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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 24704 | . . . . . 6 ⊢ ran (,) ∈ TopBases | |
| 2 | bastg 22910 | . . . . . 6 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) | |
| 3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
| 4 | retop 24705 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
| 5 | sssigagen 34302 | . . . . . 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 3943 | . . . 4 ⊢ ran (,) ⊆ (sigaGen‘(topGen‘ran (,))) |
| 8 | df-brsiga 34339 | . . . 4 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
| 9 | 7, 8 | sseqtrri 3983 | . . 3 ⊢ ran (,) ⊆ 𝔅ℝ |
| 10 | eqid 2736 | . . . . 5 ⊢ vol = vol | |
| 11 | dmvlsiga 34286 | . . . . . . 7 ⊢ dom vol ∈ (sigAlgebra‘ℝ) | |
| 12 | elrnsiga 34283 | . . . . . . 7 ⊢ (dom vol ∈ (sigAlgebra‘ℝ) → dom vol ∈ ∪ ran sigAlgebra) | |
| 13 | 11, 12 | mp1i 13 | . . . . . 6 ⊢ (vol = vol → dom vol ∈ ∪ ran sigAlgebra) |
| 14 | brsigarn 34341 | . . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
| 15 | elrnsiga 34283 | . . . . . . 7 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) | |
| 16 | 14, 15 | mp1i 13 | . . . . . 6 ⊢ (vol = vol → 𝔅ℝ ∈ ∪ ran sigAlgebra) |
| 17 | 13, 16 | ismbfm 34408 | . . . . 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 496 | . . 3 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → ∀𝑥 ∈ 𝔅ℝ (◡𝐹 “ 𝑥) ∈ dom vol) |
| 20 | ssralv 4002 | . . 3 ⊢ (ran (,) ⊆ 𝔅ℝ → (∀𝑥 ∈ 𝔅ℝ (◡𝐹 “ 𝑥) ∈ dom vol → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)) | |
| 21 | 9, 19, 20 | mpsyl 68 | . 2 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol) |
| 22 | 18 | simplbi 497 | . . 3 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ (∪ 𝔅ℝ ↑m ∪ dom vol)) |
| 23 | elmapi 8786 | . . . 4 ⊢ (𝐹 ∈ (ℝ ↑m ℝ) → 𝐹:ℝ⟶ℝ) | |
| 24 | unibrsiga 34343 | . . . . 5 ⊢ ∪ 𝔅ℝ = ℝ | |
| 25 | unidmvol 25498 | . . . . 5 ⊢ ∪ dom vol = ℝ | |
| 26 | 24, 25 | oveq12i 7370 | . . . 4 ⊢ (∪ 𝔅ℝ ↑m ∪ dom vol) = (ℝ ↑m ℝ) |
| 27 | 23, 26 | eleq2s 2854 | . . 3 ⊢ (𝐹 ∈ (∪ 𝔅ℝ ↑m ∪ dom vol) → 𝐹:ℝ⟶ℝ) |
| 28 | ismbf 25585 | . . 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 257 | 1 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ MblFn) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ⊆ wss 3901 ∪ cuni 4863 ◡ccnv 5623 dom cdm 5624 ran crn 5625 “ cima 5627 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ↑m cmap 8763 ℝcr 11025 (,)cioo 13261 topGenctg 17357 Topctop 22837 TopBasesctb 22889 volcvol 25420 MblFncmbf 25571 sigAlgebracsiga 34265 sigaGencsigagen 34295 𝔅ℝcbrsiga 34338 MblFnMcmbfm 34406 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cc 10345 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-z 12489 df-uz 12752 df-q 12862 df-rp 12906 df-xadd 13027 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-rlim 15412 df-sum 15610 df-topgen 17363 df-xmet 21302 df-met 21303 df-top 22838 df-bases 22890 df-ovol 25421 df-vol 25422 df-mbf 25576 df-siga 34266 df-sigagen 34296 df-brsiga 34339 df-mbfm 34407 |
| This theorem is referenced by: (None) |
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