![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | β’ ((πΉ β dom β«1 β§ πΊ β dom β«1 β§ πΉ βr β€ πΊ) β (β«1βπΉ) β€ (β«1βπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . 2 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1 β§ πΉ βr β€ πΊ) β πΉ β dom β«1) | |
2 | 0ss 4398 | . . 3 β’ β β β | |
3 | 2 | a1i 11 | . 2 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1 β§ πΉ βr β€ πΊ) β β β β) |
4 | ovol0 25440 | . . 3 β’ (vol*ββ ) = 0 | |
5 | 4 | a1i 11 | . 2 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1 β§ πΉ βr β€ πΊ) β (vol*ββ ) = 0) |
6 | simp2 1134 | . 2 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1 β§ πΉ βr β€ πΊ) β πΊ β dom β«1) | |
7 | simpl 481 | . . . . . . 7 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β πΉ β dom β«1) | |
8 | i1ff 25623 | . . . . . . 7 β’ (πΉ β dom β«1 β πΉ:ββΆβ) | |
9 | ffn 6725 | . . . . . . 7 β’ (πΉ:ββΆβ β πΉ Fn β) | |
10 | 7, 8, 9 | 3syl 18 | . . . . . 6 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β πΉ Fn β) |
11 | simpr 483 | . . . . . . 7 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β πΊ β dom β«1) | |
12 | i1ff 25623 | . . . . . . 7 β’ (πΊ β dom β«1 β πΊ:ββΆβ) | |
13 | ffn 6725 | . . . . . . 7 β’ (πΊ:ββΆβ β πΊ Fn β) | |
14 | 11, 12, 13 | 3syl 18 | . . . . . 6 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β πΊ Fn β) |
15 | reex 11235 | . . . . . . 7 β’ β β V | |
16 | 15 | a1i 11 | . . . . . 6 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β β β V) |
17 | inidm 4219 | . . . . . 6 β’ (β β© β) = β | |
18 | eqidd 2728 | . . . . . 6 β’ (((πΉ β dom β«1 β§ πΊ β dom β«1) β§ π₯ β β) β (πΉβπ₯) = (πΉβπ₯)) | |
19 | eqidd 2728 | . . . . . 6 β’ (((πΉ β dom β«1 β§ πΊ β dom β«1) β§ π₯ β β) β (πΊβπ₯) = (πΊβπ₯)) | |
20 | 10, 14, 16, 16, 17, 18, 19 | ofrval 7701 | . . . . 5 β’ (((πΉ β dom β«1 β§ πΊ β dom β«1) β§ πΉ βr β€ πΊ β§ π₯ β β) β (πΉβπ₯) β€ (πΊβπ₯)) |
21 | 20 | 3exp 1116 | . . . 4 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β (πΉ βr β€ πΊ β (π₯ β β β (πΉβπ₯) β€ (πΊβπ₯)))) |
22 | 21 | 3impia 1114 | . . 3 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1 β§ πΉ βr β€ πΊ) β (π₯ β β β (πΉβπ₯) β€ (πΊβπ₯))) |
23 | eldifi 4125 | . . 3 β’ (π₯ β (β β β ) β π₯ β β) | |
24 | 22, 23 | impel 504 | . 2 β’ (((πΉ β dom β«1 β§ πΊ β dom β«1 β§ πΉ βr β€ πΊ) β§ π₯ β (β β β )) β (πΉβπ₯) β€ (πΊβπ₯)) |
25 | 1, 3, 5, 6, 24 | itg1lea 25660 | 1 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1 β§ πΉ βr β€ πΊ) β (β«1βπΉ) β€ (β«1βπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3471 β cdif 3944 β wss 3947 β c0 4324 class class class wbr 5150 dom cdm 5680 Fn wfn 6546 βΆwf 6547 βcfv 6551 βr cofr 7688 βcr 11143 0cc0 11144 β€ cle 11285 vol*covol 25409 β«1citg1 25562 |
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 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 ax-addf 11223 |
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 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-disj 5116 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-ofr 7690 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-er 8729 df-map 8851 df-pm 8852 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-inf 9472 df-oi 9539 df-dju 9930 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-q 12969 df-rp 13013 df-xadd 13131 df-ioo 13366 df-ico 13368 df-icc 13369 df-fz 13523 df-fzo 13666 df-fl 13795 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 df-sum 15671 df-xmet 21277 df-met 21278 df-ovol 25411 df-vol 25412 df-mbf 25566 df-itg1 25567 |
This theorem is referenced by: itg2itg1 25684 itg2i1fseq2 25704 itg2addnclem 37149 ftc1anclem5 37175 |
Copyright terms: Public domain | W3C validator |