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Mirrors > Home > MPE Home > Th. List > mbfconst | Structured version Visualization version GIF version |
Description: A constant function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
Ref | Expression |
---|---|
mbfconst | ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 768 | . . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
2 | fconstmpt 5578 | . . . 4 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | fmptd 6855 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}):𝐴⟶ℂ) |
4 | mblss 24135 | . . . 4 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | |
5 | 4 | adantr 484 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → 𝐴 ⊆ ℝ) |
6 | cnex 10607 | . . . 4 ⊢ ℂ ∈ V | |
7 | reex 10617 | . . . 4 ⊢ ℝ ∈ V | |
8 | elpm2r 8407 | . . . 4 ⊢ (((ℂ ∈ V ∧ ℝ ∈ V) ∧ ((𝐴 × {𝐵}):𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → (𝐴 × {𝐵}) ∈ (ℂ ↑pm ℝ)) | |
9 | 6, 7, 8 | mpanl12 701 | . . 3 ⊢ (((𝐴 × {𝐵}):𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐴 × {𝐵}) ∈ (ℂ ↑pm ℝ)) |
10 | 3, 5, 9 | syl2anc 587 | . 2 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ (ℂ ↑pm ℝ)) |
11 | 2 | a1i 11 | . . . . . . . . 9 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
12 | ref 14463 | . . . . . . . . . . 11 ⊢ ℜ:ℂ⟶ℝ | |
13 | 12 | a1i 11 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → ℜ:ℂ⟶ℝ) |
14 | 13 | feqmptd 6708 | . . . . . . . . 9 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → ℜ = (𝑦 ∈ ℂ ↦ (ℜ‘𝑦))) |
15 | fveq2 6645 | . . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (ℜ‘𝑦) = (ℜ‘𝐵)) | |
16 | 1, 11, 14, 15 | fmptco 6868 | . . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (ℜ ∘ (𝐴 × {𝐵})) = (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) |
17 | fconstmpt 5578 | . . . . . . . 8 ⊢ (𝐴 × {(ℜ‘𝐵)}) = (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) | |
18 | 16, 17 | eqtr4di 2851 | . . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (ℜ ∘ (𝐴 × {𝐵})) = (𝐴 × {(ℜ‘𝐵)})) |
19 | 18 | cnveqd 5710 | . . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → ◡(ℜ ∘ (𝐴 × {𝐵})) = ◡(𝐴 × {(ℜ‘𝐵)})) |
20 | 19 | imaeq1d 5895 | . . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (◡(ℜ ∘ (𝐴 × {𝐵})) “ 𝑦) = (◡(𝐴 × {(ℜ‘𝐵)}) “ 𝑦)) |
21 | recl 14461 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ) | |
22 | mbfconstlem 24231 | . . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ (ℜ‘𝐵) ∈ ℝ) → (◡(𝐴 × {(ℜ‘𝐵)}) “ 𝑦) ∈ dom vol) | |
23 | 21, 22 | sylan2 595 | . . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (◡(𝐴 × {(ℜ‘𝐵)}) “ 𝑦) ∈ dom vol) |
24 | 20, 23 | eqeltrd 2890 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (◡(ℜ ∘ (𝐴 × {𝐵})) “ 𝑦) ∈ dom vol) |
25 | imf 14464 | . . . . . . . . . . 11 ⊢ ℑ:ℂ⟶ℝ | |
26 | 25 | a1i 11 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → ℑ:ℂ⟶ℝ) |
27 | 26 | feqmptd 6708 | . . . . . . . . 9 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → ℑ = (𝑦 ∈ ℂ ↦ (ℑ‘𝑦))) |
28 | fveq2 6645 | . . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (ℑ‘𝑦) = (ℑ‘𝐵)) | |
29 | 1, 11, 27, 28 | fmptco 6868 | . . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (ℑ ∘ (𝐴 × {𝐵})) = (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) |
30 | fconstmpt 5578 | . . . . . . . 8 ⊢ (𝐴 × {(ℑ‘𝐵)}) = (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) | |
31 | 29, 30 | eqtr4di 2851 | . . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (ℑ ∘ (𝐴 × {𝐵})) = (𝐴 × {(ℑ‘𝐵)})) |
32 | 31 | cnveqd 5710 | . . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → ◡(ℑ ∘ (𝐴 × {𝐵})) = ◡(𝐴 × {(ℑ‘𝐵)})) |
33 | 32 | imaeq1d 5895 | . . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (◡(ℑ ∘ (𝐴 × {𝐵})) “ 𝑦) = (◡(𝐴 × {(ℑ‘𝐵)}) “ 𝑦)) |
34 | imcl 14462 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℝ) | |
35 | mbfconstlem 24231 | . . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ (ℑ‘𝐵) ∈ ℝ) → (◡(𝐴 × {(ℑ‘𝐵)}) “ 𝑦) ∈ dom vol) | |
36 | 34, 35 | sylan2 595 | . . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (◡(𝐴 × {(ℑ‘𝐵)}) “ 𝑦) ∈ dom vol) |
37 | 33, 36 | eqeltrd 2890 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (◡(ℑ ∘ (𝐴 × {𝐵})) “ 𝑦) ∈ dom vol) |
38 | 24, 37 | jca 515 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → ((◡(ℜ ∘ (𝐴 × {𝐵})) “ 𝑦) ∈ dom vol ∧ (◡(ℑ ∘ (𝐴 × {𝐵})) “ 𝑦) ∈ dom vol)) |
39 | 38 | ralrimivw 3150 | . 2 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → ∀𝑦 ∈ ran (,)((◡(ℜ ∘ (𝐴 × {𝐵})) “ 𝑦) ∈ dom vol ∧ (◡(ℑ ∘ (𝐴 × {𝐵})) “ 𝑦) ∈ dom vol)) |
40 | ismbf1 24228 | . 2 ⊢ ((𝐴 × {𝐵}) ∈ MblFn ↔ ((𝐴 × {𝐵}) ∈ (ℂ ↑pm ℝ) ∧ ∀𝑦 ∈ ran (,)((◡(ℜ ∘ (𝐴 × {𝐵})) “ 𝑦) ∈ dom vol ∧ (◡(ℑ ∘ (𝐴 × {𝐵})) “ 𝑦) ∈ dom vol))) | |
41 | 10, 39, 40 | sylanbrc 586 | 1 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 {csn 4525 ↦ cmpt 5110 × cxp 5517 ◡ccnv 5518 dom cdm 5519 ran crn 5520 “ cima 5522 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑pm cpm 8390 ℂcc 10524 ℝcr 10525 (,)cioo 12726 ℜcre 14448 ℑcim 14449 volcvol 24067 MblFncmbf 24218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-xmet 20084 df-met 20085 df-ovol 24068 df-vol 24069 df-mbf 24223 |
This theorem is referenced by: mbf0 24238 mbfss 24250 mbfmulc2lem 24251 mbfpos 24255 ibl0 24390 iblconst 24421 |
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