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Theorem meadjun 38416
Description: The measure of the union of two disjoint sets is the sum of the measures, Property 112C (a) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
meadjun.m  |-  ( ph  ->  M  e. Meas )
meadjun.x  |-  S  =  dom  M
meadjun.a  |-  ( ph  ->  A  e.  S )
meadjun.b  |-  ( ph  ->  B  e.  S )
meadjun.dj  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
Assertion
Ref Expression
meadjun  |-  ( ph  ->  ( M `  ( A  u.  B )
)  =  ( ( M `  A ) +e ( M `
 B ) ) )

Proof of Theorem meadjun
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 iccssxr 11742 . . . . . . 7  |-  ( 0 [,] +oo )  C_  RR*
2 meadjun.m . . . . . . . . 9  |-  ( ph  ->  M  e. Meas )
3 meadjun.x . . . . . . . . 9  |-  S  =  dom  M
42, 3meaf 38407 . . . . . . . 8  |-  ( ph  ->  M : S --> ( 0 [,] +oo ) )
5 meadjun.b . . . . . . . 8  |-  ( ph  ->  B  e.  S )
64, 5ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( M `  B
)  e.  ( 0 [,] +oo ) )
71, 6sseldi 3416 . . . . . 6  |-  ( ph  ->  ( M `  B
)  e.  RR* )
8 xaddid2 11557 . . . . . 6  |-  ( ( M `  B )  e.  RR*  ->  ( 0 +e ( M `
 B ) )  =  ( M `  B ) )
97, 8syl 17 . . . . 5  |-  ( ph  ->  ( 0 +e
( M `  B
) )  =  ( M `  B ) )
109eqcomd 2477 . . . 4  |-  ( ph  ->  ( M `  B
)  =  ( 0 +e ( M `
 B ) ) )
1110adantr 472 . . 3  |-  ( (
ph  /\  A  =  (/) )  ->  ( M `  B )  =  ( 0 +e ( M `  B ) ) )
12 uneq1 3572 . . . . . 6  |-  ( A  =  (/)  ->  ( A  u.  B )  =  ( (/)  u.  B
) )
13 0un 37445 . . . . . . 7  |-  ( (/)  u.  B )  =  B
1413a1i 11 . . . . . 6  |-  ( A  =  (/)  ->  ( (/)  u.  B )  =  B )
1512, 14eqtrd 2505 . . . . 5  |-  ( A  =  (/)  ->  ( A  u.  B )  =  B )
1615fveq2d 5883 . . . 4  |-  ( A  =  (/)  ->  ( M `
 ( A  u.  B ) )  =  ( M `  B
) )
1716adantl 473 . . 3  |-  ( (
ph  /\  A  =  (/) )  ->  ( M `  ( A  u.  B
) )  =  ( M `  B ) )
18 fveq2 5879 . . . . . 6  |-  ( A  =  (/)  ->  ( M `
 A )  =  ( M `  (/) ) )
1918adantl 473 . . . . 5  |-  ( (
ph  /\  A  =  (/) )  ->  ( M `  A )  =  ( M `  (/) ) )
202mea0 38408 . . . . . 6  |-  ( ph  ->  ( M `  (/) )  =  0 )
2120adantr 472 . . . . 5  |-  ( (
ph  /\  A  =  (/) )  ->  ( M `  (/) )  =  0 )
2219, 21eqtrd 2505 . . . 4  |-  ( (
ph  /\  A  =  (/) )  ->  ( M `  A )  =  0 )
2322oveq1d 6323 . . 3  |-  ( (
ph  /\  A  =  (/) )  ->  ( ( M `  A ) +e ( M `
 B ) )  =  ( 0 +e ( M `  B ) ) )
2411, 17, 233eqtr4d 2515 . 2  |-  ( (
ph  /\  A  =  (/) )  ->  ( M `  ( A  u.  B
) )  =  ( ( M `  A
) +e ( M `  B ) ) )
25 simpl 464 . . 3  |-  ( (
ph  /\  -.  A  =  (/) )  ->  ph )
26 meadjun.dj . . . . . 6  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
2726ad2antrr 740 . . . . 5  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  A  =  B )  ->  ( A  i^i  B
)  =  (/) )
28 inidm 3632 . . . . . . . . . . 11  |-  ( A  i^i  A )  =  A
2928eqcomi 2480 . . . . . . . . . 10  |-  A  =  ( A  i^i  A
)
30 ineq2 3619 . . . . . . . . . 10  |-  ( A  =  B  ->  ( A  i^i  A )  =  ( A  i^i  B
) )
3129, 30syl5req 2518 . . . . . . . . 9  |-  ( A  =  B  ->  ( A  i^i  B )  =  A )
3231adantl 473 . . . . . . . 8  |-  ( ( -.  A  =  (/)  /\  A  =  B )  ->  ( A  i^i  B )  =  A )
33 neqne 2651 . . . . . . . . 9  |-  ( -.  A  =  (/)  ->  A  =/=  (/) )
3433adantr 472 . . . . . . . 8  |-  ( ( -.  A  =  (/)  /\  A  =  B )  ->  A  =/=  (/) )
3532, 34eqnetrd 2710 . . . . . . 7  |-  ( ( -.  A  =  (/)  /\  A  =  B )  ->  ( A  i^i  B )  =/=  (/) )
3635neneqd 2648 . . . . . 6  |-  ( ( -.  A  =  (/)  /\  A  =  B )  ->  -.  ( A  i^i  B )  =  (/) )
