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 32861
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 7946 . . . 4 (2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇
2 1stmbfm.1 . . . . . 6 (𝜑𝑆 ran sigAlgebra)
3 1stmbfm.2 . . . . . 6 (𝜑𝑇 ran sigAlgebra)
4 sxuni 32792 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
52, 3, 4syl2anc 584 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
65feq2d 6654 . . . 4 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇 ↔ (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇))
71, 6mpbii 232 . . 3 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇)
8 unielsiga 32727 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
93, 8syl 17 . . . 4 (𝜑 𝑇𝑇)
10 sxsiga 32790 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
112, 3, 10syl2anc 584 . . . . 5 (𝜑 → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
12 unielsiga 32727 . . . . 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 8779 . . 3 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇m (𝑆 ×s 𝑇)) ↔ (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇))
157, 14mpbird 256 . 2 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇m (𝑆 ×s 𝑇)))
16 ffn 6668 . . . . . . . 8 ((2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇 → (2nd ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇))
17 elpreima 7008 . . . . . . . 8 ((2nd ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇) → (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎)))
181, 16, 17mp2b 10 . . . . . . 7 (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎))
19 fvres 6861 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) = (2nd𝑧))
2019eleq1d 2822 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎 ↔ (2nd𝑧) ∈ 𝑎))
21 1st2nd2 7960 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
22 xp1st 7953 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → (1st𝑧) ∈ 𝑆)
23 elxp6 7955 . . . . . . . . . . . 12 (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑎)))
24 anass 469 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) ∧ (2nd𝑧) ∈ 𝑎) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑎)))
2523, 24bitr4i 277 . . . . . . . . . . 11 (𝑧 ∈ ( 𝑆 × 𝑎) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) ∧ (2nd𝑧) ∈ 𝑎))
2625baib 536 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (2nd𝑧) ∈ 𝑎))
2721, 22, 26syl2anc 584 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (2nd𝑧) ∈ 𝑎))
2820, 27bitr4d 281 . . . . . . . 8 (𝑧 ∈ ( 𝑆 × 𝑇) → (((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎𝑧 ∈ ( 𝑆 × 𝑎)))
2928pm5.32i 575 . . . . . . 7 ((𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎)))
3018, 29bitri 274 . . . . . 6 (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎)))
31 sgon 32723 . . . . . . . . . . 11 (𝑇 ran sigAlgebra → 𝑇 ∈ (sigAlgebra‘ 𝑇))
32 sigasspw 32715 . . . . . . . . . . 11 (𝑇 ∈ (sigAlgebra‘ 𝑇) → 𝑇 ⊆ 𝒫 𝑇)
33 pwssb 5061 . . . . . . . . . . . 12 (𝑇 ⊆ 𝒫 𝑇 ↔ ∀𝑎𝑇 𝑎 𝑇)
3433biimpi 215 . . . . . . . . . . 11 (𝑇 ⊆ 𝒫 𝑇 → ∀𝑎𝑇 𝑎 𝑇)
353, 31, 32, 344syl 19 . . . . . . . . . 10 (𝜑 → ∀𝑎𝑇 𝑎 𝑇)
3635r19.21bi 3234 . . . . . . . . 9 ((𝜑𝑎𝑇) → 𝑎 𝑇)
37 xpss2 5653 . . . . . . . . 9 (𝑎 𝑇 → ( 𝑆 × 𝑎) ⊆ ( 𝑆 × 𝑇))
3836, 37syl 17 . . . . . . . 8 ((𝜑𝑎𝑇) → ( 𝑆 × 𝑎) ⊆ ( 𝑆 × 𝑇))
3938sseld 3943 . . . . . . 7 ((𝜑𝑎𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) → 𝑧 ∈ ( 𝑆 × 𝑇)))
4039pm4.71rd 563 . . . . . 6 ((𝜑𝑎𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎))))
4130, 40bitr4id 289 . . . . 5 ((𝜑𝑎𝑇) → (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ 𝑧 ∈ ( 𝑆 × 𝑎)))
4241eqrdv 2734 . . . 4 ((𝜑𝑎𝑇) → ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) = ( 𝑆 × 𝑎))
432adantr 481 . . . . 5 ((𝜑𝑎𝑇) → 𝑆 ran sigAlgebra)
443adantr 481 . . . . 5 ((𝜑𝑎𝑇) → 𝑇 ran sigAlgebra)
45 eqid 2736 . . . . . . . 8 𝑆 = 𝑆
46 issgon 32722 . . . . . . . 8 (𝑆 ∈ (sigAlgebra‘ 𝑆) ↔ (𝑆 ran sigAlgebra ∧ 𝑆 = 𝑆))
472, 45, 46sylanblrc 590 . . . . . . 7 (𝜑𝑆 ∈ (sigAlgebra‘ 𝑆))
48 baselsiga 32714 . . . . . . 7 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆𝑆)
4947, 48syl 17 . . . . . 6 (𝜑 𝑆𝑆)
5049adantr 481 . . . . 5 ((𝜑𝑎𝑇) → 𝑆𝑆)
51 simpr 485 . . . . 5 ((𝜑𝑎𝑇) → 𝑎𝑇)
52 elsx 32793 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) ∧ ( 𝑆𝑆𝑎𝑇)) → ( 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
5343, 44, 50, 51, 52syl22anc 837 . . . 4 ((𝜑𝑎𝑇) → ( 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
5442, 53eqeltrd 2838 . . 3 ((𝜑𝑎𝑇) → ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5554ralrimiva 3143 . 2 (𝜑 → ∀𝑎𝑇 ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5611, 3ismbfm 32850 . 2 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇) ↔ ((2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇m (𝑆 ×s 𝑇)) ∧ ∀𝑎𝑇 ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))))
5715, 55, 56mpbir2and 711 1 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wss 3910  𝒫 cpw 4560  cop 4592   cuni 4865   × cxp 5631  ccnv 5632  ran crn 5634  cres 5635  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  m cmap 8765  sigAlgebracsiga 32707   ×s csx 32787  MblFnMcmbfm 32848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-siga 32708  df-sigagen 32738  df-sx 32788  df-mbfm 32849
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator