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Mirrors > Home > MPE Home > Th. List > i1f1 | Structured version Visualization version GIF version |
Description: Base case simple functions are indicator functions of measurable sets. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
i1f1.1 | ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0)) |
Ref | Expression |
---|---|
i1f1 | ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹 ∈ dom ∫1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | i1f1.1 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0)) | |
2 | 1 | i1f1lem 24217 | . . . . 5 ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴)) |
3 | 2 | simpli 484 | . . . 4 ⊢ 𝐹:ℝ⟶{0, 1} |
4 | 0re 10631 | . . . . 5 ⊢ 0 ∈ ℝ | |
5 | 1re 10629 | . . . . 5 ⊢ 1 ∈ ℝ | |
6 | prssi 4746 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ) | |
7 | 4, 5, 6 | mp2an 688 | . . . 4 ⊢ {0, 1} ⊆ ℝ |
8 | fss 6520 | . . . 4 ⊢ ((𝐹:ℝ⟶{0, 1} ∧ {0, 1} ⊆ ℝ) → 𝐹:ℝ⟶ℝ) | |
9 | 3, 7, 8 | mp2an 688 | . . 3 ⊢ 𝐹:ℝ⟶ℝ |
10 | 9 | a1i 11 | . 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹:ℝ⟶ℝ) |
11 | prfi 8781 | . . 3 ⊢ {0, 1} ∈ Fin | |
12 | 1ex 10625 | . . . . . . . 8 ⊢ 1 ∈ V | |
13 | 12 | prid2 4691 | . . . . . . 7 ⊢ 1 ∈ {0, 1} |
14 | c0ex 10623 | . . . . . . . 8 ⊢ 0 ∈ V | |
15 | 14 | prid1 4690 | . . . . . . 7 ⊢ 0 ∈ {0, 1} |
16 | 13, 15 | ifcli 4509 | . . . . . 6 ⊢ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} |
17 | 16 | a1i 11 | . . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1}) |
18 | 17, 1 | fmptd 6870 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹:ℝ⟶{0, 1}) |
19 | frn 6513 | . . . 4 ⊢ (𝐹:ℝ⟶{0, 1} → ran 𝐹 ⊆ {0, 1}) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ran 𝐹 ⊆ {0, 1}) |
21 | ssfi 8726 | . . 3 ⊢ (({0, 1} ∈ Fin ∧ ran 𝐹 ⊆ {0, 1}) → ran 𝐹 ∈ Fin) | |
22 | 11, 20, 21 | sylancr 587 | . 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ran 𝐹 ∈ Fin) |
23 | 3, 19 | ax-mp 5 | . . . . . . . . . . 11 ⊢ ran 𝐹 ⊆ {0, 1} |
24 | df-pr 4560 | . . . . . . . . . . . 12 ⊢ {0, 1} = ({0} ∪ {1}) | |
25 | 24 | equncomi 4128 | . . . . . . . . . . 11 ⊢ {0, 1} = ({1} ∪ {0}) |
26 | 23, 25 | sseqtri 4000 | . . . . . . . . . 10 ⊢ ran 𝐹 ⊆ ({1} ∪ {0}) |
27 | ssdif 4113 | . . . . . . . . . 10 ⊢ (ran 𝐹 ⊆ ({1} ∪ {0}) → (ran 𝐹 ∖ {0}) ⊆ (({1} ∪ {0}) ∖ {0})) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ (ran 𝐹 ∖ {0}) ⊆ (({1} ∪ {0}) ∖ {0}) |
29 | difun2 4425 | . . . . . . . . . 10 ⊢ (({1} ∪ {0}) ∖ {0}) = ({1} ∖ {0}) | |
30 | difss 4105 | . . . . . . . . . 10 ⊢ ({1} ∖ {0}) ⊆ {1} | |
31 | 29, 30 | eqsstri 3998 | . . . . . . . . 9 ⊢ (({1} ∪ {0}) ∖ {0}) ⊆ {1} |
32 | 28, 31 | sstri 3973 | . . . . . . . 8 ⊢ (ran 𝐹 ∖ {0}) ⊆ {1} |
33 | 32 | sseli 3960 | . . . . . . 7 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 ∈ {1}) |
34 | elsni 4574 | . . . . . . 7 ⊢ (𝑦 ∈ {1} → 𝑦 = 1) | |
35 | 33, 34 | syl 17 | . . . . . 6 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 = 1) |
36 | 35 | sneqd 4569 | . . . . 5 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → {𝑦} = {1}) |
37 | 36 | imaeq2d 5922 | . . . 4 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {1})) |
38 | 2 | simpri 486 | . . . . 5 ⊢ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴) |
39 | 38 | adantr 481 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (◡𝐹 “ {1}) = 𝐴) |
40 | 37, 39 | sylan9eqr 2875 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) = 𝐴) |
41 | simpll 763 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐴 ∈ dom vol) | |
42 | 40, 41 | eqeltrd 2910 | . 2 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) ∈ dom vol) |
43 | 40 | fveq2d 6667 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) = (vol‘𝐴)) |
44 | simplr 765 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘𝐴) ∈ ℝ) | |
45 | 43, 44 | eqeltrd 2910 | . 2 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ) |
46 | 10, 22, 42, 45 | i1fd 24209 | 1 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹 ∈ dom ∫1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ∪ cun 3931 ⊆ wss 3933 ifcif 4463 {csn 4557 {cpr 4559 ↦ cmpt 5137 ◡ccnv 5547 dom cdm 5548 ran crn 5549 “ cima 5551 ⟶wf 6344 ‘cfv 6348 Fincfn 8497 ℝcr 10524 0cc0 10525 1c1 10526 volcvol 23991 ∫1citg1 24143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 df-xmet 20466 df-met 20467 df-ovol 23992 df-vol 23993 df-mbf 24147 df-itg1 24148 |
This theorem is referenced by: itg11 24219 itg2const 24268 itg2addnclem 34824 |
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