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Theorem mbfmco 30311
Description: The composition of two measurable functions is measurable. ( cf. cnmpt11 21460) (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 30302 . . . 4 (𝜑𝐺: 𝑆 𝑇)
5 mbfmco.1 . . . . 5 (𝜑𝑅 ran sigAlgebra)
6 mbfmco.4 . . . . 5 (𝜑𝐹 ∈ (𝑅MblFnM𝑆))
75, 1, 6mbfmf 30302 . . . 4 (𝜑𝐹: 𝑅 𝑆)
8 fco 6056 . . . 4 ((𝐺: 𝑆 𝑇𝐹: 𝑅 𝑆) → (𝐺𝐹): 𝑅 𝑇)
94, 7, 8syl2anc 693 . . 3 (𝜑 → (𝐺𝐹): 𝑅 𝑇)
10 unielsiga 30176 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
112, 10syl 17 . . . 4 (𝜑 𝑇𝑇)
12 unielsiga 30176 . . . . 5 (𝑅 ran sigAlgebra → 𝑅𝑅)
135, 12syl 17 . . . 4 (𝜑 𝑅𝑅)
1411, 13elmapd 7868 . . 3 (𝜑 → ((𝐺𝐹) ∈ ( 𝑇𝑚 𝑅) ↔ (𝐺𝐹): 𝑅 𝑇))
159, 14mpbird 247 . 2 (𝜑 → (𝐺𝐹) ∈ ( 𝑇𝑚 𝑅))
16 cnvco 5306 . . . . . 6 (𝐺𝐹) = (𝐹𝐺)
1716imaeq1i 5461 . . . . 5 ((𝐺𝐹) “ 𝑎) = ((𝐹𝐺) “ 𝑎)
18 imaco 5638 . . . . 5 ((𝐹𝐺) “ 𝑎) = (𝐹 “ (𝐺𝑎))
1917, 18eqtri 2643 . . . 4 ((𝐺𝐹) “ 𝑎) = (𝐹 “ (𝐺𝑎))
205adantr 481 . . . . 5 ((𝜑𝑎𝑇) → 𝑅 ran sigAlgebra)
211adantr 481 . . . . 5 ((𝜑𝑎𝑇) → 𝑆 ran sigAlgebra)
226adantr 481 . . . . 5 ((𝜑𝑎𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆))
232adantr 481 . . . . . 6 ((𝜑𝑎𝑇) → 𝑇 ran sigAlgebra)
243adantr 481 . . . . . 6 ((𝜑𝑎𝑇) → 𝐺 ∈ (𝑆MblFnM𝑇))
25 simpr 477 . . . . . 6 ((𝜑𝑎𝑇) → 𝑎𝑇)
2621, 23, 24, 25mbfmcnvima 30304 . . . . 5 ((𝜑𝑎𝑇) → (𝐺𝑎) ∈ 𝑆)
2720, 21, 22, 26mbfmcnvima 30304 . . . 4 ((𝜑𝑎𝑇) → (𝐹 “ (𝐺𝑎)) ∈ 𝑅)
2819, 27syl5eqel 2704 . . 3 ((𝜑𝑎𝑇) → ((𝐺𝐹) “ 𝑎) ∈ 𝑅)
2928ralrimiva 2965 . 2 (𝜑 → ∀𝑎𝑇 ((𝐺𝐹) “ 𝑎) ∈ 𝑅)
305, 2ismbfm 30299 . 2 (𝜑 → ((𝐺𝐹) ∈ (𝑅MblFnM𝑇) ↔ ((𝐺𝐹) ∈ ( 𝑇𝑚 𝑅) ∧ ∀𝑎𝑇 ((𝐺𝐹) “ 𝑎) ∈ 𝑅)))
3115, 29, 30mpbir2and 957 1 (𝜑 → (𝐺𝐹) ∈ (𝑅MblFnM𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1989  wral 2911   cuni 4434  ccnv 5111  ran crn 5113  cima 5115  ccom 5116  wf 5882  (class class class)co 6647  𝑚 cmap 7854  sigAlgebracsiga 30155  MblFnMcmbfm 30297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-map 7856  df-siga 30156  df-mbfm 30298
This theorem is referenced by:  rrvadd  30499  rrvmulc  30500
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