![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > itg1val | Structured version Visualization version GIF version |
Description: The value of the integral on simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
itg1val | β’ (πΉ β dom β«1 β (β«1βπΉ) = Ξ£π₯ β (ran πΉ β {0})(π₯ Β· (volβ(β‘πΉ β {π₯})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rneq 5896 | . . . . 5 β’ (π = πΉ β ran π = ran πΉ) | |
2 | 1 | difeq1d 4086 | . . . 4 β’ (π = πΉ β (ran π β {0}) = (ran πΉ β {0})) |
3 | cnveq 5834 | . . . . . . . 8 β’ (π = πΉ β β‘π = β‘πΉ) | |
4 | 3 | imaeq1d 6017 | . . . . . . 7 β’ (π = πΉ β (β‘π β {π₯}) = (β‘πΉ β {π₯})) |
5 | 4 | fveq2d 6851 | . . . . . 6 β’ (π = πΉ β (volβ(β‘π β {π₯})) = (volβ(β‘πΉ β {π₯}))) |
6 | 5 | oveq2d 7378 | . . . . 5 β’ (π = πΉ β (π₯ Β· (volβ(β‘π β {π₯}))) = (π₯ Β· (volβ(β‘πΉ β {π₯})))) |
7 | 6 | adantr 482 | . . . 4 β’ ((π = πΉ β§ π₯ β (ran π β {0})) β (π₯ Β· (volβ(β‘π β {π₯}))) = (π₯ Β· (volβ(β‘πΉ β {π₯})))) |
8 | 2, 7 | sumeq12dv 15598 | . . 3 β’ (π = πΉ β Ξ£π₯ β (ran π β {0})(π₯ Β· (volβ(β‘π β {π₯}))) = Ξ£π₯ β (ran πΉ β {0})(π₯ Β· (volβ(β‘πΉ β {π₯})))) |
9 | df-itg1 25000 | . . 3 β’ β«1 = (π β {π β MblFn β£ (π:ββΆβ β§ ran π β Fin β§ (volβ(β‘π β (β β {0}))) β β)} β¦ Ξ£π₯ β (ran π β {0})(π₯ Β· (volβ(β‘π β {π₯})))) | |
10 | sumex 15579 | . . 3 β’ Ξ£π₯ β (ran πΉ β {0})(π₯ Β· (volβ(β‘πΉ β {π₯}))) β V | |
11 | 8, 9, 10 | fvmpt 6953 | . 2 β’ (πΉ β {π β MblFn β£ (π:ββΆβ β§ ran π β Fin β§ (volβ(β‘π β (β β {0}))) β β)} β (β«1βπΉ) = Ξ£π₯ β (ran πΉ β {0})(π₯ Β· (volβ(β‘πΉ β {π₯})))) |
12 | sumex 15579 | . . 3 β’ Ξ£π₯ β (ran π β {0})(π₯ Β· (volβ(β‘π β {π₯}))) β V | |
13 | 12, 9 | dmmpti 6650 | . 2 β’ dom β«1 = {π β MblFn β£ (π:ββΆβ β§ ran π β Fin β§ (volβ(β‘π β (β β {0}))) β β)} |
14 | 11, 13 | eleq2s 2856 | 1 β’ (πΉ β dom β«1 β (β«1βπΉ) = Ξ£π₯ β (ran πΉ β {0})(π₯ Β· (volβ(β‘πΉ β {π₯})))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 {crab 3410 β cdif 3912 {csn 4591 β‘ccnv 5637 dom cdm 5638 ran crn 5639 β cima 5641 βΆwf 6497 βcfv 6501 (class class class)co 7362 Fincfn 8890 βcr 11057 0cc0 11058 Β· cmul 11063 Ξ£csu 15577 volcvol 24843 MblFncmbf 24994 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-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 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 |
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-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-iun 4961 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-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-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-seq 13914 df-sum 15578 df-itg1 25000 |
This theorem is referenced by: itg1val2 25064 itg1cl 25065 itg1ge0 25066 itg10 25068 itg11 25071 itg1addlem5 25081 itg1mulc 25085 itg10a 25091 itg1ge0a 25092 itg1climres 25095 |
Copyright terms: Public domain | W3C validator |