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Theorem mbfmco 34267
Description: The composition of two measurable functions is measurable. See cnmpt11 23672. (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 34256 . . . 4 (𝜑𝐺: 𝑆 𝑇)
5 mbfmco.1 . . . . 5 (𝜑𝑅 ran sigAlgebra)
6 mbfmco.4 . . . . 5 (𝜑𝐹 ∈ (𝑅MblFnM𝑆))
75, 1, 6mbfmf 34256 . . . 4 (𝜑𝐹: 𝑅 𝑆)
8 fco 6759 . . . 4 ((𝐺: 𝑆 𝑇𝐹: 𝑅 𝑆) → (𝐺𝐹): 𝑅 𝑇)
94, 7, 8syl2anc 584 . . 3 (𝜑 → (𝐺𝐹): 𝑅 𝑇)
10 unielsiga 34130 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
112, 10syl 17 . . . 4 (𝜑 𝑇𝑇)
12 unielsiga 34130 . . . . 5 (𝑅 ran sigAlgebra → 𝑅𝑅)
135, 12syl 17 . . . 4 (𝜑 𝑅𝑅)
1411, 13elmapd 8881 . . 3 (𝜑 → ((𝐺𝐹) ∈ ( 𝑇m 𝑅) ↔ (𝐺𝐹): 𝑅 𝑇))
159, 14mpbird 257 . 2 (𝜑 → (𝐺𝐹) ∈ ( 𝑇m 𝑅))
16 cnvco 5895 . . . . . 6 (𝐺𝐹) = (𝐹𝐺)
1716imaeq1i 6074 . . . . 5 ((𝐺𝐹) “ 𝑎) = ((𝐹𝐺) “ 𝑎)
18 imaco 6270 . . . . 5 ((𝐹𝐺) “ 𝑎) = (𝐹 “ (𝐺𝑎))
1917, 18eqtri 2764 . . . 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 34258 . . . . 5 ((𝜑𝑎𝑇) → (𝐺𝑎) ∈ 𝑆)
2720, 21, 22, 26mbfmcnvima 34258 . . . 4 ((𝜑𝑎𝑇) → (𝐹 “ (𝐺𝑎)) ∈ 𝑅)
2819, 27eqeltrid 2844 . . 3 ((𝜑𝑎𝑇) → ((𝐺𝐹) “ 𝑎) ∈ 𝑅)
2928ralrimiva 3145 . 2 (𝜑 → ∀𝑎𝑇 ((𝐺𝐹) “ 𝑎) ∈ 𝑅)
305, 2ismbfm 34253 . 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 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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