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Mirrors > Home > MPE Home > Th. List > itg1lea | Structured version Visualization version GIF version |
Description: Approximate version of itg1le 25094. If πΉ β€ πΊ for almost all π₯, then β«1πΉ β€ β«1πΊ. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.) |
Ref | Expression |
---|---|
itg10a.1 | β’ (π β πΉ β dom β«1) |
itg10a.2 | β’ (π β π΄ β β) |
itg10a.3 | β’ (π β (vol*βπ΄) = 0) |
itg1lea.4 | β’ (π β πΊ β dom β«1) |
itg1lea.5 | β’ ((π β§ π₯ β (β β π΄)) β (πΉβπ₯) β€ (πΊβπ₯)) |
Ref | Expression |
---|---|
itg1lea | β’ (π β (β«1βπΉ) β€ (β«1βπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg1lea.4 | . . . . 5 β’ (π β πΊ β dom β«1) | |
2 | itg10a.1 | . . . . 5 β’ (π β πΉ β dom β«1) | |
3 | i1fsub 25089 | . . . . 5 β’ ((πΊ β dom β«1 β§ πΉ β dom β«1) β (πΊ βf β πΉ) β dom β«1) | |
4 | 1, 2, 3 | syl2anc 585 | . . . 4 β’ (π β (πΊ βf β πΉ) β dom β«1) |
5 | itg10a.2 | . . . 4 β’ (π β π΄ β β) | |
6 | itg10a.3 | . . . 4 β’ (π β (vol*βπ΄) = 0) | |
7 | itg1lea.5 | . . . . . 6 β’ ((π β§ π₯ β (β β π΄)) β (πΉβπ₯) β€ (πΊβπ₯)) | |
8 | eldifi 4091 | . . . . . . 7 β’ (π₯ β (β β π΄) β π₯ β β) | |
9 | i1ff 25056 | . . . . . . . . . 10 β’ (πΊ β dom β«1 β πΊ:ββΆβ) | |
10 | 1, 9 | syl 17 | . . . . . . . . 9 β’ (π β πΊ:ββΆβ) |
11 | 10 | ffvelcdmda 7040 | . . . . . . . 8 β’ ((π β§ π₯ β β) β (πΊβπ₯) β β) |
12 | i1ff 25056 | . . . . . . . . . 10 β’ (πΉ β dom β«1 β πΉ:ββΆβ) | |
13 | 2, 12 | syl 17 | . . . . . . . . 9 β’ (π β πΉ:ββΆβ) |
14 | 13 | ffvelcdmda 7040 | . . . . . . . 8 β’ ((π β§ π₯ β β) β (πΉβπ₯) β β) |
15 | 11, 14 | subge0d 11752 | . . . . . . 7 β’ ((π β§ π₯ β β) β (0 β€ ((πΊβπ₯) β (πΉβπ₯)) β (πΉβπ₯) β€ (πΊβπ₯))) |
16 | 8, 15 | sylan2 594 | . . . . . 6 β’ ((π β§ π₯ β (β β π΄)) β (0 β€ ((πΊβπ₯) β (πΉβπ₯)) β (πΉβπ₯) β€ (πΊβπ₯))) |
17 | 7, 16 | mpbird 257 | . . . . 5 β’ ((π β§ π₯ β (β β π΄)) β 0 β€ ((πΊβπ₯) β (πΉβπ₯))) |
18 | 10 | ffnd 6674 | . . . . . . 7 β’ (π β πΊ Fn β) |
19 | 13 | ffnd 6674 | . . . . . . 7 β’ (π β πΉ Fn β) |
20 | reex 11149 | . . . . . . . 8 β’ β β V | |
21 | 20 | a1i 11 | . . . . . . 7 β’ (π β β β V) |
22 | inidm 4183 | . . . . . . 7 β’ (β β© β) = β | |
23 | eqidd 2738 | . . . . . . 7 β’ ((π β§ π₯ β β) β (πΊβπ₯) = (πΊβπ₯)) | |
24 | eqidd 2738 | . . . . . . 7 β’ ((π β§ π₯ β β) β (πΉβπ₯) = (πΉβπ₯)) | |
25 | 18, 19, 21, 21, 22, 23, 24 | ofval 7633 | . . . . . 6 β’ ((π β§ π₯ β β) β ((πΊ βf β πΉ)βπ₯) = ((πΊβπ₯) β (πΉβπ₯))) |
26 | 8, 25 | sylan2 594 | . . . . 5 β’ ((π β§ π₯ β (β β π΄)) β ((πΊ βf β πΉ)βπ₯) = ((πΊβπ₯) β (πΉβπ₯))) |
27 | 17, 26 | breqtrrd 5138 | . . . 4 β’ ((π β§ π₯ β (β β π΄)) β 0 β€ ((πΊ βf β πΉ)βπ₯)) |
28 | 4, 5, 6, 27 | itg1ge0a 25092 | . . 3 β’ (π β 0 β€ (β«1β(πΊ βf β πΉ))) |
29 | itg1sub 25090 | . . . 4 β’ ((πΊ β dom β«1 β§ πΉ β dom β«1) β (β«1β(πΊ βf β πΉ)) = ((β«1βπΊ) β (β«1βπΉ))) | |
30 | 1, 2, 29 | syl2anc 585 | . . 3 β’ (π β (β«1β(πΊ βf β πΉ)) = ((β«1βπΊ) β (β«1βπΉ))) |
31 | 28, 30 | breqtrd 5136 | . 2 β’ (π β 0 β€ ((β«1βπΊ) β (β«1βπΉ))) |
32 | itg1cl 25065 | . . . 4 β’ (πΊ β dom β«1 β (β«1βπΊ) β β) | |
33 | 1, 32 | syl 17 | . . 3 β’ (π β (β«1βπΊ) β β) |
34 | itg1cl 25065 | . . . 4 β’ (πΉ β dom β«1 β (β«1βπΉ) β β) | |
35 | 2, 34 | syl 17 | . . 3 β’ (π β (β«1βπΉ) β β) |
36 | 33, 35 | subge0d 11752 | . 2 β’ (π β (0 β€ ((β«1βπΊ) β (β«1βπΉ)) β (β«1βπΉ) β€ (β«1βπΊ))) |
37 | 31, 36 | mpbid 231 | 1 β’ (π β (β«1βπΉ) β€ (β«1βπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3448 β cdif 3912 β wss 3915 class class class wbr 5110 dom cdm 5638 βΆwf 6497 βcfv 6501 (class class class)co 7362 βf cof 7620 βcr 11057 0cc0 11058 β€ cle 11197 β cmin 11392 vol*covol 24842 β«1citg1 24995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-disj 5076 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-xadd 13041 df-ioo 13275 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 df-xmet 20805 df-met 20806 df-ovol 24844 df-vol 24845 df-mbf 24999 df-itg1 25000 |
This theorem is referenced by: itg1le 25094 itg2uba 25124 itg2splitlem 25129 |
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