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| Mirrors > Home > MPE Home > Th. List > itg1lea | Structured version Visualization version GIF version | ||
| Description: Approximate version of itg1le 25748. 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 25743 | . . . . 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 4131 | . . . . . . 7 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ) | |
| 9 | i1ff 25711 | . . . . . . . . . 10 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ) | |
| 10 | 1, 9 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ) |
| 11 | 10 | ffvelcdmda 7104 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℝ) |
| 12 | i1ff 25711 | . . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
| 13 | 2, 12 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
| 14 | 13 | ffvelcdmda 7104 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
| 15 | 11, 14 | subge0d 11853 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥))) |
| 16 | 8, 15 | sylan2 593 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥)) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥))) |
| 17 | 7, 16 | mpbird 257 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ ((𝐺‘𝑥) − (𝐹‘𝑥))) |
| 18 | 10 | ffnd 6737 | . . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ) |
| 19 | 13 | ffnd 6737 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ) |
| 20 | reex 11246 | . . . . . . . 8 ⊢ ℝ ∈ V | |
| 21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ∈ V) |
| 22 | inidm 4227 | . . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ | |
| 23 | eqidd 2738 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
| 24 | eqidd 2738 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
| 25 | 18, 19, 21, 21, 22, 23, 24 | ofval 7708 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ∘f − 𝐹)‘𝑥) = ((𝐺‘𝑥) − (𝐹‘𝑥))) |
| 26 | 8, 25 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ((𝐺 ∘f − 𝐹)‘𝑥) = ((𝐺‘𝑥) − (𝐹‘𝑥))) |
| 27 | 17, 26 | breqtrrd 5171 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ ((𝐺 ∘f − 𝐹)‘𝑥)) |
| 28 | 4, 5, 6, 27 | itg1ge0a 25746 | . . 3 ⊢ (𝜑 → 0 ≤ (∫1‘(𝐺 ∘f − 𝐹))) |
| 29 | itg1sub 25744 | . . . 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 5169 | . 2 ⊢ (𝜑 → 0 ≤ ((∫1‘𝐺) − (∫1‘𝐹))) |
| 32 | itg1cl 25720 | . . . 4 ⊢ (𝐺 ∈ dom ∫1 → (∫1‘𝐺) ∈ ℝ) | |
| 33 | 1, 32 | syl 17 | . . 3 ⊢ (𝜑 → (∫1‘𝐺) ∈ ℝ) |
| 34 | itg1cl 25720 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℝ) | |
| 35 | 2, 34 | syl 17 | . . 3 ⊢ (𝜑 → (∫1‘𝐹) ∈ ℝ) |
| 36 | 33, 35 | subge0d 11853 | . 2 ⊢ (𝜑 → (0 ≤ ((∫1‘𝐺) − (∫1‘𝐹)) ↔ (∫1‘𝐹) ≤ (∫1‘𝐺))) |
| 37 | 31, 36 | mpbid 232 | 1 ⊢ (𝜑 → (∫1‘𝐹) ≤ (∫1‘𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 class class class wbr 5143 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 ℝcr 11154 0cc0 11155 ≤ cle 11296 − cmin 11492 vol*covol 25497 ∫1citg1 25650 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xadd 13155 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-xmet 21357 df-met 21358 df-ovol 25499 df-vol 25500 df-mbf 25654 df-itg1 25655 |
| This theorem is referenced by: itg1le 25748 itg2uba 25778 itg2splitlem 25783 |
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