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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 25709 | . . . . 5 ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴)) |
3 | 2 | simpli 482 | . . . 4 ⊢ 𝐹:ℝ⟶{0, 1} |
4 | 0re 11266 | . . . . 5 ⊢ 0 ∈ ℝ | |
5 | 1re 11264 | . . . . 5 ⊢ 1 ∈ ℝ | |
6 | prssi 4830 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ) | |
7 | 4, 5, 6 | mp2an 690 | . . . 4 ⊢ {0, 1} ⊆ ℝ |
8 | fss 6744 | . . . 4 ⊢ ((𝐹:ℝ⟶{0, 1} ∧ {0, 1} ⊆ ℝ) → 𝐹:ℝ⟶ℝ) | |
9 | 3, 7, 8 | mp2an 690 | . . 3 ⊢ 𝐹:ℝ⟶ℝ |
10 | 9 | a1i 11 | . 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹:ℝ⟶ℝ) |
11 | prfi 9365 | . . 3 ⊢ {0, 1} ∈ Fin | |
12 | 1ex 11260 | . . . . . . . 8 ⊢ 1 ∈ V | |
13 | 12 | prid2 4772 | . . . . . . 7 ⊢ 1 ∈ {0, 1} |
14 | c0ex 11258 | . . . . . . . 8 ⊢ 0 ∈ V | |
15 | 14 | prid1 4771 | . . . . . . 7 ⊢ 0 ∈ {0, 1} |
16 | 13, 15 | ifcli 4580 | . . . . . 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 7128 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹:ℝ⟶{0, 1}) |
19 | frn 6735 | . . . 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 9211 | . . 3 ⊢ (({0, 1} ∈ Fin ∧ ran 𝐹 ⊆ {0, 1}) → ran 𝐹 ∈ Fin) | |
22 | 11, 20, 21 | sylancr 585 | . 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ran 𝐹 ∈ Fin) |
23 | 3, 19 | ax-mp 5 | . . . . . . . . . . 11 ⊢ ran 𝐹 ⊆ {0, 1} |
24 | df-pr 4636 | . . . . . . . . . . . 12 ⊢ {0, 1} = ({0} ∪ {1}) | |
25 | 24 | equncomi 4155 | . . . . . . . . . . 11 ⊢ {0, 1} = ({1} ∪ {0}) |
26 | 23, 25 | sseqtri 4016 | . . . . . . . . . 10 ⊢ ran 𝐹 ⊆ ({1} ∪ {0}) |
27 | ssdif 4139 | . . . . . . . . . 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 4485 | . . . . . . . . . 10 ⊢ (({1} ∪ {0}) ∖ {0}) = ({1} ∖ {0}) | |
30 | difss 4131 | . . . . . . . . . 10 ⊢ ({1} ∖ {0}) ⊆ {1} | |
31 | 29, 30 | eqsstri 4014 | . . . . . . . . 9 ⊢ (({1} ∪ {0}) ∖ {0}) ⊆ {1} |
32 | 28, 31 | sstri 3989 | . . . . . . . 8 ⊢ (ran 𝐹 ∖ {0}) ⊆ {1} |
33 | 32 | sseli 3975 | . . . . . . 7 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 ∈ {1}) |
34 | elsni 4650 | . . . . . . 7 ⊢ (𝑦 ∈ {1} → 𝑦 = 1) | |
35 | 33, 34 | syl 17 | . . . . . 6 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 = 1) |
36 | 35 | sneqd 4645 | . . . . 5 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → {𝑦} = {1}) |
37 | 36 | imaeq2d 6069 | . . . 4 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {1})) |
38 | 2 | simpri 484 | . . . . 5 ⊢ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴) |
39 | 38 | adantr 479 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (◡𝐹 “ {1}) = 𝐴) |
40 | 37, 39 | sylan9eqr 2788 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) = 𝐴) |
41 | simpll 765 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐴 ∈ dom vol) | |
42 | 40, 41 | eqeltrd 2826 | . 2 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) ∈ dom vol) |
43 | 40 | fveq2d 6905 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) = (vol‘𝐴)) |
44 | simplr 767 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘𝐴) ∈ ℝ) | |
45 | 43, 44 | eqeltrd 2826 | . 2 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ) |
46 | 10, 22, 42, 45 | i1fd 25701 | 1 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹 ∈ dom ∫1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∖ cdif 3944 ∪ cun 3945 ⊆ wss 3947 ifcif 4533 {csn 4633 {cpr 4635 ↦ cmpt 5236 ◡ccnv 5681 dom cdm 5682 ran crn 5683 “ cima 5685 ⟶wf 6550 ‘cfv 6554 Fincfn 8974 ℝcr 11157 0cc0 11158 1c1 11159 volcvol 25483 ∫1citg1 25635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 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 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-oi 9553 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-q 12985 df-rp 13029 df-xadd 13147 df-ioo 13382 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-sum 15691 df-xmet 21336 df-met 21337 df-ovol 25484 df-vol 25485 df-mbf 25639 df-itg1 25640 |
This theorem is referenced by: itg11 25711 itg2const 25761 itg2addnclem 37372 |
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