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Theorem mbfmco2 31422
Description: The pair building of two measurable functions is measurable. ( cf. cnmpt1t 22201). (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 31412 . . . . . 6 (𝜑𝐹: 𝑅 𝑆)
54ffvelrnda 6843 . . . . 5 ((𝜑𝑥 𝑅) → (𝐹𝑥) ∈ 𝑆)
6 mbfmco.3 . . . . . . 7 (𝜑𝑇 ran sigAlgebra)
7 mbfmco2.5 . . . . . . 7 (𝜑𝐺 ∈ (𝑅MblFnM𝑇))
81, 6, 7mbfmf 31412 . . . . . 6 (𝜑𝐺: 𝑅 𝑇)
98ffvelrnda 6843 . . . . 5 ((𝜑𝑥 𝑅) → (𝐺𝑥) ∈ 𝑇)
10 opelxpi 5585 . . . . 5 (((𝐹𝑥) ∈ 𝑆 ∧ (𝐺𝑥) ∈ 𝑇) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ ( 𝑆 × 𝑇))
115, 9, 10syl2anc 584 . . . 4 ((𝜑𝑥 𝑅) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ ( 𝑆 × 𝑇))
12 sxuni 31351 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
132, 6, 12syl2anc 584 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
1413adantr 481 . . . 4 ((𝜑𝑥 𝑅) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
1511, 14eleqtrd 2912 . . 3 ((𝜑𝑥 𝑅) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑆 ×s 𝑇))
16 mbfmco2.6 . . 3 𝐻 = (𝑥 𝑅 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
1715, 16fmptd 6870 . 2 (𝜑𝐻: 𝑅 (𝑆 ×s 𝑇))
18 eqid 2818 . . . . 5 (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) = (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))
19 vex 3495 . . . . . 6 𝑎 ∈ V
20 vex 3495 . . . . . 6 𝑏 ∈ V
2119, 20xpex 7465 . . . . 5 (𝑎 × 𝑏) ∈ V
2218, 21elrnmpo 7276 . . . 4 (𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) ↔ ∃𝑎𝑆𝑏𝑇 𝑐 = (𝑎 × 𝑏))
23 simp3 1130 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑐 = (𝑎 × 𝑏))
2423imaeq2d 5922 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (𝐻𝑐) = (𝐻 “ (𝑎 × 𝑏)))
25 simp1 1128 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝜑)
26 simp2l 1191 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑎𝑆)
27 simp2r 1192 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑏𝑇)
284, 8, 16xppreima2 30323 . . . . . . . . . . 11 (𝜑 → (𝐻 “ (𝑎 × 𝑏)) = ((𝐹𝑎) ∩ (𝐺𝑏)))
29283ad2ant1 1125 . . . . . . . . . 10 ((𝜑𝑎𝑆𝑏𝑇) → (𝐻 “ (𝑎 × 𝑏)) = ((𝐹𝑎) ∩ (𝐺𝑏)))
3013ad2ant1 1125 . . . . . . . . . . 11 ((𝜑𝑎𝑆𝑏𝑇) → 𝑅 ran sigAlgebra)
3123ad2ant1 1125 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑆 ran sigAlgebra)
3233ad2ant1 1125 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆))
33 simp2 1129 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑎𝑆)
3430, 31, 32, 33mbfmcnvima 31414 . . . . . . . . . . 11 ((𝜑𝑎𝑆𝑏𝑇) → (𝐹𝑎) ∈ 𝑅)
3563ad2ant1 1125 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑇 ran sigAlgebra)
3673ad2ant1 1125 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝐺 ∈ (𝑅MblFnM𝑇))
37 simp3 1130 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑏𝑇)
3830, 35, 36, 37mbfmcnvima 31414 . . . . . . . . . . 11 ((𝜑𝑎𝑆𝑏𝑇) → (𝐺𝑏) ∈ 𝑅)
39 inelsiga 31293 . . . . . . . . . . 11 ((𝑅 ran sigAlgebra ∧ (𝐹𝑎) ∈ 𝑅 ∧ (𝐺𝑏) ∈ 𝑅) → ((𝐹𝑎) ∩ (𝐺𝑏)) ∈ 𝑅)
4030, 34, 38, 39syl3anc 1363 . . . . . . . . . 10 ((𝜑𝑎𝑆𝑏𝑇) → ((𝐹𝑎) ∩ (𝐺𝑏)) ∈ 𝑅)
4129, 40eqeltrd 2910 . . . . . . . . 9 ((𝜑𝑎𝑆𝑏𝑇) → (𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
4225, 26, 27, 41syl3anc 1363 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
4324, 42eqeltrd 2910 . . . . . . 7 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (𝐻𝑐) ∈ 𝑅)
44433expia 1113 . . . . . 6 ((𝜑 ∧ (𝑎𝑆𝑏𝑇)) → (𝑐 = (𝑎 × 𝑏) → (𝐻𝑐) ∈ 𝑅))
4544rexlimdvva 3291 . . . . 5 (𝜑 → (∃𝑎𝑆𝑏𝑇 𝑐 = (𝑎 × 𝑏) → (𝐻𝑐) ∈ 𝑅))
4645imp 407 . . . 4 ((𝜑 ∧ ∃𝑎𝑆𝑏𝑇 𝑐 = (𝑎 × 𝑏)) → (𝐻𝑐) ∈ 𝑅)
4722, 46sylan2b 593 . . 3 ((𝜑𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))) → (𝐻𝑐) ∈ 𝑅)
4847ralrimiva 3179 . 2 (𝜑 → ∀𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))(𝐻𝑐) ∈ 𝑅)
49 eqid 2818 . . . . 5 ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) = ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))
5049txbasex 22102 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) ∈ V)
512, 6, 50syl2anc 584 . . 3 (𝜑 → ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) ∈ V)
5249sxval 31348 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))))
532, 6, 52syl2anc 584 . . 3 (𝜑 → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))))
5451, 1, 53imambfm 31419 . 2 (𝜑 → (𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)) ↔ (𝐻: 𝑅 (𝑆 ×s 𝑇) ∧ ∀𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))(𝐻𝑐) ∈ 𝑅)))
5517, 48, 54mpbir2and 709 1 (𝜑𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  cin 3932  cop 4563   cuni 4830  cmpt 5137   × cxp 5546  ccnv 5547  ran crn 5549  cima 5551  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  sigAlgebracsiga 31266  sigaGencsigagen 31296   ×s csx 31346  MblFnMcmbfm 31407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-ac2 9873
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-ac 9530  df-siga 31267  df-sigagen 31297  df-sx 31347  df-mbfm 31408
This theorem is referenced by:  rrvadd  31609
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