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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 22824. (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 32230 | . . . 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 32230 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝑅⟶∪ 𝑆) |
8 | fco 6616 | . . . 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 32104 | . . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇) | |
11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇) |
12 | unielsiga 32104 | . . . . 5 ⊢ (𝑅 ∈ ∪ ran sigAlgebra → ∪ 𝑅 ∈ 𝑅) | |
13 | 5, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ∪ 𝑅 ∈ 𝑅) |
14 | 11, 13 | elmapd 8616 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ↔ (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇)) |
15 | 9, 14 | mpbird 256 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅)) |
16 | cnvco 5787 | . . . . . 6 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺) | |
17 | 16 | imaeq1i 5959 | . . . . 5 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = ((◡𝐹 ∘ ◡𝐺) “ 𝑎) |
18 | imaco 6148 | . . . . 5 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎)) | |
19 | 17, 18 | eqtri 2766 | . . . 4 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎)) |
20 | 5 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑅 ∈ ∪ ran sigAlgebra) |
21 | 1 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra) |
22 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆)) |
23 | 2 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra) |
24 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐺 ∈ (𝑆MblFnM𝑇)) |
25 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ 𝑇) | |
26 | 21, 23, 24, 25 | mbfmcnvima 32232 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐺 “ 𝑎) ∈ 𝑆) |
27 | 20, 21, 22, 26 | mbfmcnvima 32232 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐹 “ (◡𝐺 “ 𝑎)) ∈ 𝑅) |
28 | 19, 27 | eqeltrid 2843 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅) |
29 | 28 | ralrimiva 3108 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅) |
30 | 5, 2 | ismbfm 32227 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ∧ ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅))) |
31 | 15, 29, 30 | mpbir2and 710 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3064 ∪ cuni 4839 ◡ccnv 5583 ran crn 5585 “ cima 5587 ∘ ccom 5588 ⟶wf 6422 (class class class)co 7267 ↑m cmap 8602 sigAlgebracsiga 32084 MblFnMcmbfm 32225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-fv 6434 df-ov 7270 df-oprab 7271 df-mpo 7272 df-1st 7820 df-2nd 7821 df-map 8604 df-siga 32085 df-mbfm 32226 |
This theorem is referenced by: rrvadd 32427 rrvmulc 32428 |
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