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Theorem measres 33220
Description: Building a measure restricted to a smaller sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measres ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ (𝑀 β†Ύ 𝑇) ∈ (measuresβ€˜π‘‡))
Proof of Theorem measres
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1138 . 2 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ 𝑇 ∈ βˆͺ ran sigAlgebra)
2 measfrge0 33201 . . . . 5 (𝑀 ∈ (measuresβ€˜π‘†) β†’ 𝑀:π‘†βŸΆ(0[,]+∞))
323ad2ant1 1134 . . . 4 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ 𝑀:π‘†βŸΆ(0[,]+∞))
4 simp3 1139 . . . 4 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ 𝑇 βŠ† 𝑆)
53, 4fssresd 6759 . . 3 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ (𝑀 β†Ύ 𝑇):π‘‡βŸΆ(0[,]+∞))
6 0elsiga 33112 . . . . 5 (𝑇 ∈ βˆͺ ran sigAlgebra β†’ βˆ… ∈ 𝑇)
7 fvres 6911 . . . . 5 (βˆ… ∈ 𝑇 β†’ ((𝑀 β†Ύ 𝑇)β€˜βˆ…) = (π‘€β€˜βˆ…))
81, 6, 73syl 18 . . . 4 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ ((𝑀 β†Ύ 𝑇)β€˜βˆ…) = (π‘€β€˜βˆ…))
9 measvnul 33204 . . . . 5 (𝑀 ∈ (measuresβ€˜π‘†) β†’ (π‘€β€˜βˆ…) = 0)
1093ad2ant1 1134 . . . 4 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ (π‘€β€˜βˆ…) = 0)
118, 10eqtrd 2773 . . 3 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ ((𝑀 β†Ύ 𝑇)β€˜βˆ…) = 0)
12 simp11 1204 . . . . . . 7 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ 𝑀 ∈ (measuresβ€˜π‘†))
13 simp13 1206 . . . . . . . 8 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ 𝑇 βŠ† 𝑆)
14 simp2 1138 . . . . . . . 8 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ π‘₯ ∈ 𝒫 𝑇)
15 sspw 4614 . . . . . . . . 9 (𝑇 βŠ† 𝑆 β†’ 𝒫 𝑇 βŠ† 𝒫 𝑆)
1615sselda 3983 . . . . . . . 8 ((𝑇 βŠ† 𝑆 ∧ π‘₯ ∈ 𝒫 𝑇) β†’ π‘₯ ∈ 𝒫 𝑆)
1713, 14, 16syl2anc 585 . . . . . . 7 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ π‘₯ ∈ 𝒫 𝑆)
18 simp3 1139 . . . . . . 7 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦))
19 measvun 33207 . . . . . . 7 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ π‘₯ ∈ 𝒫 𝑆 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ (π‘€β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘€β€˜π‘¦))
2012, 17, 18, 19syl3anc 1372 . . . . . 6 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ (π‘€β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘€β€˜π‘¦))
2113ad2ant1 1134 . . . . . . . 8 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ 𝑇 ∈ βˆͺ ran sigAlgebra)
22 simp3l 1202 . . . . . . . 8 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ π‘₯ β‰Ό Ο‰)
23 sigaclcu 33115 . . . . . . . 8 ((𝑇 ∈ βˆͺ ran sigAlgebra ∧ π‘₯ ∈ 𝒫 𝑇 ∧ π‘₯ β‰Ό Ο‰) β†’ βˆͺ π‘₯ ∈ 𝑇)
2421, 14, 22, 23syl3anc 1372 . . . . . . 7 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ βˆͺ π‘₯ ∈ 𝑇)
2524fvresd 6912 . . . . . 6 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ ((𝑀 β†Ύ 𝑇)β€˜βˆͺ π‘₯) = (π‘€β€˜βˆͺ π‘₯))
26 elpwi 4610 . . . . . . . . . . 11 (π‘₯ ∈ 𝒫 𝑇 β†’ π‘₯ βŠ† 𝑇)
2726sselda 3983 . . . . . . . . . 10 ((π‘₯ ∈ 𝒫 𝑇 ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ 𝑇)
2827adantll 713 . . . . . . . . 9 ((((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇) ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ 𝑇)
2928fvresd 6912 . . . . . . . 8 ((((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇) ∧ 𝑦 ∈ π‘₯) β†’ ((𝑀 β†Ύ 𝑇)β€˜π‘¦) = (π‘€β€˜π‘¦))
3029esumeq2dv 33036 . . . . . . 7 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇) β†’ Ξ£*𝑦 ∈ π‘₯((𝑀 β†Ύ 𝑇)β€˜π‘¦) = Ξ£*𝑦 ∈ π‘₯(π‘€β€˜π‘¦))
31303adant3 1133 . . . . . 6 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ Ξ£*𝑦 ∈ π‘₯((𝑀 β†Ύ 𝑇)β€˜π‘¦) = Ξ£*𝑦 ∈ π‘₯(π‘€β€˜π‘¦))
3220, 25, 313eqtr4d 2783 . . . . 5 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇 ∧ (π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦)) β†’ ((𝑀 β†Ύ 𝑇)β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯((𝑀 β†Ύ 𝑇)β€˜π‘¦))
33323expia 1122 . . . 4 (((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇) β†’ ((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ ((𝑀 β†Ύ 𝑇)β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯((𝑀 β†Ύ 𝑇)β€˜π‘¦)))
3433ralrimiva 3147 . . 3 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ βˆ€π‘₯ ∈ 𝒫 𝑇((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ ((𝑀 β†Ύ 𝑇)β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯((𝑀 β†Ύ 𝑇)β€˜π‘¦)))
355, 11, 343jca 1129 . 2 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ ((𝑀 β†Ύ 𝑇):π‘‡βŸΆ(0[,]+∞) ∧ ((𝑀 β†Ύ 𝑇)β€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑇((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ ((𝑀 β†Ύ 𝑇)β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯((𝑀 β†Ύ 𝑇)β€˜π‘¦))))
36 ismeas 33197 . . 3 (𝑇 ∈ βˆͺ ran sigAlgebra β†’ ((𝑀 β†Ύ 𝑇) ∈ (measuresβ€˜π‘‡) ↔ ((𝑀 β†Ύ 𝑇):π‘‡βŸΆ(0[,]+∞) ∧ ((𝑀 β†Ύ 𝑇)β€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑇((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ ((𝑀 β†Ύ 𝑇)β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯((𝑀 β†Ύ 𝑇)β€˜π‘¦)))))
3736biimprd 247 . 2 (𝑇 ∈ βˆͺ ran sigAlgebra β†’ (((𝑀 β†Ύ 𝑇):π‘‡βŸΆ(0[,]+∞) ∧ ((𝑀 β†Ύ 𝑇)β€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑇((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ ((𝑀 β†Ύ 𝑇)β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯((𝑀 β†Ύ 𝑇)β€˜π‘¦))) β†’ (𝑀 β†Ύ 𝑇) ∈ (measuresβ€˜π‘‡)))
381, 35, 37sylc 65 1 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ (𝑀 β†Ύ 𝑇) ∈ (measuresβ€˜π‘‡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆͺ cuni 4909  Disj wdisj 5114   class class class wbr 5149  ran crn 5678   β†Ύ cres 5679  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Ο‰com 7855   β‰Ό cdom 8937  0cc0 11110  +∞cpnf 11245  [,]cicc 13327  Ξ£*cesum 33025  sigAlgebracsiga 33106  measurescmeas 33193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-esum 33026  df-siga 33107  df-meas 33194
This theorem is referenced by:  measinb2  33221
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