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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 5928 | . . . . 5 β’ (π = πΉ β ran π = ran πΉ) | |
2 | 1 | difeq1d 4116 | . . . 4 β’ (π = πΉ β (ran π β {0}) = (ran πΉ β {0})) |
3 | cnveq 5866 | . . . . . . . 8 β’ (π = πΉ β β‘π = β‘πΉ) | |
4 | 3 | imaeq1d 6051 | . . . . . . 7 β’ (π = πΉ β (β‘π β {π₯}) = (β‘πΉ β {π₯})) |
5 | 4 | fveq2d 6888 | . . . . . 6 β’ (π = πΉ β (volβ(β‘π β {π₯})) = (volβ(β‘πΉ β {π₯}))) |
6 | 5 | oveq2d 7420 | . . . . 5 β’ (π = πΉ β (π₯ Β· (volβ(β‘π β {π₯}))) = (π₯ Β· (volβ(β‘πΉ β {π₯})))) |
7 | 6 | adantr 480 | . . . 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 15655 | . . 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 25499 | . . 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 15637 | . . 3 β’ Ξ£π₯ β (ran πΉ β {0})(π₯ Β· (volβ(β‘πΉ β {π₯}))) β V | |
11 | 8, 9, 10 | fvmpt 6991 | . 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 15637 | . . 3 β’ Ξ£π₯ β (ran π β {0})(π₯ Β· (volβ(β‘π β {π₯}))) β V | |
13 | 12, 9 | dmmpti 6687 | . 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 2845 | 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 1084 = wceq 1533 β wcel 2098 {crab 3426 β cdif 3940 {csn 4623 β‘ccnv 5668 dom cdm 5669 ran crn 5670 β cima 5672 βΆwf 6532 βcfv 6536 (class class class)co 7404 Fincfn 8938 βcr 11108 0cc0 11109 Β· cmul 11114 Ξ£csu 15635 volcvol 25342 MblFncmbf 25493 β«1citg1 25494 |
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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 |
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-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-iun 4992 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-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-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-seq 13970 df-sum 15636 df-itg1 25499 |
This theorem is referenced by: itg1val2 25563 itg1cl 25564 itg1ge0 25565 itg10 25567 itg11 25570 itg1addlem5 25580 itg1mulc 25584 itg10a 25590 itg1ge0a 25591 itg1climres 25594 |
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