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Mirrors > Home > MPE Home > Th. List > Mathboxes > mbfmco | Structured version Visualization version GIF version |
Description: The composition of two measurable functions is measurable. See cnmpt11 22268. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
Ref | Expression |
---|---|
mbfmco.1 | ⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra) |
mbfmco.2 | ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
mbfmco.3 | ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) |
mbfmco.4 | ⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆)) |
mbfmco.5 | ⊢ (𝜑 → 𝐺 ∈ (𝑆MblFnM𝑇)) |
Ref | Expression |
---|---|
mbfmco | ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfmco.2 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) | |
2 | mbfmco.3 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) | |
3 | mbfmco.5 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑆MblFnM𝑇)) | |
4 | 1, 2, 3 | mbfmf 31623 | . . . 4 ⊢ (𝜑 → 𝐺:∪ 𝑆⟶∪ 𝑇) |
5 | mbfmco.1 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra) | |
6 | mbfmco.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆)) | |
7 | 5, 1, 6 | mbfmf 31623 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝑅⟶∪ 𝑆) |
8 | fco 6505 | . . . 4 ⊢ ((𝐺:∪ 𝑆⟶∪ 𝑇 ∧ 𝐹:∪ 𝑅⟶∪ 𝑆) → (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇) | |
9 | 4, 7, 8 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇) |
10 | unielsiga 31497 | . . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇) | |
11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇) |
12 | unielsiga 31497 | . . . . 5 ⊢ (𝑅 ∈ ∪ ran sigAlgebra → ∪ 𝑅 ∈ 𝑅) | |
13 | 5, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ∪ 𝑅 ∈ 𝑅) |
14 | 11, 13 | elmapd 8403 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ↔ (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇)) |
15 | 9, 14 | mpbird 260 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅)) |
16 | cnvco 5720 | . . . . . 6 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺) | |
17 | 16 | imaeq1i 5893 | . . . . 5 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = ((◡𝐹 ∘ ◡𝐺) “ 𝑎) |
18 | imaco 6071 | . . . . 5 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎)) | |
19 | 17, 18 | eqtri 2821 | . . . 4 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎)) |
20 | 5 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑅 ∈ ∪ ran sigAlgebra) |
21 | 1 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra) |
22 | 6 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆)) |
23 | 2 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra) |
24 | 3 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐺 ∈ (𝑆MblFnM𝑇)) |
25 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ 𝑇) | |
26 | 21, 23, 24, 25 | mbfmcnvima 31625 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐺 “ 𝑎) ∈ 𝑆) |
27 | 20, 21, 22, 26 | mbfmcnvima 31625 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐹 “ (◡𝐺 “ 𝑎)) ∈ 𝑅) |
28 | 19, 27 | eqeltrid 2894 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅) |
29 | 28 | ralrimiva 3149 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅) |
30 | 5, 2 | ismbfm 31620 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ∧ ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅))) |
31 | 15, 29, 30 | mpbir2and 712 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 ∪ cuni 4800 ◡ccnv 5518 ran crn 5520 “ cima 5522 ∘ ccom 5523 ⟶wf 6320 (class class class)co 7135 ↑m cmap 8389 sigAlgebracsiga 31477 MblFnMcmbfm 31618 |
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-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-ral 3111 df-rex 3112 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-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 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-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-siga 31478 df-mbfm 31619 |
This theorem is referenced by: rrvadd 31820 rrvmulc 31821 |
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