Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elmbfmvol2 | Structured version Visualization version GIF version |
Description: Measurable functions with respect to the Lebesgue measure. We only have the inclusion, since MblFn includes complex-valued functions. (Contributed by Thierry Arnoux, 26-Jan-2017.) |
Ref | Expression |
---|---|
elmbfmvol2 | ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retopbas 22374 | . . . . . 6 ⊢ ran (,) ∈ TopBases | |
2 | bastg 20581 | . . . . . 6 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
4 | retop 22375 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
5 | sssigagen 29535 | . . . . . 6 ⊢ ((topGen‘ran (,)) ∈ Top → (topGen‘ran (,)) ⊆ (sigaGen‘(topGen‘ran (,)))) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ (topGen‘ran (,)) ⊆ (sigaGen‘(topGen‘ran (,))) |
7 | 3, 6 | sstri 3577 | . . . 4 ⊢ ran (,) ⊆ (sigaGen‘(topGen‘ran (,))) |
8 | df-brsiga 29572 | . . . 4 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
9 | 7, 8 | sseqtr4i 3601 | . . 3 ⊢ ran (,) ⊆ 𝔅ℝ |
10 | eqid 2610 | . . . . 5 ⊢ vol = vol | |
11 | dmvlsiga 29519 | . . . . . . 7 ⊢ dom vol ∈ (sigAlgebra‘ℝ) | |
12 | elrnsiga 29516 | . . . . . . 7 ⊢ (dom vol ∈ (sigAlgebra‘ℝ) → dom vol ∈ ∪ ran sigAlgebra) | |
13 | 11, 12 | mp1i 13 | . . . . . 6 ⊢ (vol = vol → dom vol ∈ ∪ ran sigAlgebra) |
14 | brsigarn 29574 | . . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
15 | elrnsiga 29516 | . . . . . . 7 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) | |
16 | 14, 15 | mp1i 13 | . . . . . 6 ⊢ (vol = vol → 𝔅ℝ ∈ ∪ ran sigAlgebra) |
17 | 13, 16 | ismbfm 29641 | . . . . 5 ⊢ (vol = vol → (𝐹 ∈ (dom volMblFnM𝔅ℝ) ↔ (𝐹 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ dom vol) ∧ ∀𝑥 ∈ 𝔅ℝ (◡𝐹 “ 𝑥) ∈ dom vol))) |
18 | 10, 17 | ax-mp 5 | . . . 4 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) ↔ (𝐹 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ dom vol) ∧ ∀𝑥 ∈ 𝔅ℝ (◡𝐹 “ 𝑥) ∈ dom vol)) |
19 | 18 | simprbi 479 | . . 3 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → ∀𝑥 ∈ 𝔅ℝ (◡𝐹 “ 𝑥) ∈ dom vol) |
20 | ssralv 3629 | . . 3 ⊢ (ran (,) ⊆ 𝔅ℝ → (∀𝑥 ∈ 𝔅ℝ (◡𝐹 “ 𝑥) ∈ dom vol → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)) | |
21 | 9, 19, 20 | mpsyl 66 | . 2 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol) |
22 | 18 | simplbi 475 | . . 3 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ dom vol)) |
23 | elmapi 7765 | . . . 4 ⊢ (𝐹 ∈ (ℝ ↑𝑚 ℝ) → 𝐹:ℝ⟶ℝ) | |
24 | unibrsiga 29576 | . . . . 5 ⊢ ∪ 𝔅ℝ = ℝ | |
25 | unidmvol 23116 | . . . . 5 ⊢ ∪ dom vol = ℝ | |
26 | 24, 25 | oveq12i 6561 | . . . 4 ⊢ (∪ 𝔅ℝ ↑𝑚 ∪ dom vol) = (ℝ ↑𝑚 ℝ) |
27 | 23, 26 | eleq2s 2706 | . . 3 ⊢ (𝐹 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ dom vol) → 𝐹:ℝ⟶ℝ) |
28 | ismbf 23203 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)) | |
29 | 22, 27, 28 | 3syl 18 | . 2 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)) |
30 | 21, 29 | mpbird 246 | 1 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 ∪ cuni 4372 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℝcr 9814 (,)cioo 12046 topGenctg 15921 Topctop 20517 TopBasesctb 20520 volcvol 23039 MblFncmbf 23189 sigAlgebracsiga 29497 sigaGencsigagen 29528 𝔅ℝcbrsiga 29571 MblFnMcmbfm 29639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xadd 11823 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-topgen 15927 df-xmet 19560 df-met 19561 df-top 20521 df-bases 20522 df-ovol 23040 df-vol 23041 df-mbf 23194 df-siga 29498 df-sigagen 29529 df-brsiga 29572 df-mbfm 29640 |
This theorem is referenced by: (None) |
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