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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 | fdm 5964 | . . . . 5 ⊢ (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
4 | mbfmulc2re.1 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ MblFn) | |
5 | dmexg 6989 | . . . . 5 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ V) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐹 ∈ V) |
7 | 3, 6 | eqeltrrd 2689 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
8 | mbfmulc2re.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
10 | 1 | ffvelrnda 6267 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℂ) |
11 | fconstmpt 5085 | . . . 4 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
13 | 1 | feqmptd 6159 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
14 | 7, 9, 10, 12, 13 | offval2 6812 | . 2 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥)))) |
15 | 9, 10 | remul2d 13815 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 · (𝐹‘𝑥))) = (𝐵 · (ℜ‘(𝐹‘𝑥)))) |
16 | 15 | mpteq2dva 4672 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 · (𝐹‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (ℜ‘(𝐹‘𝑥))))) |
17 | 10 | recld 13782 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐹‘𝑥)) ∈ ℝ) |
18 | eqidd 2611 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥)))) | |
19 | 7, 9, 17, 12, 18 | offval2 6812 | . . . . 5 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (ℜ‘(𝐹‘𝑥))))) |
20 | 16, 19 | eqtr4d 2647 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 · (𝐹‘𝑥)))) = ((𝐴 × {𝐵}) ∘𝑓 · (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))))) |
21 | 13, 4 | eqeltrrd 2689 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ∈ MblFn) |
22 | 10 | ismbfcn2 23212 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))) ∈ MblFn))) |
23 | 21, 22 | mpbid 221 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))) ∈ MblFn)) |
24 | 23 | simpld 474 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))) ∈ MblFn) |
25 | eqid 2610 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))) | |
26 | 17, 25 | fmptd 6292 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥))):𝐴⟶ℝ) |
27 | 24, 8, 26 | mbfmulc2lem 23220 | . . . 4 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑥)))) ∈ MblFn) |
28 | 20, 27 | eqeltrd 2688 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 · (𝐹‘𝑥)))) ∈ MblFn) |
29 | 9, 10 | immul2d 13816 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘(𝐵 · (𝐹‘𝑥))) = (𝐵 · (ℑ‘(𝐹‘𝑥)))) |
30 | 29 | mpteq2dva 4672 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 · (𝐹‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (ℑ‘(𝐹‘𝑥))))) |
31 | 10 | imcld 13783 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘(𝐹‘𝑥)) ∈ ℝ) |
32 | eqidd 2611 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥)))) | |
33 | 7, 9, 31, 12, 32 | offval2 6812 | . . . . 5 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (ℑ‘(𝐹‘𝑥))))) |
34 | 30, 33 | eqtr4d 2647 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 · (𝐹‘𝑥)))) = ((𝐴 × {𝐵}) ∘𝑓 · (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))))) |
35 | 23 | simprd 478 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))) ∈ MblFn) |
36 | eqid 2610 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))) | |
37 | 31, 36 | fmptd 6292 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥))):𝐴⟶ℝ) |
38 | 35, 8, 37 | mbfmulc2lem 23220 | . . . 4 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑥)))) ∈ MblFn) |
39 | 34, 38 | eqeltrd 2688 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 · (𝐹‘𝑥)))) ∈ MblFn) |
40 | 8 | recnd 9947 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
41 | 40 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
42 | 41, 10 | mulcld 9939 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 · (𝐹‘𝑥)) ∈ ℂ) |
43 | 42 | ismbfcn2 23212 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 · (𝐹‘𝑥)))) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 · (𝐹‘𝑥)))) ∈ MblFn))) |
44 | 28, 39, 43 | mpbir2and 959 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ MblFn) |
45 | 14, 44 | eqeltrd 2688 | 1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 · 𝐹) ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 ↦ cmpt 4643 × cxp 5036 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 ℂcc 9813 ℝcr 9814 · cmul 9820 ℜcre 13685 ℑcim 13686 MblFncmbf 23189 |
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-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-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-sum 14265 df-xmet 19560 df-met 19561 df-ovol 23040 df-vol 23041 df-mbf 23194 |
This theorem is referenced by: mbfneg 23223 mbfmulc2 23236 itgmulc2nclem2 32647 itgmulc2nc 32648 itgabsnc 32649 |
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