidd 2623 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ ℝ) → (𝑓‘𝑥) = (𝑓‘𝑥)) | |
32 | eqidd 2623 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
33 | 25, 27, 29, 29, 30, 31, 32 | ofrfval 6905 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → (𝑓 ∘𝑟 ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ (𝑓‘𝑥) ≤ (𝐹‘𝑥))) |
34 | 23, 33 | mpbid 222 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → ∀𝑥 ∈ ℝ (𝑓‘𝑥) ≤ (𝐹‘𝑥)) |
35 | 34 | r19.21bi 2932 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ ℝ) → (𝑓‘𝑥) ≤ (𝐹‘𝑥)) |
36 | 10, 35 | sylan2 491 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝑓‘𝑥) ≤ (𝐹‘𝑥)) |
37 | itg2lea.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) | |
38 | 37 | adantlr 751 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) |
39 | 13, 19, 22, 36, 38 | xrletrd 11993 | . . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝑓‘𝑥) ≤ (𝐺‘𝑥)) |
40 | 2, 3, 5, 7, 39 | itg2uba 23510 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐹)) → (∫1‘𝑓) ≤ (∫2‘𝐺)) |
41 | 40 | expr 643 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ dom ∫1) → (𝑓 ∘𝑟 ≤ 𝐹 → (∫1‘𝑓) ≤ (∫2‘𝐺))) |
42 | 41 | ralrimiva 2966 | . 2 ⊢ (𝜑 → ∀𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 → (∫1‘𝑓) ≤ (∫2‘𝐺))) |
43 | itg2cl 23499 | . . . 4 ⊢ (𝐺:ℝ⟶(0[,]+∞) → (∫2‘𝐺) ∈ ℝ*) | |
44 | 1, 43 | syl 17 | . . 3 ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ*) |
45 | itg2leub 23501 | . . 3 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (∫2‘𝐺) ∈ ℝ*) → ((∫2‘𝐹) ≤ (∫2‘𝐺) ↔ ∀𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 → (∫1‘𝑓) ≤ (∫2‘𝐺)))) | |
46 | 15, 44, 45 | syl2anc 693 | . 2 ⊢ (𝜑 → ((∫2‘𝐹) ≤ (∫2‘𝐺) ↔ ∀𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 → (∫1‘𝑓) ≤ (∫2‘𝐺)))) |
47 | 42, 46 | mpbird 247 | 1 ⊢ (𝜑 → (∫2‘𝐹) ≤ (∫2‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 class class class wbr 4653 dom cdm 5114 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑟 cofr 6896 ℝcr 9935 0cc0 9936 +∞cpnf 10071 ℝ*cxr 10073 ≤ cle 10075 [,]cicc 12178 vol*covol 23231 ∫1citg1 23384 ∫2citg2 23385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-rest 16083 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cmp 21190 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 df-itg2 23390 |
This theorem is referenced by: itg2eqa 23512 |
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