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Theorem mbfmco2 30455
Description: The pair building of two measurable functions is measurable. ( cf. cnmpt1t 21516). (Contributed by Thierry Arnoux, 6-Jun-2017.)
Hypotheses
Ref Expression
mbfmco.1 (𝜑𝑅 ran sigAlgebra)
mbfmco.2 (𝜑𝑆 ran sigAlgebra)
mbfmco.3 (𝜑𝑇 ran sigAlgebra)
mbfmco2.4 (𝜑𝐹 ∈ (𝑅MblFnM𝑆))
mbfmco2.5 (𝜑𝐺 ∈ (𝑅MblFnM𝑇))
mbfmco2.6 𝐻 = (𝑥 𝑅 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
Assertion
Ref Expression
mbfmco2 (𝜑𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))
Distinct variable groups:   𝑥,𝑅   𝑥,𝑆   𝑥,𝑇   𝜑,𝑥   𝑥,𝐹   𝑥,𝐺   𝑥,𝐻

Proof of Theorem mbfmco2
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mbfmco.1 . . . . . . 7 (𝜑𝑅 ran sigAlgebra)
2 mbfmco.2 . . . . . . 7 (𝜑𝑆 ran sigAlgebra)
3 mbfmco2.4 . . . . . . 7 (𝜑𝐹 ∈ (𝑅MblFnM𝑆))
41, 2, 3mbfmf 30445 . . . . . 6 (𝜑𝐹: 𝑅 𝑆)
54ffvelrnda 6399 . . . . 5 ((𝜑𝑥 𝑅) → (𝐹𝑥) ∈ 𝑆)
6 mbfmco.3 . . . . . . 7 (𝜑𝑇 ran sigAlgebra)
7 mbfmco2.5 . . . . . . 7 (𝜑𝐺 ∈ (𝑅MblFnM𝑇))
81, 6, 7mbfmf 30445 . . . . . 6 (𝜑𝐺: 𝑅 𝑇)
98ffvelrnda 6399 . . . . 5 ((𝜑𝑥 𝑅) → (𝐺𝑥) ∈ 𝑇)
10 opelxpi 5182 . . . . 5 (((𝐹𝑥) ∈ 𝑆 ∧ (𝐺𝑥) ∈ 𝑇) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ ( 𝑆 × 𝑇))
115, 9, 10syl2anc 694 . . . 4 ((𝜑𝑥 𝑅) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ ( 𝑆 × 𝑇))
12 sxuni 30384 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
132, 6, 12syl2anc 694 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
1413adantr 480 . . . 4 ((𝜑𝑥 𝑅) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
1511, 14eleqtrd 2732 . . 3 ((𝜑𝑥 𝑅) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑆 ×s 𝑇))
16 mbfmco2.6 . . 3 𝐻 = (𝑥 𝑅 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
1715, 16fmptd 6425 . 2 (𝜑𝐻: 𝑅 (𝑆 ×s 𝑇))
18 eqid 2651 . . . . 5 (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) = (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))
19 vex 3234 . . . . . 6 𝑎 ∈ V
20 vex 3234 . . . . . 6 𝑏 ∈ V
2119, 20xpex 7004 . . . . 5 (𝑎 × 𝑏) ∈ V
2218, 21elrnmpt2 6815 . . . 4 (𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) ↔ ∃𝑎𝑆𝑏𝑇 𝑐 = (𝑎 × 𝑏))
23 simp3 1083 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑐 = (𝑎 × 𝑏))
2423imaeq2d 5501 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (𝐻𝑐) = (𝐻 “ (𝑎 × 𝑏)))
25 simp1 1081 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝜑)
26 simp2l 1107 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑎𝑆)
27 simp2r 1108 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑏𝑇)
284, 8, 16xppreima2 29578 . . . . . . . . . . 11 (𝜑 → (𝐻 “ (𝑎 × 𝑏)) = ((𝐹𝑎) ∩ (𝐺𝑏)))
29283ad2ant1 1102 . . . . . . . . . 10 ((𝜑𝑎𝑆𝑏𝑇) → (𝐻 “ (𝑎 × 𝑏)) = ((𝐹𝑎) ∩ (𝐺𝑏)))
3013ad2ant1 1102 . . . . . . . . . . 11 ((𝜑𝑎𝑆𝑏𝑇) → 𝑅 ran sigAlgebra)
