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Theorem mbfmco 34246
Description: The composition of two measurable functions is measurable. See cnmpt11 23687. (Contributed by Thierry Arnoux, 4-Jun-2017.)
Hypotheses
Ref Expression
mbfmco.1 (𝜑𝑅 ran sigAlgebra)
mbfmco.2 (𝜑𝑆 ran sigAlgebra)
mbfmco.3 (𝜑𝑇 ran sigAlgebra)
mbfmco.4 (𝜑𝐹 ∈ (𝑅MblFnM𝑆))
mbfmco.5 (𝜑𝐺 ∈ (𝑆MblFnM𝑇))
Assertion
Ref Expression
mbfmco (𝜑 → (𝐺𝐹) ∈ (𝑅MblFnM𝑇))

Proof of Theorem mbfmco
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 mbfmco.2 . . . . 5 (𝜑𝑆 ran sigAlgebra)
2 mbfmco.3 . . . . 5 (𝜑𝑇 ran sigAlgebra)
3 mbfmco.5 . . . . 5 (𝜑𝐺 ∈ (𝑆MblFnM𝑇))
41, 2, 3mbfmf 34235 . . . 4 (𝜑𝐺: 𝑆 𝑇)
5 mbfmco.1 . . . . 5 (𝜑𝑅 ran sigAlgebra)
6 mbfmco.4 . . . . 5 (𝜑𝐹 ∈ (𝑅MblFnM𝑆))
75, 1, 6mbfmf 34235 . . . 4 (𝜑𝐹: 𝑅 𝑆)
8 fco 6761 . . . 4 ((𝐺: 𝑆 𝑇𝐹: 𝑅 𝑆) → (𝐺𝐹): 𝑅 𝑇)
94, 7, 8syl2anc 584 . . 3 (𝜑 → (𝐺𝐹): 𝑅 𝑇)
10 unielsiga 34109 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
112, 10syl 17 . . . 4 (𝜑 𝑇𝑇)
12 unielsiga 34109 . . . . 5 (𝑅 ran sigAlgebra → 𝑅𝑅)
135, 12syl 17 . . . 4 (𝜑 𝑅𝑅)
1411, 13elmapd 8879 . . 3 (𝜑 → ((𝐺𝐹) ∈ ( 𝑇m 𝑅) ↔ (𝐺𝐹): 𝑅 𝑇))
159, 14mpbird 257 . 2 (𝜑 → (𝐺𝐹) ∈ ( 𝑇m 𝑅))
16 cnvco 5899 . . . . . 6 (𝐺𝐹) = (𝐹𝐺)
1716imaeq1i 6077 . . . . 5 ((𝐺𝐹) “ 𝑎) = ((𝐹𝐺) “ 𝑎)
18 imaco 6273 . . . . 5 ((𝐹𝐺) “ 𝑎) = (𝐹 “ (𝐺𝑎))
1917, 18eqtri 2763 . . . 4 ((𝐺𝐹) “ 𝑎) = (𝐹 “ (𝐺𝑎))
205adantr 480 . . . . 5 ((𝜑𝑎𝑇) → 𝑅 ran sigAlgebra)
211adantr 480 . . . . 5 ((𝜑𝑎𝑇) → 𝑆 ran sigAlgebra)
226adantr 480 . . . . 5 ((𝜑𝑎𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆))
232adantr 480 . . . . . 6 ((𝜑𝑎𝑇) → 𝑇 ran sigAlgebra)
243adantr 480 . . . . . 6 ((𝜑𝑎𝑇) → 𝐺 ∈ (𝑆MblFnM𝑇))
25 simpr 484 . . . . . 6 ((𝜑𝑎𝑇) → 𝑎𝑇)
2621, 23, 24, 25mbfmcnvima 34237 . . . . 5 ((𝜑𝑎𝑇) → (𝐺𝑎) ∈ 𝑆)
2720, 21, 22, 26mbfmcnvima 34237 . . . 4 ((𝜑𝑎𝑇) → (𝐹 “ (𝐺𝑎)) ∈ 𝑅)
2819, 27eqeltrid 2843 . . 3 ((𝜑𝑎𝑇) → ((𝐺𝐹) “ 𝑎) ∈ 𝑅)
2928ralrimiva 3144 . 2 (𝜑 → ∀𝑎𝑇 ((𝐺𝐹) “ 𝑎) ∈ 𝑅)
305, 2ismbfm 34232 . 2 (𝜑 → ((𝐺𝐹) ∈ (𝑅MblFnM𝑇) ↔ ((𝐺𝐹) ∈ ( 𝑇m 𝑅) ∧ ∀𝑎𝑇 ((𝐺𝐹) “ 𝑎) ∈ 𝑅)))
3115, 29, 30mpbir2and 713 1 (𝜑 → (𝐺𝐹) ∈ (𝑅MblFnM𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wral 3059   cuni 4912  ccnv 5688  ran crn 5690  cima 5692  ccom 5693  wf 6559  (class class class)co 7431  m cmap 8865  sigAlgebracsiga 34089  MblFnMcmbfm 34230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-siga 34090  df-mbfm 34231
This theorem is referenced by:  rrvadd  34434  rrvmulc  34435
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