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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 23672. (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 34256 | . . . 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 34256 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝑅⟶∪ 𝑆) | 
| 8 | fco 6759 | . . . 4 ⊢ ((𝐺:∪ 𝑆⟶∪ 𝑇 ∧ 𝐹:∪ 𝑅⟶∪ 𝑆) → (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇) | |
| 9 | 4, 7, 8 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇) | 
| 10 | unielsiga 34130 | . . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇) | |
| 11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇) | 
| 12 | unielsiga 34130 | . . . . 5 ⊢ (𝑅 ∈ ∪ ran sigAlgebra → ∪ 𝑅 ∈ 𝑅) | |
| 13 | 5, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ∪ 𝑅 ∈ 𝑅) | 
| 14 | 11, 13 | elmapd 8881 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ↔ (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇)) | 
| 15 | 9, 14 | mpbird 257 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅)) | 
| 16 | cnvco 5895 | . . . . . 6 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺) | |
| 17 | 16 | imaeq1i 6074 | . . . . 5 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = ((◡𝐹 ∘ ◡𝐺) “ 𝑎) | 
| 18 | imaco 6270 | . . . . 5 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎)) | |
| 19 | 17, 18 | eqtri 2764 | . . . 4 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎)) | 
| 20 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑅 ∈ ∪ ran sigAlgebra) | 
| 21 | 1 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra) | 
| 22 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆)) | 
| 23 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra) | 
| 24 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐺 ∈ (𝑆MblFnM𝑇)) | 
| 25 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ 𝑇) | |
| 26 | 21, 23, 24, 25 | mbfmcnvima 34258 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐺 “ 𝑎) ∈ 𝑆) | 
| 27 | 20, 21, 22, 26 | mbfmcnvima 34258 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐹 “ (◡𝐺 “ 𝑎)) ∈ 𝑅) | 
| 28 | 19, 27 | eqeltrid 2844 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅) | 
| 29 | 28 | ralrimiva 3145 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅) | 
| 30 | 5, 2 | ismbfm 34253 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ∧ ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅))) | 
| 31 | 15, 29, 30 | mpbir2and 713 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ∀wral 3060 ∪ cuni 4906 ◡ccnv 5683 ran crn 5685 “ cima 5687 ∘ ccom 5688 ⟶wf 6556 (class class class)co 7432 ↑m cmap 8867 sigAlgebracsiga 34110 MblFnMcmbfm 34251 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-map 8869 df-siga 34111 df-mbfm 34252 | 
| This theorem is referenced by: rrvadd 34455 rrvmulc 34456 | 
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