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Theorem mbfmco2 32865
Description: The pair building of two measurable functions is measurable. ( cf. cnmpt1t 23016). (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 32853 . . . . . 6 (𝜑𝐹: 𝑅 𝑆)
54ffvelcdmda 7035 . . . . 5 ((𝜑𝑥 𝑅) → (𝐹𝑥) ∈ 𝑆)
6 mbfmco.3 . . . . . . 7 (𝜑𝑇 ran sigAlgebra)
7 mbfmco2.5 . . . . . . 7 (𝜑𝐺 ∈ (𝑅MblFnM𝑇))
81, 6, 7mbfmf 32853 . . . . . 6 (𝜑𝐺: 𝑅 𝑇)
98ffvelcdmda 7035 . . . . 5 ((𝜑𝑥 𝑅) → (𝐺𝑥) ∈ 𝑇)
10 opelxpi 5670 . . . . 5 (((𝐹𝑥) ∈ 𝑆 ∧ (𝐺𝑥) ∈ 𝑇) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ ( 𝑆 × 𝑇))
115, 9, 10syl2anc 584 . . . 4 ((𝜑𝑥 𝑅) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ ( 𝑆 × 𝑇))
12 sxuni 32792 . . . . . 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 2840 . . 3 ((𝜑𝑥 𝑅) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑆 ×s 𝑇))
16 mbfmco2.6 . . 3 𝐻 = (𝑥 𝑅 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
1715, 16fmptd 7062 . 2 (𝜑𝐻: 𝑅 (𝑆 ×s 𝑇))
18 eqid 2736 . . . . 5 (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) = (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))
19 vex 3449 . . . . . 6 𝑎 ∈ V
20 vex 3449 . . . . . 6 𝑏 ∈ V
2119, 20xpex 7687 . . . . 5 (𝑎 × 𝑏) ∈ V
2218, 21elrnmpo 7492 . . . 4 (𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) ↔ ∃𝑎𝑆𝑏𝑇 𝑐 = (𝑎 × 𝑏))
23 simp3 1138 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑐 = (𝑎 × 𝑏))
2423imaeq2d 6013 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (𝐻𝑐) = (𝐻 “ (𝑎 × 𝑏)))
25 simp1 1136 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝜑)
26 simp2l 1199 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑎𝑆)
27 simp2r 1200 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑏𝑇)
284, 8, 16xppreima2 31567 . . . . . . . . . . 11 (𝜑 → (𝐻 “ (𝑎 × 𝑏)) = ((𝐹𝑎) ∩ (𝐺𝑏)))
29283ad2ant1 1133 . . . . . . . . . 10 ((𝜑𝑎𝑆𝑏𝑇) → (𝐻 “ (𝑎 × 𝑏)) = ((𝐹𝑎) ∩ (𝐺𝑏)))
3013ad2ant1 1133 . . . . . . . . . . 11 ((𝜑𝑎𝑆𝑏𝑇) → 𝑅 ran sigAlgebra)
3123ad2ant1 1133 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑆 ran sigAlgebra)
3233ad2ant1 1133 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆))
33 simp2 1137 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑎𝑆)
3430, 31, 32, 33mbfmcnvima 32855 . . . . . . . . . . 11 ((𝜑𝑎𝑆𝑏𝑇) → (𝐹𝑎) ∈ 𝑅)
3563ad2ant1 1133 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑇 ran sigAlgebra)
3673ad2ant1 1133 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝐺 ∈ (𝑅MblFnM𝑇))
37 simp3 1138 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑏𝑇)
3830, 35, 36, 37mbfmcnvima 32855 . . . . . . . . . . 11 ((𝜑𝑎𝑆𝑏𝑇) → (𝐺𝑏) ∈ 𝑅)
39 inelsiga 32734 . . . . . . . . . . 11 ((𝑅 ran sigAlgebra ∧ (𝐹𝑎) ∈ 𝑅 ∧ (𝐺𝑏) ∈ 𝑅) → ((𝐹𝑎) ∩ (𝐺𝑏)) ∈ 𝑅)
4030, 34, 38, 39syl3anc 1371 . . . . . . . . . 10 ((𝜑𝑎𝑆𝑏𝑇) → ((𝐹𝑎) ∩ (𝐺𝑏)) ∈ 𝑅)
4129, 40eqeltrd 2838 . . . . . . . . 9 ((𝜑𝑎𝑆𝑏𝑇) → (𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
4225, 26, 27, 41syl3anc 1371 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
4324, 42eqeltrd 2838 . . . . . . 7 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (𝐻𝑐) ∈ 𝑅)
44433expia 1121 . . . . . 6 ((𝜑 ∧ (𝑎𝑆𝑏𝑇)) → (𝑐 = (𝑎 × 𝑏) → (𝐻𝑐) ∈ 𝑅))
4544rexlimdvva 3205 . . . . 5 (𝜑 → (∃𝑎𝑆𝑏𝑇 𝑐 = (𝑎 × 𝑏) → (𝐻𝑐) ∈ 𝑅))
4645imp 407 . . . 4 ((𝜑 ∧ ∃𝑎𝑆𝑏𝑇 𝑐 = (𝑎 × 𝑏)) → (𝐻𝑐) ∈ 𝑅)
4722, 46sylan2b 594 . . 3 ((𝜑𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))) → (𝐻𝑐) ∈ 𝑅)
4847ralrimiva 3143 . 2 (𝜑 → ∀𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))(𝐻𝑐) ∈ 𝑅)
49 eqid 2736 . . . . 5 ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) = ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))
5049txbasex 22917 . . . 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 32789 . . . 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 32862 . 2 (𝜑 → (𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)) ↔ (𝐻: 𝑅 (𝑆 ×s 𝑇) ∧ ∀𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))(𝐻𝑐) ∈ 𝑅)))
5517, 48, 54mpbir2and 711 1 (𝜑𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  cin 3909  cop 4592   cuni 4865  cmpt 5188   × cxp 5631  ccnv 5632  ran crn 5634  cima 5636  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  sigAlgebracsiga 32707  sigaGencsigagen 32737   ×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  ax-inf2 9577  ax-ac2 10399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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-rmo 3353  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-pss 3929  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-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  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-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  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-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-siga 32708  df-sigagen 32738  df-sx 32788  df-mbfm 32849
This theorem is referenced by:  rrvadd  33052
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