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Theorem mbfmco2 30861
Description: The pair building of two measurable functions is measurable. ( cf. cnmpt1t 21839). (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 30851 . . . . . 6 (𝜑𝐹: 𝑅 𝑆)
54ffvelrnda 6608 . . . . 5 ((𝜑𝑥 𝑅) → (𝐹𝑥) ∈ 𝑆)
6 mbfmco.3 . . . . . . 7 (𝜑𝑇 ran sigAlgebra)
7 mbfmco2.5 . . . . . . 7 (𝜑𝐺 ∈ (𝑅MblFnM𝑇))
81, 6, 7mbfmf 30851 . . . . . 6 (𝜑𝐺: 𝑅 𝑇)
98ffvelrnda 6608 . . . . 5 ((𝜑𝑥 𝑅) → (𝐺𝑥) ∈ 𝑇)
10 opelxpi 5379 . . . . 5 (((𝐹𝑥) ∈ 𝑆 ∧ (𝐺𝑥) ∈ 𝑇) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ ( 𝑆 × 𝑇))
115, 9, 10syl2anc 579 . . . 4 ((𝜑𝑥 𝑅) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ ( 𝑆 × 𝑇))
12 sxuni 30790 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
132, 6, 12syl2anc 579 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
1413adantr 474 . . . 4 ((𝜑𝑥 𝑅) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
1511, 14eleqtrd 2908 . . 3 ((𝜑𝑥 𝑅) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑆 ×s 𝑇))
16 mbfmco2.6 . . 3 𝐻 = (𝑥 𝑅 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
1715, 16fmptd 6633 . 2 (𝜑𝐻: 𝑅 (𝑆 ×s 𝑇))
18 eqid 2825 . . . . 5 (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) = (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))
19 vex 3417 . . . . . 6 𝑎 ∈ V
20 vex 3417 . . . . . 6 𝑏 ∈ V
2119, 20xpex 7223 . . . . 5 (𝑎 × 𝑏) ∈ V
2218, 21elrnmpt2 7033 . . . 4 (𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) ↔ ∃𝑎𝑆𝑏𝑇 𝑐 = (𝑎 × 𝑏))
23 simp3 1172 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑐 = (𝑎 × 𝑏))
2423imaeq2d 5707 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (𝐻𝑐) = (𝐻 “ (𝑎 × 𝑏)))
25 simp1 1170 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝜑)
26 simp2l 1260 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑎𝑆)
27 simp2r 1261 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑏𝑇)
284, 8, 16xppreima2 29988 . . . . . . . . . . 11 (𝜑 → (𝐻 “ (𝑎 × 𝑏)) = ((𝐹𝑎) ∩ (𝐺𝑏)))
29283ad2ant1 1167 . . . . . . . . . 10 ((𝜑𝑎𝑆𝑏𝑇) → (𝐻 “ (𝑎 × 𝑏)) = ((𝐹𝑎) ∩ (𝐺𝑏)))
3013ad2ant1 1167 . . . . . . . . . . 11 ((𝜑𝑎𝑆𝑏𝑇) → 𝑅 ran sigAlgebra)
3123ad2ant1 1167 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑆 ran sigAlgebra)
3233ad2ant1 1167 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆))
33 simp2 1171 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑎𝑆)
3430, 31, 32, 33mbfmcnvima 30853 . . . . . . . . . . 11 ((𝜑𝑎𝑆𝑏𝑇) → (𝐹𝑎) ∈ 𝑅)
3563ad2ant1 1167 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑇 ran sigAlgebra)
3673ad2ant1 1167 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝐺 ∈ (𝑅MblFnM𝑇))
37 simp3 1172 . . . . . . . . . . . 12 ((𝜑𝑎𝑆𝑏𝑇) → 𝑏𝑇)
3830, 35, 36, 37mbfmcnvima 30853 . . . . . . . . . . 11 ((𝜑𝑎𝑆𝑏𝑇) → (𝐺𝑏) ∈ 𝑅)
39 inelsiga 30732 . . . . . . . . . . 11 ((𝑅 ran sigAlgebra ∧ (𝐹𝑎) ∈ 𝑅 ∧ (𝐺𝑏) ∈ 𝑅) → ((𝐹𝑎) ∩ (𝐺𝑏)) ∈ 𝑅)
4030, 34, 38, 39syl3anc 1494 . . . . . . . . . 10 ((𝜑𝑎𝑆𝑏𝑇) → ((𝐹𝑎) ∩ (𝐺𝑏)) ∈ 𝑅)
4129, 40eqeltrd 2906 . . . . . . . . 9 ((𝜑𝑎𝑆𝑏𝑇) → (𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
4225, 26, 27, 41syl3anc 1494 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
4324, 42eqeltrd 2906 . . . . . . 7 ((𝜑 ∧ (𝑎𝑆𝑏𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (𝐻𝑐) ∈ 𝑅)
44433expia 1154 . . . . . 6 ((𝜑 ∧ (𝑎𝑆𝑏𝑇)) → (𝑐 = (𝑎 × 𝑏) → (𝐻𝑐) ∈ 𝑅))
4544rexlimdvva 3248 . . . . 5 (𝜑 → (∃𝑎𝑆𝑏𝑇 𝑐 = (𝑎 × 𝑏) → (𝐻𝑐) ∈ 𝑅))
4645imp 397 . . . 4 ((𝜑 ∧ ∃𝑎𝑆𝑏𝑇 𝑐 = (𝑎 × 𝑏)) → (𝐻𝑐) ∈ 𝑅)
4722, 46sylan2b 587 . . 3 ((𝜑𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))) → (𝐻𝑐) ∈ 𝑅)
4847ralrimiva 3175 . 2 (𝜑 → ∀𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))(𝐻𝑐) ∈ 𝑅)
49 eqid 2825 . . . . 5 ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) = ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))
5049txbasex 21740 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) ∈ V)
512, 6, 50syl2anc 579 . . 3 (𝜑 → ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏)) ∈ V)
5249sxval 30787 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))))
532, 6, 52syl2anc 579 . . 3 (𝜑 → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))))
5451, 1, 53imambfm 30858 . 2 (𝜑 → (𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)) ↔ (𝐻: 𝑅 (𝑆 ×s 𝑇) ∧ ∀𝑐 ∈ ran (𝑎𝑆, 𝑏𝑇 ↦ (𝑎 × 𝑏))(𝐻𝑐) ∈ 𝑅)))
5517, 48, 54mpbir2and 704 1 (𝜑𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1111   = wceq 1656  wcel 2164  wral 3117  wrex 3118  Vcvv 3414  cin 3797  cop 4403   cuni 4658  cmpt 4952   × cxp 5340  ccnv 5341  ran crn 5343  cima 5345  wf 6119  cfv 6123  (class class class)co 6905  cmpt2 6907  sigAlgebracsiga 30704  sigaGencsigagen 30735   ×s csx 30785  MblFnMcmbfm 30846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815  ax-ac2 9600
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-map 8124  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-oi 8684  df-card 9078  df-acn 9081  df-ac 9252  df-cda 9305  df-siga 30705  df-sigagen 30736  df-sx 30786  df-mbfm 30847
This theorem is referenced by:  rrvadd  31049
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