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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 5942 | . . . . 5 β’ (π = πΉ β ran π = ran πΉ) | |
2 | 1 | difeq1d 4121 | . . . 4 β’ (π = πΉ β (ran π β {0}) = (ran πΉ β {0})) |
3 | cnveq 5880 | . . . . . . . 8 β’ (π = πΉ β β‘π = β‘πΉ) | |
4 | 3 | imaeq1d 6067 | . . . . . . 7 β’ (π = πΉ β (β‘π β {π₯}) = (β‘πΉ β {π₯})) |
5 | 4 | fveq2d 6906 | . . . . . 6 β’ (π = πΉ β (volβ(β‘π β {π₯})) = (volβ(β‘πΉ β {π₯}))) |
6 | 5 | oveq2d 7442 | . . . . 5 β’ (π = πΉ β (π₯ Β· (volβ(β‘π β {π₯}))) = (π₯ Β· (volβ(β‘πΉ β {π₯})))) |
7 | 6 | adantr 479 | . . . 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 15692 | . . 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 25569 | . . 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 15674 | . . 3 β’ Ξ£π₯ β (ran πΉ β {0})(π₯ Β· (volβ(β‘πΉ β {π₯}))) β V | |
11 | 8, 9, 10 | fvmpt 7010 | . 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 15674 | . . 3 β’ Ξ£π₯ β (ran π β {0})(π₯ Β· (volβ(β‘π β {π₯}))) β V | |
13 | 12, 9 | dmmpti 6704 | . 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 2847 | 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 3430 β cdif 3946 {csn 4632 β‘ccnv 5681 dom cdm 5682 ran crn 5683 β cima 5685 βΆwf 6549 βcfv 6553 (class class class)co 7426 Fincfn 8970 βcr 11145 0cc0 11146 Β· cmul 11151 Ξ£csu 15672 volcvol 25412 MblFncmbf 25563 β«1citg1 25564 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-seq 14007 df-sum 15673 df-itg1 25569 |
This theorem is referenced by: itg1val2 25633 itg1cl 25634 itg1ge0 25635 itg10 25637 itg11 25640 itg1addlem5 25650 itg1mulc 25654 itg10a 25660 itg1ge0a 25661 itg1climres 25664 |
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