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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 25724 | . . . . 5 ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴)) |
| 3 | 2 | simpli 483 | . . . 4 ⊢ 𝐹:ℝ⟶{0, 1} |
| 4 | 0re 11263 | . . . . 5 ⊢ 0 ∈ ℝ | |
| 5 | 1re 11261 | . . . . 5 ⊢ 1 ∈ ℝ | |
| 6 | prssi 4821 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ) | |
| 7 | 4, 5, 6 | mp2an 692 | . . . 4 ⊢ {0, 1} ⊆ ℝ |
| 8 | fss 6752 | . . . 4 ⊢ ((𝐹:ℝ⟶{0, 1} ∧ {0, 1} ⊆ ℝ) → 𝐹:ℝ⟶ℝ) | |
| 9 | 3, 7, 8 | mp2an 692 | . . 3 ⊢ 𝐹:ℝ⟶ℝ |
| 10 | 9 | a1i 11 | . 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹:ℝ⟶ℝ) |
| 11 | prfi 9363 | . . 3 ⊢ {0, 1} ∈ Fin | |
| 12 | 1ex 11257 | . . . . . . . 8 ⊢ 1 ∈ V | |
| 13 | 12 | prid2 4763 | . . . . . . 7 ⊢ 1 ∈ {0, 1} |
| 14 | c0ex 11255 | . . . . . . . 8 ⊢ 0 ∈ V | |
| 15 | 14 | prid1 4762 | . . . . . . 7 ⊢ 0 ∈ {0, 1} |
| 16 | 13, 15 | ifcli 4573 | . . . . . 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 7134 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹:ℝ⟶{0, 1}) |
| 19 | frn 6743 | . . . 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 9213 | . . 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 4629 | . . . . . . . . . . . 12 ⊢ {0, 1} = ({0} ∪ {1}) | |
| 25 | 24 | equncomi 4160 | . . . . . . . . . . 11 ⊢ {0, 1} = ({1} ∪ {0}) |
| 26 | 23, 25 | sseqtri 4032 | . . . . . . . . . 10 ⊢ ran 𝐹 ⊆ ({1} ∪ {0}) |
| 27 | ssdif 4144 | . . . . . . . . . 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 4481 | . . . . . . . . . 10 ⊢ (({1} ∪ {0}) ∖ {0}) = ({1} ∖ {0}) | |
| 30 | difss 4136 | . . . . . . . . . 10 ⊢ ({1} ∖ {0}) ⊆ {1} | |
| 31 | 29, 30 | eqsstri 4030 | . . . . . . . . 9 ⊢ (({1} ∪ {0}) ∖ {0}) ⊆ {1} |
| 32 | 28, 31 | sstri 3993 | . . . . . . . 8 ⊢ (ran 𝐹 ∖ {0}) ⊆ {1} |
| 33 | 32 | sseli 3979 | . . . . . . 7 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 ∈ {1}) |
| 34 | elsni 4643 | . . . . . . 7 ⊢ (𝑦 ∈ {1} → 𝑦 = 1) | |
| 35 | 33, 34 | syl 17 | . . . . . 6 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 = 1) |
| 36 | 35 | sneqd 4638 | . . . . 5 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → {𝑦} = {1}) |
| 37 | 36 | imaeq2d 6078 | . . . 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 2799 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) = 𝐴) |
| 41 | simpll 767 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐴 ∈ dom vol) | |
| 42 | 40, 41 | eqeltrd 2841 | . 2 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) ∈ dom vol) |
| 43 | 40 | fveq2d 6910 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) = (vol‘𝐴)) |
| 44 | simplr 769 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘𝐴) ∈ ℝ) | |
| 45 | 43, 44 | eqeltrd 2841 | . 2 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ) |
| 46 | 10, 22, 42, 45 | i1fd 25716 | 1 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹 ∈ dom ∫1) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 ifcif 4525 {csn 4626 {cpr 4628 ↦ cmpt 5225 ◡ccnv 5684 dom cdm 5685 ran crn 5686 “ cima 5688 ⟶wf 6557 ‘cfv 6561 Fincfn 8985 ℝcr 11154 0cc0 11155 1c1 11156 volcvol 25498 ∫1citg1 25650 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xadd 13155 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-xmet 21357 df-met 21358 df-ovol 25499 df-vol 25500 df-mbf 25654 df-itg1 25655 |
| This theorem is referenced by: itg11 25726 itg2const 25775 itg2addnclem 37678 |
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