![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > itg1sub | Structured version Visualization version GIF version |
Description: The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.) |
Ref | Expression |
---|---|
itg1sub | β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β (β«1β(πΉ βf β πΊ)) = ((β«1βπΉ) β (β«1βπΊ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . . 4 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β πΉ β dom β«1) | |
2 | simpr 486 | . . . . 5 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β πΊ β dom β«1) | |
3 | neg1rr 12275 | . . . . . 6 β’ -1 β β | |
4 | 3 | a1i 11 | . . . . 5 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β -1 β β) |
5 | 2, 4 | i1fmulc 25084 | . . . 4 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β ((β Γ {-1}) βf Β· πΊ) β dom β«1) |
6 | 1, 5 | itg1add 25082 | . . 3 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β (β«1β(πΉ βf + ((β Γ {-1}) βf Β· πΊ))) = ((β«1βπΉ) + (β«1β((β Γ {-1}) βf Β· πΊ)))) |
7 | 2, 4 | itg1mulc 25085 | . . . . 5 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β (β«1β((β Γ {-1}) βf Β· πΊ)) = (-1 Β· (β«1βπΊ))) |
8 | itg1cl 25065 | . . . . . . . 8 β’ (πΊ β dom β«1 β (β«1βπΊ) β β) | |
9 | 8 | recnd 11190 | . . . . . . 7 β’ (πΊ β dom β«1 β (β«1βπΊ) β β) |
10 | 2, 9 | syl 17 | . . . . . 6 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β (β«1βπΊ) β β) |
11 | 10 | mulm1d 11614 | . . . . 5 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β (-1 Β· (β«1βπΊ)) = -(β«1βπΊ)) |
12 | 7, 11 | eqtrd 2777 | . . . 4 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β (β«1β((β Γ {-1}) βf Β· πΊ)) = -(β«1βπΊ)) |
13 | 12 | oveq2d 7378 | . . 3 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β ((β«1βπΉ) + (β«1β((β Γ {-1}) βf Β· πΊ))) = ((β«1βπΉ) + -(β«1βπΊ))) |
14 | 6, 13 | eqtrd 2777 | . 2 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β (β«1β(πΉ βf + ((β Γ {-1}) βf Β· πΊ))) = ((β«1βπΉ) + -(β«1βπΊ))) |
15 | reex 11149 | . . . 4 β’ β β V | |
16 | i1ff 25056 | . . . . 5 β’ (πΉ β dom β«1 β πΉ:ββΆβ) | |
17 | ax-resscn 11115 | . . . . 5 β’ β β β | |
18 | fss 6690 | . . . . 5 β’ ((πΉ:ββΆβ β§ β β β) β πΉ:ββΆβ) | |
19 | 16, 17, 18 | sylancl 587 | . . . 4 β’ (πΉ β dom β«1 β πΉ:ββΆβ) |
20 | i1ff 25056 | . . . . 5 β’ (πΊ β dom β«1 β πΊ:ββΆβ) | |
21 | fss 6690 | . . . . 5 β’ ((πΊ:ββΆβ β§ β β β) β πΊ:ββΆβ) | |
22 | 20, 17, 21 | sylancl 587 | . . . 4 β’ (πΊ β dom β«1 β πΊ:ββΆβ) |
23 | ofnegsub 12158 | . . . 4 β’ ((β β V β§ πΉ:ββΆβ β§ πΊ:ββΆβ) β (πΉ βf + ((β Γ {-1}) βf Β· πΊ)) = (πΉ βf β πΊ)) | |
24 | 15, 19, 22, 23 | mp3an3an 1468 | . . 3 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β (πΉ βf + ((β Γ {-1}) βf Β· πΊ)) = (πΉ βf β πΊ)) |
25 | 24 | fveq2d 6851 | . 2 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β (β«1β(πΉ βf + ((β Γ {-1}) βf Β· πΊ))) = (β«1β(πΉ βf β πΊ))) |
26 | itg1cl 25065 | . . . 4 β’ (πΉ β dom β«1 β (β«1βπΉ) β β) | |
27 | 26 | recnd 11190 | . . 3 β’ (πΉ β dom β«1 β (β«1βπΉ) β β) |
28 | negsub 11456 | . . 3 β’ (((β«1βπΉ) β β β§ (β«1βπΊ) β β) β ((β«1βπΉ) + -(β«1βπΊ)) = ((β«1βπΉ) β (β«1βπΊ))) | |
29 | 27, 9, 28 | syl2an 597 | . 2 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β ((β«1βπΉ) + -(β«1βπΊ)) = ((β«1βπΉ) β (β«1βπΊ))) |
30 | 14, 25, 29 | 3eqtr3d 2785 | 1 β’ ((πΉ β dom β«1 β§ πΊ β dom β«1) β (β«1β(πΉ βf β πΊ)) = ((β«1βπΉ) β (β«1βπΊ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3448 β wss 3915 {csn 4591 Γ cxp 5636 dom cdm 5638 βΆwf 6497 βcfv 6501 (class class class)co 7362 βf cof 7620 βcc 11056 βcr 11057 1c1 11059 + caddc 11061 Β· cmul 11063 β cmin 11392 -cneg 11393 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: itg1lea 25093 itgitg1 25189 itg2addnclem 36158 |
Copyright terms: Public domain | W3C validator |