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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 | ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rneq 5800 | . . . . 5 ⊢ (𝑓 = 𝐹 → ran 𝑓 = ran 𝐹) | |
2 | 1 | difeq1d 4097 | . . . 4 ⊢ (𝑓 = 𝐹 → (ran 𝑓 ∖ {0}) = (ran 𝐹 ∖ {0})) |
3 | cnveq 5738 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) | |
4 | 3 | imaeq1d 5922 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {𝑥}) = (◡𝐹 “ {𝑥})) |
5 | 4 | fveq2d 6668 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (vol‘(◡𝑓 “ {𝑥})) = (vol‘(◡𝐹 “ {𝑥}))) |
6 | 5 | oveq2d 7166 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑥 · (vol‘(◡𝑓 “ {𝑥}))) = (𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 ∈ (ran 𝑓 ∖ {0})) → (𝑥 · (vol‘(◡𝑓 “ {𝑥}))) = (𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
8 | 2, 7 | sumeq12dv 15057 | . . 3 ⊢ (𝑓 = 𝐹 → Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥}))) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
9 | df-itg1 24215 | . . 3 ⊢ ∫1 = (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} ↦ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥})))) | |
10 | sumex 15038 | . . 3 ⊢ Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥}))) ∈ V | |
11 | 8, 9, 10 | fvmpt 6762 | . 2 ⊢ (𝐹 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} → (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
12 | sumex 15038 | . . 3 ⊢ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥}))) ∈ V | |
13 | 12, 9 | dmmpti 6486 | . 2 ⊢ dom ∫1 = {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} |
14 | 11, 13 | eleq2s 2931 | 1 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {crab 3142 ∖ cdif 3932 {csn 4560 ◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 ℝcr 10530 0cc0 10531 · cmul 10536 Σcsu 15036 volcvol 24058 MblFncmbf 24209 ∫1citg1 24210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-seq 13364 df-sum 15037 df-itg1 24215 |
This theorem is referenced by: itg1val2 24279 itg1cl 24280 itg1ge0 24281 itg10 24283 itg11 24286 itg1addlem5 24295 itg1mulc 24299 itg10a 24305 itg1ge0a 24306 itg1climres 24309 |
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