3123ad2ant1 1102 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑆 ran sigAlgebra)
3233ad2ant1 1102 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆))
33 simp2 1082 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑎𝑆)
3430, 31, 32, 33mbfmcnvima 30447 . . . . . . . . . . 11 ((𝜑𝑎𝑆𝑏𝑇) → (𝐹𝑎) ∈ 𝑅)
3563ad2ant1 1102 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑇 ran sigAlgebra)
3673ad2ant1 1102 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝐺 ∈ (𝑅MblFnM𝑇))
37 simp3 1083 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑏𝑇)
3830, 35, 36, 37mbfmcnvima 30447 . . . . . . . . . . 11 ((𝜑𝑎𝑆𝑏𝑇) → (𝐺𝑏) ∈ 𝑅)
39 inelsiga 30326 . . . . . . . . . . 11 ((𝑅 ran sigAlgebra ∧ (𝐹𝑎) ∈ 𝑅 ∧ (𝐺𝑏) ∈ 𝑅) → ((𝐹𝑎) ∩ (𝐺𝑏)) ∈ 𝑅)
4030, 34, 38, 39syl3anc 1366 . . . . . . . . . 10 ((𝜑𝑎𝑆𝑏𝑇) → ((𝐹𝑎) ∩ (𝐺𝑏)) ∈ 𝑅)
4129, 40eqeltrd 2730 . . . . . . . . 9 ((𝜑𝑎𝑆𝑏𝑇) → (𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
4225, 26, 27, 41syl3anc 1366 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
4324, 42eqeltrd 2730 . . . . . . 7 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (𝐻𝑐) ∈ 𝑅)
44433expia 1286 . . . . . 6 ((𝜑 ∧ (𝑎𝑆𝑏𝑇)) → (𝑐 = (𝑎 × 𝑏) → (𝐻𝑐) ∈ 𝑅))
4544rexlimdvva 3067 . . . . 5 (𝜑 → (∃𝑎𝑆𝑏𝑇 𝑐 = (𝑎 × 𝑏) → (𝐻𝑐) ∈ 𝑅))
4645imp 444 . . . 4 ((𝜑 ∧ ∃𝑎𝑆𝑏𝑇 𝑐 = (𝑎 × 𝑏)) → (𝐻𝑐) ∈ 𝑅)
4722, 46sylan2b 491 . . 3 ((𝜑𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))) → (𝐻𝑐) ∈ 𝑅)
4847ralrimiva 2995 . 2 (𝜑 → ∀𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))(𝐻𝑐) ∈ 𝑅)
49 eqid 2651 . . . . 5 ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) = ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))
5049txbasex 21417 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) ∈ V)
512, 6, 50syl2anc 694 . . 3 (𝜑 → ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) ∈ V)
5249sxval 30381 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))))
532, 6, 52syl2anc 694 . . 3 (𝜑 → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))))
5451, 1, 53imambfm 30452 . 2 (𝜑 → (𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)) ↔ (𝐻: 𝑅 (𝑆 ×s 𝑇) ∧ ∀𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))(𝐻𝑐) ∈ 𝑅)))
5517, 48, 54mpbir2and 977 1 (𝜑𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cin 3606  cop 4216   cuni 4468  cmpt 4762   × cxp 5141  ccnv 5142  ran crn 5144  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  sigAlgebracsiga 30298  sigaGencsigagen 30329   ×s csx 30379  MblFnMcmbfm 30440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-ac2 9323
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-card 8803  df-acn 8806  df-ac 8977  df-cda 9028  df-siga 30299  df-sigagen 30330  df-sx 30380  df-mbfm 30441
This theorem is referenced by:  rrvadd  30642
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