Theorem mea0 40671

Description: The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
mea0.1	|- ( ph -> M e. Meas )

Assertion

Ref	Expression
mea0	|- ( ph -> ( M ` (/) ) = 0 )

Proof of Theorem mea0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mea0.1	. . 3 |- ( ph -> M e. Meas )
2		ismea 40668	. . 3 |- ( M e. Meas <-> ( ( ( M : dom M --> ( 0 [,] +oo ) /\ dom M e. SAlg ) /\ ( M ` (/) ) = 0 ) /\ A. x e. ~P dom M ( ( x ~<_ om /\ Disj y e. x y ) -> ( M ` U. x ) = (Σ^ ` ( M |` x ) ) ) ) )
3	1, 2	sylib 208	. 2 |- ( ph -> ( ( ( M : dom M --> ( 0 [,] +oo ) /\ dom M e. SAlg ) /\ ( M ` (/) ) = 0 ) /\ A. x e. ~P dom M ( ( x ~<_ om /\ Disj y e. x y ) -> ( M ` U. x ) = (Σ^ ` ( M |` x ) ) ) ) )
4	3	simplrd 793	1 |- ( ph -> ( M ` (/) ) = 0 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   ~Pcpw 4158   U.cuni 4436  Disj wdisj 4620   class class class wbr 4653   dom cdm 5114   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   0cc0 9936   +oocpnf 10071   [,]cicc 12178  SAlgcsalg 40528  Σ^{^}csumge0 40579  Meascmea 40666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-mea 40667

This theorem is referenced by:  meadjun  40679  meadjiunlem  40682  vonioo  40896  vonicc  40899