![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 24293 | . . . . 5 ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴)) |
3 | 2 | simpli 487 | . . . 4 ⊢ 𝐹:ℝ⟶{0, 1} |
4 | 0re 10632 | . . . . 5 ⊢ 0 ∈ ℝ | |
5 | 1re 10630 | . . . . 5 ⊢ 1 ∈ ℝ | |
6 | prssi 4714 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ) | |
7 | 4, 5, 6 | mp2an 691 | . . . 4 ⊢ {0, 1} ⊆ ℝ |
8 | fss 6501 | . . . 4 ⊢ ((𝐹:ℝ⟶{0, 1} ∧ {0, 1} ⊆ ℝ) → 𝐹:ℝ⟶ℝ) | |
9 | 3, 7, 8 | mp2an 691 | . . 3 ⊢ 𝐹:ℝ⟶ℝ |
10 | 9 | a1i 11 | . 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹:ℝ⟶ℝ) |
11 | prfi 8777 | . . 3 ⊢ {0, 1} ∈ Fin | |
12 | 1ex 10626 | . . . . . . . 8 ⊢ 1 ∈ V | |
13 | 12 | prid2 4659 | . . . . . . 7 ⊢ 1 ∈ {0, 1} |
14 | c0ex 10624 | . . . . . . . 8 ⊢ 0 ∈ V | |
15 | 14 | prid1 4658 | . . . . . . 7 ⊢ 0 ∈ {0, 1} |
16 | 13, 15 | ifcli 4471 | . . . . . 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 6855 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹:ℝ⟶{0, 1}) |
19 | frn 6493 | . . . 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 8722 | . . 3 ⊢ (({0, 1} ∈ Fin ∧ ran 𝐹 ⊆ {0, 1}) → ran 𝐹 ∈ Fin) | |
22 | 11, 20, 21 | sylancr 590 | . 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ran 𝐹 ∈ Fin) |
23 | 3, 19 | ax-mp 5 | . . . . . . . . . . 11 ⊢ ran 𝐹 ⊆ {0, 1} |
24 | df-pr 4528 | . . . . . . . . . . . 12 ⊢ {0, 1} = ({0} ∪ {1}) | |
25 | 24 | equncomi 4082 | . . . . . . . . . . 11 ⊢ {0, 1} = ({1} ∪ {0}) |
26 | 23, 25 | sseqtri 3951 | . . . . . . . . . 10 ⊢ ran 𝐹 ⊆ ({1} ∪ {0}) |
27 | ssdif 4067 | . . . . . . . . . 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 4387 | . . . . . . . . . 10 ⊢ (({1} ∪ {0}) ∖ {0}) = ({1} ∖ {0}) | |
30 | difss 4059 | . . . . . . . . . 10 ⊢ ({1} ∖ {0}) ⊆ {1} | |
31 | 29, 30 | eqsstri 3949 | . . . . . . . . 9 ⊢ (({1} ∪ {0}) ∖ {0}) ⊆ {1} |
32 | 28, 31 | sstri 3924 | . . . . . . . 8 ⊢ (ran 𝐹 ∖ {0}) ⊆ {1} |
33 | 32 | sseli 3911 | . . . . . . 7 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 ∈ {1}) |
34 | elsni 4542 | . . . . . . 7 ⊢ (𝑦 ∈ {1} → 𝑦 = 1) | |
35 | 33, 34 | syl 17 | . . . . . 6 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 = 1) |
36 | 35 | sneqd 4537 | . . . . 5 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → {𝑦} = {1}) |
37 | 36 | imaeq2d 5896 | . . . 4 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {1})) |
38 | 2 | simpri 489 | . . . . 5 ⊢ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴) |
39 | 38 | adantr 484 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (◡𝐹 “ {1}) = 𝐴) |
40 | 37, 39 | sylan9eqr 2855 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) = 𝐴) |
41 | simpll 766 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐴 ∈ dom vol) | |
42 | 40, 41 | eqeltrd 2890 | . 2 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) ∈ dom vol) |
43 | 40 | fveq2d 6649 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) = (vol‘𝐴)) |
44 | simplr 768 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘𝐴) ∈ ℝ) | |
45 | 43, 44 | eqeltrd 2890 | . 2 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ) |
46 | 10, 22, 42, 45 | i1fd 24285 | 1 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹 ∈ dom ∫1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ∪ cun 3879 ⊆ wss 3881 ifcif 4425 {csn 4525 {cpr 4527 ↦ cmpt 5110 ◡ccnv 5518 dom cdm 5519 ran crn 5520 “ cima 5522 ⟶wf 6320 ‘cfv 6324 Fincfn 8492 ℝcr 10525 0cc0 10526 1c1 10527 volcvol 24067 ∫1citg1 24219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-xmet 20084 df-met 20085 df-ovol 24068 df-vol 24069 df-mbf 24223 df-itg1 24224 |
This theorem is referenced by: itg11 24295 itg2const 24344 itg2addnclem 35108 |
Copyright terms: Public domain | W3C validator |