Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1stmbfm Structured version   Unicode version

Theorem 1stmbfm 24612
 Description: The first projection map is measurable with regard to the product sigma algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
Hypotheses
Ref Expression
1stmbfm.1 sigAlgebra
1stmbfm.2 sigAlgebra
Assertion
Ref Expression
1stmbfm ×s MblFnM

Proof of Theorem 1stmbfm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1stres 6370 . . . 4
2 1stmbfm.1 . . . . . 6 sigAlgebra
3 1stmbfm.2 . . . . . 6 sigAlgebra
4 sxuni 24549 . . . . . 6 sigAlgebra sigAlgebra ×s
52, 3, 4syl2anc 644 . . . . 5 ×s
65feq2d 5583 . . . 4 ×s
71, 6mpbii 204 . . 3 ×s
8 unielsiga 24513 . . . . 5 sigAlgebra
92, 8syl 16 . . . 4
10 sxsiga 24547 . . . . . 6 sigAlgebra sigAlgebra ×s sigAlgebra
112, 3, 10syl2anc 644 . . . . 5 ×s sigAlgebra
12 unielsiga 24513 . . . . 5 ×s sigAlgebra ×s ×s
1311, 12syl 16 . . . 4 ×s ×s
14 elmapg 7033 . . . 4 ×s ×s ×s ×s
159, 13, 14syl2anc 644 . . 3 ×s ×s
167, 15mpbird 225 . 2 ×s
17 sgon 24509 . . . . . . . . . . 11 sigAlgebra sigAlgebra
18 sigasspw 24501 . . . . . . . . . . 11 sigAlgebra
19 pwssb 4179 . . . . . . . . . . . 12
2019biimpi 188 . . . . . . . . . . 11
212, 17, 18, 204syl 20 . . . . . . . . . 10
2221r19.21bi 2806 . . . . . . . . 9
23 xpss1 4986 . . . . . . . . 9
2422, 23syl 16 . . . . . . . 8
2524sseld 3349 . . . . . . 7
2625pm4.71rd 618 . . . . . 6
27 ffn 5593 . . . . . . . 8
28 elpreima 5852 . . . . . . . 8
291, 27, 28mp2b 10 . . . . . . 7
30 fvres 5747 . . . . . . . . . 10
3130eleq1d 2504 . . . . . . . . 9
32 1st2nd2 6388 . . . . . . . . . 10
33 xp2nd 6379 . . . . . . . . . 10
34 elxp6 6380 . . . . . . . . . . . 12
35 anass 632 . . . . . . . . . . . 12
36 an32 775 . . . . . . . . . . . 12
3734, 35, 363bitr2i 266 . . . . . . . . . . 11
3837baib 873 . . . . . . . . . 10
3932, 33, 38syl2anc 644 . . . . . . . . 9
4031, 39bitr4d 249 . . . . . . . 8
4140pm5.32i 620 . . . . . . 7
4229, 41bitri 242 . . . . . 6
4326, 42syl6rbbr 257 . . . . 5
4443eqrdv 2436 . . . 4
452adantr 453 . . . . 5 sigAlgebra
463adantr 453 . . . . 5 sigAlgebra
47 simpr 449 . . . . 5
48 eqid 2438 . . . . . . . 8
49 issgon 24508 . . . . . . . . 9 sigAlgebra sigAlgebra
5049biimpri 199 . . . . . . . 8 sigAlgebra sigAlgebra
513, 48, 50sylancl 645 . . . . . . 7 sigAlgebra
52 baselsiga 24500 . . . . . . 7 sigAlgebra
5351, 52syl 16 . . . . . 6
5453adantr 453 . . . . 5
55 elsx 24550 . . . . 5 sigAlgebra sigAlgebra ×s
5645, 46, 47, 54, 55syl22anc 1186 . . . 4 ×s
5744, 56eqeltrd 2512 . . 3 ×s
5857ralrimiva 2791 . 2 ×s
5911, 2ismbfm 24604 . 2 ×s MblFnM ×s ×s
6016, 58, 59mpbir2and 890 1 ×s MblFnM
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707   wss 3322  cpw 3801  cop 3819  cuni 4017   cxp 4878  ccnv 4879   crn 4881   cres 4882  cima 4883   wfn 5451  wf 5452  cfv 5456  (class class class)co 6083  c1st 6349  c2nd 6350   cmap 7020  sigAlgebracsiga 24492   ×s csx 24544  MblFnMcmbfm 24602 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-map 7022  df-siga 24493  df-sigagen 24524  df-sx 24545  df-mbfm 24603
 Copyright terms: Public domain W3C validator