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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 24881 | . . . . 5 ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴)) |
3 | 2 | simpli 483 | . . . 4 ⊢ 𝐹:ℝ⟶{0, 1} |
4 | 0re 11005 | . . . . 5 ⊢ 0 ∈ ℝ | |
5 | 1re 11003 | . . . . 5 ⊢ 1 ∈ ℝ | |
6 | prssi 4757 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ) | |
7 | 4, 5, 6 | mp2an 688 | . . . 4 ⊢ {0, 1} ⊆ ℝ |
8 | fss 6635 | . . . 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 9117 | . . 3 ⊢ {0, 1} ∈ Fin | |
12 | 1ex 10999 | . . . . . . . 8 ⊢ 1 ∈ V | |
13 | 12 | prid2 4702 | . . . . . . 7 ⊢ 1 ∈ {0, 1} |
14 | c0ex 10997 | . . . . . . . 8 ⊢ 0 ∈ V | |
15 | 14 | prid1 4701 | . . . . . . 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 7008 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹:ℝ⟶{0, 1}) |
19 | frn 6625 | . . . 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 8981 | . . 3 ⊢ (({0, 1} ∈ Fin ∧ ran 𝐹 ⊆ {0, 1}) → ran 𝐹 ∈ Fin) | |
22 | 11, 20, 21 | sylancr 586 | . 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ran 𝐹 ∈ Fin) |
23 | 3, 19 | ax-mp 5 | . . . . . . . . . . 11 ⊢ ran 𝐹 ⊆ {0, 1} |
24 | df-pr 4567 | . . . . . . . . . . . 12 ⊢ {0, 1} = ({0} ∪ {1}) | |
25 | 24 | equncomi 4092 | . . . . . . . . . . 11 ⊢ {0, 1} = ({1} ∪ {0}) |
26 | 23, 25 | sseqtri 3959 | . . . . . . . . . 10 ⊢ ran 𝐹 ⊆ ({1} ∪ {0}) |
27 | ssdif 4077 | . . . . . . . . . 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 4417 | . . . . . . . . . 10 ⊢ (({1} ∪ {0}) ∖ {0}) = ({1} ∖ {0}) | |
30 | difss 4069 | . . . . . . . . . 10 ⊢ ({1} ∖ {0}) ⊆ {1} | |
31 | 29, 30 | eqsstri 3957 | . . . . . . . . 9 ⊢ (({1} ∪ {0}) ∖ {0}) ⊆ {1} |
32 | 28, 31 | sstri 3932 | . . . . . . . 8 ⊢ (ran 𝐹 ∖ {0}) ⊆ {1} |
33 | 32 | sseli 3919 | . . . . . . 7 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 ∈ {1}) |
34 | elsni 4581 | . . . . . . 7 ⊢ (𝑦 ∈ {1} → 𝑦 = 1) | |
35 | 33, 34 | syl 17 | . . . . . 6 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 = 1) |
36 | 35 | sneqd 4576 | . . . . 5 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → {𝑦} = {1}) |
37 | 36 | imaeq2d 5970 | . . . 4 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {1})) |
38 | 2 | simpri 485 | . . . . 5 ⊢ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴) |
39 | 38 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (◡𝐹 “ {1}) = 𝐴) |
40 | 37, 39 | sylan9eqr 2795 | . . 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 2834 | . 2 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) ∈ dom vol) |
43 | 40 | fveq2d 6796 | . . 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 2834 | . 2 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ) |
46 | 10, 22, 42, 45 | i1fd 24873 | 1 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹 ∈ dom ∫1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 ∖ cdif 3886 ∪ cun 3887 ⊆ wss 3889 ifcif 4462 {csn 4564 {cpr 4566 ↦ cmpt 5160 ◡ccnv 5590 dom cdm 5591 ran crn 5592 “ cima 5594 ⟶wf 6443 ‘cfv 6447 Fincfn 8753 ℝcr 10898 0cc0 10899 1c1 10900 volcvol 24655 ∫1citg1 24807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-inf2 9427 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-isom 6456 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-of 7553 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-2o 8318 df-er 8518 df-map 8637 df-pm 8638 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-sup 9229 df-inf 9230 df-oi 9297 df-dju 9687 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-2 12064 df-3 12065 df-n0 12262 df-z 12348 df-uz 12611 df-q 12717 df-rp 12759 df-xadd 12877 df-ioo 13111 df-ico 13113 df-icc 13114 df-fz 13268 df-fzo 13411 df-fl 13540 df-seq 13750 df-exp 13811 df-hash 14073 df-cj 14838 df-re 14839 df-im 14840 df-sqrt 14974 df-abs 14975 df-clim 15225 df-sum 15426 df-xmet 20618 df-met 20619 df-ovol 24656 df-vol 24657 df-mbf 24811 df-itg1 24812 |
This theorem is referenced by: itg11 24883 itg2const 24933 itg2addnclem 35856 |
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