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Mirrors > Home > MPE Home > Th. List > mbfmulc2re | Structured version Visualization version GIF version |
Description: Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 15-Aug-2014.) |
Ref | Expression |
---|---|
mbfmulc2re.1 | ⊢ (𝜑 → 𝐹 ∈ MblFn) |
mbfmulc2re.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
mbfmulc2re.3 | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
Ref | Expression |
---|---|
mbfmulc2re | ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · 𝐹) ∈ MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfmulc2re.3 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
2 | 1 | fdmd 6266 | . . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
3 | mbfmulc2re.1 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ MblFn) | |
4 | 3 | dmexd 7334 | . . . 4 ⊢ (𝜑 → dom 𝐹 ∈ V) |
5 | 2, 4 | eqeltrrd 2880 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
6 | mbfmulc2re.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
7 | 6 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
8 | 1 | ffvelrnda 6586 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℂ) |
9 | fconstmpt 5369 | . . . 4 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
11 | 1 | feqmptd 6475 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
12 | 5, 7, 8, 10, 11 | offval2 7149 | . 2 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥)))) |
13 | 7, 8 | remul2d 14307 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 · (𝐹‘𝑥))) = (𝐵 · (ℜ‘(𝐹‘𝑥)))) |
14 | 13 | mpteq2dva 4938 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 · (𝐹‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (ℜ‘(𝐹‘𝑥))))) |
15 | 8 | recld 14274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐹‘𝑥)) ∈ ℝ) |
16 | eqidd 2801 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥)))) | |
17 | 5, 7, 15, 10, 16 | offval2 7149 | . . . . 5 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (ℜ‘(𝐹‘𝑥))))) |
18 | 14, 17 | eqtr4d 2837 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 · (𝐹‘𝑥)))) = ((𝐴 × {𝐵}) ∘𝑓 · (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))))) |
19 | 11, 3 | eqeltrrd 2880 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ∈ MblFn) |
20 | 8 | ismbfcn2 23745 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))) ∈ MblFn))) |
21 | 19, 20 | mpbid 224 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))) ∈ MblFn)) |
22 | 21 | simpld 489 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))) ∈ MblFn) |
23 | 15 | fmpttd 6612 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))):𝐴⟶ℝ) |
24 | 22, 6, 23 | mbfmulc2lem 23754 | . . . 4 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥)))) ∈ MblFn) |
25 | 18, 24 | eqeltrd 2879 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 · (𝐹‘𝑥)))) ∈ MblFn) |
26 | 7, 8 | immul2d 14308 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘(𝐵 · (𝐹‘𝑥))) = (𝐵 · (ℑ‘(𝐹‘𝑥)))) |
27 | 26 | mpteq2dva 4938 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 · (𝐹‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (ℑ‘(𝐹‘𝑥))))) |
28 | 8 | imcld 14275 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘(𝐹‘𝑥)) ∈ ℝ) |
29 | eqidd 2801 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥)))) | |
30 | 5, 7, 28, 10, 29 | offval2 7149 | . . . . 5 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (ℑ‘(𝐹‘𝑥))))) |
31 | 27, 30 | eqtr4d 2837 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 · (𝐹‘𝑥)))) = ((𝐴 × {𝐵}) ∘𝑓 · (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))))) |
32 | 21 | simprd 490 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))) ∈ MblFn) |
33 | 28 | fmpttd 6612 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))):𝐴⟶ℝ) |
34 | 32, 6, 33 | mbfmulc2lem 23754 | . . . 4 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥)))) ∈ MblFn) |
35 | 31, 34 | eqeltrd 2879 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 · (𝐹‘𝑥)))) ∈ MblFn) |
36 | 6 | recnd 10358 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
37 | 36 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
38 | 37, 8 | mulcld 10350 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 · (𝐹‘𝑥)) ∈ ℂ) |
39 | 38 | ismbfcn2 23745 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 · (𝐹‘𝑥)))) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 · (𝐹‘𝑥)))) ∈ MblFn))) |
40 | 25, 35, 39 | mpbir2and 705 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ MblFn) |
41 | 12, 40 | eqeltrd 2879 | 1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · 𝐹) ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3386 {csn 4369 ↦ cmpt 4923 × cxp 5311 dom cdm 5313 ⟶wf 6098 ‘cfv 6102 (class class class)co 6879 ∘𝑓 cof 7130 ℂcc 10223 ℝcr 10224 · cmul 10230 ℜcre 14177 ℑcim 14178 MblFncmbf 23721 |
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 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-inf2 8789 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 ax-pre-sup 10303 |
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 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-se 5273 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-isom 6111 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-of 7132 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-2o 7801 df-oadd 7804 df-er 7983 df-map 8098 df-pm 8099 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-sup 8591 df-inf 8592 df-oi 8658 df-card 9052 df-cda 9279 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-div 10978 df-nn 11314 df-2 11375 df-3 11376 df-n0 11580 df-z 11666 df-uz 11930 df-q 12033 df-rp 12074 df-xadd 12193 df-ioo 12427 df-ico 12429 df-icc 12430 df-fz 12580 df-fzo 12720 df-fl 12847 df-seq 13055 df-exp 13114 df-hash 13370 df-cj 14179 df-re 14180 df-im 14181 df-sqrt 14315 df-abs 14316 df-clim 14559 df-sum 14757 df-xmet 20060 df-met 20061 df-ovol 23571 df-vol 23572 df-mbf 23726 |
This theorem is referenced by: mbfneg 23757 mbfmulc2 23770 itgmulc2nclem2 33964 itgmulc2nc 33965 itgabsnc 33966 |
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