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Mirrors > Home > MPE Home > Th. List > ismbfcn2 | Structured version Visualization version GIF version |
Description: A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 13-Aug-2014.) |
Ref | Expression |
---|---|
ismbfcn2.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
ismbfcn2 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismbfcn2.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
2 | 1 | fmpttd 6609 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
3 | ismbfcn 23734 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((ℜ ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ MblFn ∧ (ℑ ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ MblFn))) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((ℜ ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ MblFn ∧ (ℑ ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ MblFn))) |
5 | eqidd 2798 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
6 | ref 14190 | . . . . . . 7 ⊢ ℜ:ℂ⟶ℝ | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℜ:ℂ⟶ℝ) |
8 | 7 | feqmptd 6472 | . . . . 5 ⊢ (𝜑 → ℜ = (𝑦 ∈ ℂ ↦ (ℜ‘𝑦))) |
9 | fveq2 6409 | . . . . 5 ⊢ (𝑦 = 𝐵 → (ℜ‘𝑦) = (ℜ‘𝐵)) | |
10 | 1, 5, 8, 9 | fmptco 6621 | . . . 4 ⊢ (𝜑 → (ℜ ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) |
11 | 10 | eleq1d 2861 | . . 3 ⊢ (𝜑 → ((ℜ ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ MblFn ↔ (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn)) |
12 | imf 14191 | . . . . . . 7 ⊢ ℑ:ℂ⟶ℝ | |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℑ:ℂ⟶ℝ) |
14 | 13 | feqmptd 6472 | . . . . 5 ⊢ (𝜑 → ℑ = (𝑦 ∈ ℂ ↦ (ℑ‘𝑦))) |
15 | fveq2 6409 | . . . . 5 ⊢ (𝑦 = 𝐵 → (ℑ‘𝑦) = (ℑ‘𝐵)) | |
16 | 1, 5, 14, 15 | fmptco 6621 | . . . 4 ⊢ (𝜑 → (ℑ ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) |
17 | 16 | eleq1d 2861 | . . 3 ⊢ (𝜑 → ((ℑ ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ MblFn ↔ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn)) |
18 | 11, 17 | anbi12d 625 | . 2 ⊢ (𝜑 → (((ℜ ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ MblFn ∧ (ℑ ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ MblFn) ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn))) |
19 | 4, 18 | bitrd 271 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∈ wcel 2157 ↦ cmpt 4920 ∘ ccom 5314 ⟶wf 6095 ‘cfv 6099 ℂcc 10220 ℝcr 10221 ℜcre 14175 ℑcim 14176 MblFncmbf 23719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-inf2 8786 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-se 5270 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-2o 7798 df-oadd 7801 df-er 7980 df-map 8095 df-pm 8096 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-sup 8588 df-inf 8589 df-oi 8655 df-card 9049 df-cda 9276 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-n0 11577 df-z 11663 df-uz 11927 df-q 12030 df-rp 12071 df-xadd 12190 df-ioo 12424 df-ico 12426 df-icc 12427 df-fz 12577 df-fzo 12717 df-fl 12844 df-seq 13052 df-exp 13111 df-hash 13367 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-clim 14557 df-sum 14755 df-xmet 20058 df-met 20059 df-ovol 23569 df-vol 23570 df-mbf 23724 |
This theorem is referenced by: mbfeqa 23748 mbfss 23751 mbfmulc2re 23753 mbfadd 23766 mbfmulc2 23768 mbflim 23773 mbfmul 23831 iblcn 23903 ibladd 23925 ibladdnc 33947 ftc1anclem2 33966 ftc1anclem5 33969 ftc1anclem6 33970 ftc1anclem8 33972 |
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