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Theorem measres 33518
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 1135 . 2 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ 𝑇 ∈ βˆͺ ran sigAlgebra)
2 measfrge0 33499 . . . . 5 (𝑀 ∈ (measuresβ€˜π‘†) β†’ 𝑀:π‘†βŸΆ(0[,]+∞))
323ad2ant1 1131 . . . 4 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ 𝑀:π‘†βŸΆ(0[,]+∞))
4 simp3 1136 . . . 4 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ 𝑇 βŠ† 𝑆)
53, 4fssresd 6757 . . 3 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ (𝑀 β†Ύ 𝑇):π‘‡βŸΆ(0[,]+∞))
6 0elsiga 33410 . . . . 5 (𝑇 ∈ βˆͺ ran sigAlgebra β†’ βˆ… ∈ 𝑇)
7 fvres 6909 . . . . 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 33502 . . . . 5 (𝑀 ∈ (measuresβ€˜π‘†) β†’ (π‘€β€˜βˆ…) = 0)
1093ad2ant1 1131 . . . 4 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ (π‘€β€˜βˆ…) = 0)
118, 10eqtrd 2770 . . 3 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) β†’ ((𝑀 β†Ύ 𝑇)β€˜βˆ…) = 0)
12 simp11 1201 . . . . . . 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 1203 . . . . . . . 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 1135 . . . . . . . 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 4612 . . . . . . . . 9 (𝑇 βŠ† 𝑆 β†’ 𝒫 𝑇 βŠ† 𝒫 𝑆)
1615sselda 3981 . . . . . . . 8 ((𝑇 βŠ† 𝑆 ∧ π‘₯ ∈ 𝒫 𝑇) β†’ π‘₯ ∈ 𝒫 𝑆)
1713, 14, 16syl2anc 582 . . . . . . 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 1136 . . . . . . 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 33505 . . . . . . 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 1369 . . . . . 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 1131 . . . . . . . 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 1199 . . . . . . . 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 33413 . . . . . . . 8 ((𝑇 ∈ βˆͺ ran sigAlgebra ∧ π‘₯ ∈ 𝒫 𝑇 ∧ π‘₯ β‰Ό Ο‰) β†’ βˆͺ π‘₯ ∈ 𝑇)
2421, 14, 22, 23syl3anc 1369 . . . . . . 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 6910 . . . . . 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 4608 . . . . . . . . . . 11 (π‘₯ ∈ 𝒫 𝑇 β†’ π‘₯ βŠ† 𝑇)
2726sselda 3981 . . . . . . . . . 10 ((π‘₯ ∈ 𝒫 𝑇 ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ 𝑇)
2827adantll 710 . . . . . . . . 9 ((((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝑇 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 βŠ† 𝑆) ∧ π‘₯ ∈ 𝒫 𝑇) ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ 𝑇)
2928fvresd 6910 . . . . . . . 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 33334 . . . . . . 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 1130 . . . . . 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 2780 . . . . 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 1119 . . . 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 3144 . . 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 1126 . 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 33495 . . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  βˆͺ cuni 4907  Disj wdisj 5112   class class class wbr 5147  ran crn 5676   β†Ύ cres 5677  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  Ο‰com 7857   β‰Ό cdom 8939  0cc0 11112  +∞cpnf 11249  [,]cicc 13331  Ξ£*cesum 33323  sigAlgebracsiga 33404  measurescmeas 33491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-esum 33324  df-siga 33405  df-meas 33492
This theorem is referenced by:  measinb2  33519
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