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Mirrors > Home > MPE Home > Th. List > itg1lea | Structured version Visualization version GIF version |
Description: Approximate version of itg1le 25663. 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 25658 | . . . . 5 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐹 ∈ dom ∫1) → (𝐺 ∘f − 𝐹) ∈ dom ∫1) | |
4 | 1, 2, 3 | syl2anc 582 | . . . 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 4127 | . . . . . . 7 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ) | |
9 | i1ff 25625 | . . . . . . . . . 10 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ) | |
10 | 1, 9 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ) |
11 | 10 | ffvelcdmda 7099 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℝ) |
12 | i1ff 25625 | . . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
13 | 2, 12 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
14 | 13 | ffvelcdmda 7099 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
15 | 11, 14 | subge0d 11842 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥))) |
16 | 8, 15 | sylan2 591 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥))) |
17 | 7, 16 | mpbird 256 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥))) |
18 | 10 | ffnd 6728 | . . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ) |
19 | 13 | ffnd 6728 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ) |
20 | reex 11237 | . . . . . . . 8 ⊢ ℝ ∈ V | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ∈ V) |
22 | inidm 4221 | . . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ | |
23 | eqidd 2729 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
24 | eqidd 2729 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
25 | 18, 19, 21, 21, 22, 23, 24 | ofval 7702 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ∘f − 𝐹)‘𝑥) = ((𝐺‘𝑥) − (𝐹‘𝑥))) |
26 | 8, 25 | sylan2 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ((𝐺 ∘f − 𝐹)‘𝑥) = ((𝐺‘𝑥) − (𝐹‘𝑥))) |
27 | 17, 26 | breqtrrd 5180 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ ((𝐺 ∘f − 𝐹)‘𝑥)) |
28 | 4, 5, 6, 27 | itg1ge0a 25661 | . . 3 ⊢ (𝜑 → 0 ≤ (∫1‘(𝐺 ∘f − 𝐹))) |
29 | itg1sub 25659 | . . . 4 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐹 ∈ dom ∫1) → (∫1‘(𝐺 ∘f − 𝐹)) = ((∫1‘𝐺) − (∫1‘𝐹))) | |
30 | 1, 2, 29 | syl2anc 582 | . . 3 ⊢ (𝜑 → (∫1‘(𝐺 ∘f − 𝐹)) = ((∫1‘𝐺) − (∫1‘𝐹))) |
31 | 28, 30 | breqtrd 5178 | . 2 ⊢ (𝜑 → 0 ≤ ((∫1‘𝐺) − (∫1‘𝐹))) |
32 | itg1cl 25634 | . . . 4 ⊢ (𝐺 ∈ dom ∫1 → (∫1‘𝐺) ∈ ℝ) | |
33 | 1, 32 | syl 17 | . . 3 ⊢ (𝜑 → (∫1‘𝐺) ∈ ℝ) |
34 | itg1cl 25634 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℝ) | |
35 | 2, 34 | syl 17 | . . 3 ⊢ (𝜑 → (∫1‘𝐹) ∈ ℝ) |
36 | 33, 35 | subge0d 11842 | . 2 ⊢ (𝜑 → (0 ≤ ((∫1‘𝐺) − (∫1‘𝐹)) ↔ (∫1‘𝐹) ≤ (∫1‘𝐺))) |
37 | 31, 36 | mpbid 231 | 1 ⊢ (𝜑 → (∫1‘𝐹) ≤ (∫1‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ∖ cdif 3946 ⊆ wss 3949 class class class wbr 5152 dom cdm 5682 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ∘f cof 7689 ℝcr 11145 0cc0 11146 ≤ cle 11287 − cmin 11482 vol*covol 25411 ∫1citg1 25564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-disj 5118 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xadd 13133 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 df-xmet 21279 df-met 21280 df-ovol 25413 df-vol 25414 df-mbf 25568 df-itg1 25569 |
This theorem is referenced by: itg1le 25663 itg2uba 25693 itg2splitlem 25698 |
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