Theorem mbfmco 32131

Description: The composition of two measurable functions is measurable. See cnmpt11 22722. (Contributed by Thierry Arnoux, 4-Jun-2017.)

Hypotheses

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))

Assertion

Ref	Expression
mbfmco	⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇))

Proof of Theorem mbfmco

Dummy variable 𝑎 is distinct from all other variables.

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 32122	. . . 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 32122	. . . 4 ⊢ (𝜑 → 𝐹:∪ 𝑅⟶∪ 𝑆)
8		fco 6608	. . . 4 ⊢ ((𝐺:∪ 𝑆⟶∪ 𝑇 ∧ 𝐹:∪ 𝑅⟶∪ 𝑆) → (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇)
9	4, 7, 8	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇)
10		unielsiga 31996	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇)
11	2, 10	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
12		unielsiga 31996	. . . . 5 ⊢ (𝑅 ∈ ∪ ran sigAlgebra → ∪ 𝑅 ∈ 𝑅)
13	5, 12	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑅 ∈ 𝑅)
14	11, 13	elmapd 8587	. . 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 5783	. . . . . 6 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺)
17	16	imaeq1i 5955	. . . . 5 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = ((◡𝐹 ∘ ◡𝐺) “ 𝑎)
18		imaco 6144	. . . . 5 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎))
19	17, 18	eqtri 2766	. . . 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 32124	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐺 “ 𝑎) ∈ 𝑆)
27	20, 21, 22, 26	mbfmcnvima 32124	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐹 “ (◡𝐺 “ 𝑎)) ∈ 𝑅)
28	19, 27	eqeltrid 2843	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅)
29	28	ralrimiva 3107	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅)
30	5, 2	ismbfm 32119	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ∧ ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅)))
31	15, 29, 30	mpbir2and 709	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3063  ∪ cuni 4836  ^◡ccnv 5579  ran crn 5581   “ cima 5583   ∘ ccom 5584  ⟶wf 6414  (class class class)co 7255   ↑_m cmap 8573  sigAlgebracsiga 31976  MblFnMcmbfm 32117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-siga 31977  df-mbfm 32118

This theorem is referenced by:  rrvadd  32319  rrvmulc  32320