Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2ndmbfm Structured version   Visualization version   GIF version

Theorem 2ndmbfm 29650
Description: The second projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
Hypotheses
Ref Expression
1stmbfm.1 (𝜑𝑆 ran sigAlgebra)
1stmbfm.2 (𝜑𝑇 ran sigAlgebra)
Assertion
Ref Expression
2ndmbfm (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))

Proof of Theorem 2ndmbfm
Dummy variables 𝑧 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f2ndres 7082 . . . 4 (2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇
2 1stmbfm.1 . . . . . 6 (𝜑𝑆 ran sigAlgebra)
3 1stmbfm.2 . . . . . 6 (𝜑𝑇 ran sigAlgebra)
4 sxuni 29583 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
52, 3, 4syl2anc 691 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
65feq2d 5944 . . . 4 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇 ↔ (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇))
71, 6mpbii 222 . . 3 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇)
8 unielsiga 29518 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
93, 8syl 17 . . . 4 (𝜑 𝑇𝑇)
10 sxsiga 29581 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
112, 3, 10syl2anc 691 . . . . 5 (𝜑 → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
12 unielsiga 29518 . . . . 5 ((𝑆 ×s 𝑇) ∈ ran sigAlgebra → (𝑆 ×s 𝑇) ∈ (𝑆 ×s 𝑇))
1311, 12syl 17 . . . 4 (𝜑 (𝑆 ×s 𝑇) ∈ (𝑆 ×s 𝑇))
149, 13elmapd 7758 . . 3 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇𝑚 (𝑆 ×s 𝑇)) ↔ (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇))
157, 14mpbird 246 . 2 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇𝑚 (𝑆 ×s 𝑇)))
16 sgon 29514 . . . . . . . . . . 11 (𝑇 ran sigAlgebra → 𝑇 ∈ (sigAlgebra‘ 𝑇))
17 sigasspw 29506 . . . . . . . . . . 11 (𝑇 ∈ (sigAlgebra‘ 𝑇) → 𝑇 ⊆ 𝒫 𝑇)
18 pwssb 4548 . . . . . . . . . . . 12 (𝑇 ⊆ 𝒫 𝑇 ↔ ∀𝑎𝑇 𝑎 𝑇)
1918biimpi 205 . . . . . . . . . . 11 (𝑇 ⊆ 𝒫 𝑇 → ∀𝑎𝑇 𝑎 𝑇)
203, 16, 17, 194syl 19 . . . . . . . . . 10 (𝜑 → ∀𝑎𝑇 𝑎 𝑇)
2120r19.21bi 2916 . . . . . . . . 9 ((𝜑𝑎𝑇) → 𝑎 𝑇)
22 xpss2 5152 . . . . . . . . 9 (𝑎 𝑇 → ( 𝑆 × 𝑎) ⊆ ( 𝑆 × 𝑇))
2321, 22syl 17 . . . . . . . 8 ((𝜑𝑎𝑇) → ( 𝑆 × 𝑎) ⊆ ( 𝑆 × 𝑇))
2423sseld 3567 . . . . . . 7 ((𝜑𝑎𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) → 𝑧 ∈ ( 𝑆 × 𝑇)))
2524pm4.71rd 665 . . . . . 6 ((𝜑𝑎𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎))))
26 ffn 5958 . . . . . . . 8 ((2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇 → (2nd ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇))
27 elpreima 6245 . . . . . . . 8 ((2nd ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇) → (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎)))
281, 26, 27mp2b 10 . . . . . . 7 (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎))
29 fvres 6117 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) = (2nd𝑧))
3029eleq1d 2672 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎 ↔ (2nd𝑧) ∈ 𝑎))
31 1st2nd2 7096 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
32 xp1st 7089 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → (1st𝑧) ∈ 𝑆)
33 elxp6 7091 . . . . . . . . . . . 12 (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑎)))
34 anass 679 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) ∧ (2nd𝑧) ∈ 𝑎) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑎)))
3533, 34bitr4i 266 . . . . . . . . . . 11 (𝑧 ∈ ( 𝑆 × 𝑎) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) ∧ (2nd𝑧) ∈ 𝑎))
3635baib 942 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (2nd𝑧) ∈ 𝑎))
3731, 32, 36syl2anc 691 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (2nd𝑧) ∈ 𝑎))
3830, 37bitr4d 270 . . . . . . . 8 (𝑧 ∈ ( 𝑆 × 𝑇) → (((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎𝑧 ∈ ( 𝑆 × 𝑎)))
3938pm5.32i 667 . . . . . . 7 ((𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎)))
4028, 39bitri 263 . . . . . 6 (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎)))
4125, 40syl6rbbr 278 . . . . 5 ((𝜑𝑎𝑇) → (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ 𝑧 ∈ ( 𝑆 × 𝑎)))
4241eqrdv 2608 . . . 4 ((𝜑𝑎𝑇) → ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) = ( 𝑆 × 𝑎))
432adantr 480 . . . . 5 ((𝜑𝑎𝑇) → 𝑆 ran sigAlgebra)
443adantr 480 . . . . 5 ((𝜑𝑎𝑇) → 𝑇 ran sigAlgebra)
45 eqid 2610 . . . . . . . 8 𝑆 = 𝑆
46 issgon 29513 . . . . . . . . 9 (𝑆 ∈ (sigAlgebra‘ 𝑆) ↔ (𝑆 ran sigAlgebra ∧ 𝑆 = 𝑆))
4746biimpri 217 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝑆 = 𝑆) → 𝑆 ∈ (sigAlgebra‘ 𝑆))
482, 45, 47sylancl 693 . . . . . . 7 (𝜑𝑆 ∈ (sigAlgebra‘ 𝑆))
49 baselsiga 29505 . . . . . . 7 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆𝑆)
5048, 49syl 17 . . . . . 6 (𝜑 𝑆𝑆)
5150adantr 480 . . . . 5 ((𝜑𝑎𝑇) → 𝑆𝑆)
52 simpr 476 . . . . 5 ((𝜑𝑎𝑇) → 𝑎𝑇)
53 elsx 29584 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) ∧ ( 𝑆𝑆𝑎𝑇)) → ( 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
5443, 44, 51, 52, 53syl22anc 1319 . . . 4 ((𝜑𝑎𝑇) → ( 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
5542, 54eqeltrd 2688 . . 3 ((𝜑𝑎𝑇) → ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5655ralrimiva 2949 . 2 (𝜑 → ∀𝑎𝑇 ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5711, 3ismbfm 29641 . 2 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇) ↔ ((2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇𝑚 (𝑆 ×s 𝑇)) ∧ ∀𝑎𝑇 ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))))
5815, 56, 57mpbir2and 959 1 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wss 3540  𝒫 cpw 4108  cop 4131   cuni 4372   × cxp 5036  ccnv 5037  ran crn 5039  cres 5040  cima 5041   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  𝑚 cmap 7744  sigAlgebracsiga 29497   ×s csx 29578  MblFnMcmbfm 29639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-siga 29498  df-sigagen 29529  df-sx 29579  df-mbfm 29640
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator