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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 | 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 4391 | . . 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 25373 | . . 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 482 | . . . . . . 7 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β πΉ β dom β«1) | |
8 | i1ff 25556 | . . . . . . 7 β’ (πΉ β dom β«1 β πΉ:ββΆβ) | |
9 | ffn 6710 | . . . . . . 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 484 | . . . . . . 7 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β πΊ β dom β«1) | |
12 | i1ff 25556 | . . . . . . 7 β’ (πΊ β dom β«1 β πΊ:ββΆβ) | |
13 | ffn 6710 | . . . . . . 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 11200 | . . . . . . 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 4213 | . . . . . 6 β’ (β β© β) = β | |
18 | eqidd 2727 | . . . . . 6 β’ (((πΉ β dom β«1 β§ πΊ β dom β«1) β§ π₯ β β) β (πΉβπ₯) = (πΉβπ₯)) | |
19 | eqidd 2727 | . . . . . 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 7678 | . . . . 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 4121 | . . 3 β’ (π₯ β (β β β ) β π₯ β β) | |
24 | 22, 23 | impel 505 | . 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 25593 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3468 β cdif 3940 β wss 3943 β c0 4317 class class class wbr 5141 dom cdm 5669 Fn wfn 6531 βΆwf 6532 βcfv 6536 βr cofr 7665 βcr 11108 0cc0 11109 β€ cle 11250 vol*covol 25342 β«1citg1 25495 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xadd 13096 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14031 df-hash 14294 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-sum 15637 df-xmet 21229 df-met 21230 df-ovol 25344 df-vol 25345 df-mbf 25499 df-itg1 25500 |
This theorem is referenced by: itg2itg1 25617 itg2i1fseq2 25637 itg2addnclem 37050 ftc1anclem5 37076 |
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