3736adantll 728 . . . . 5  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  A  =  B )  ->  -.  ( A  i^i  B )  =  (/) )
3827, 37pm2.65da 586 . . . 4  |-  ( (
ph  /\  -.  A  =  (/) )  ->  -.  A  =  B )
3938neqned 2650 . . 3  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A  =/=  B )
40 meadjun.a . . . . . . . 8  |-  ( ph  ->  A  e.  S )
41 uniprg 4204 . . . . . . . 8  |-  ( ( A  e.  S  /\  B  e.  S )  ->  U. { A ,  B }  =  ( A  u.  B )
)
4240, 5, 41syl2anc 673 . . . . . . 7  |-  ( ph  ->  U. { A ,  B }  =  ( A  u.  B )
)
4342eqcomd 2477 . . . . . 6  |-  ( ph  ->  ( A  u.  B
)  =  U. { A ,  B }
)
4443fveq2d 5883 . . . . 5  |-  ( ph  ->  ( M `  ( A  u.  B )
)  =  ( M `
 U. { A ,  B } ) )
4544adantr 472 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( M `  ( A  u.  B
) )  =  ( M `  U. { A ,  B }
) )
4640, 5prssd 4120 . . . . . 6  |-  ( ph  ->  { A ,  B }  C_  S )
47 prfi 7864 . . . . . . . 8  |-  { A ,  B }  e.  Fin
48 isfinite 8175 . . . . . . . . . 10  |-  ( { A ,  B }  e.  Fin  <->  { A ,  B }  ~<  om )
4948biimpi 199 . . . . . . . . 9  |-  ( { A ,  B }  e.  Fin  ->  { A ,  B }  ~<  om )
50 sdomdom 7615 . . . . . . . . 9  |-  ( { A ,  B }  ~<  om  ->  { A ,  B }  ~<_  om )
5149, 50syl 17 . . . . . . . 8  |-  ( { A ,  B }  e.  Fin  ->  { A ,  B }  ~<_  om )
5247, 51ax-mp 5 . . . . . . 7  |-  { A ,  B }  ~<_  om
5352a1i 11 . . . . . 6  |-  ( ph  ->  { A ,  B }  ~<_  om )
54 disjxsn 4389 . . . . . . . . . 10  |- Disj  x  e. 
{ B } x
5554a1i 11 . . . . . . . . 9  |-  ( A  =  B  -> Disj  x  e. 
{ B } x
)
56 preq1 4042 . . . . . . . . . . 11  |-  ( A  =  B  ->  { A ,  B }  =  { B ,  B }
)
57 dfsn2 3972 . . . . . . . . . . . . 13  |-  { B }  =  { B ,  B }
5857eqcomi 2480 . . . . . . . . . . . 12  |-  { B ,  B }  =  { B }
5958a1i 11 . . . . . . . . . . 11  |-  ( A  =  B  ->  { B ,  B }  =  { B } )
6056, 59eqtrd 2505 . . . . . . . . . 10  |-  ( A  =  B  ->  { A ,  B }  =  { B } )
6160disjeq1d 4374 . . . . . . . . 9  |-  ( A  =  B  ->  (Disj  x  e.  { A ,  B } x  <-> Disj  x  e.  { B } x ) )
6255, 61mpbird 240 . . . . . . . 8  |-  ( A  =  B  -> Disj  x  e. 
{ A ,  B } x )
6362adantl 473 . . . . . . 7  |-  ( (
ph  /\  A  =  B )  -> Disj  x  e. 
{ A ,  B } x )
64 simpl 464 . . . . . . . 8  |-  ( (
ph  /\  -.  A  =  B )  ->  ph )
65 neqne 2651 . . . . . . . . 9  |-  ( -.  A  =  B  ->  A  =/=  B )
6665adantl 473 . . . . . . . 8  |-  ( (
ph  /\  -.  A  =  B )  ->  A  =/=  B )
6726adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  B )  ->  ( A  i^i  B )  =  (/) )
6840adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  B )  ->  A  e.  S )
695adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  B )  ->  B  e.  S )
70 simpr 468 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  B )  ->  A  =/=  B )
71 id 22 . . . . . . . . . . 11  |-  ( x  =  A  ->  x  =  A )
72 id 22 . . . . . . . . . . 11  |-  ( x  =  B  ->  x  =  B )
7371, 72disjprg 4391 . . . . . . . . . 10  |-  ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  -> 
(Disj  x  e.  { A ,  B } x  <->  ( A  i^i  B )  =  (/) ) )
7468, 69, 70, 73syl3anc 1292 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  B )  ->  (Disj  x  e. 
{ A ,  B } x  <->  ( A  i^i  B )  =  (/) ) )
7567, 74mpbird 240 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  B )  -> Disj  x  e.  { A ,  B }
x )
7664, 66, 75syl2anc 673 . . . . . . 7  |-  ( (
ph  /\  -.  A  =  B )  -> Disj  x  e. 
{ A ,  B } x )
7763, 76pm2.61dan 808 . . . . . 6  |-  ( ph  -> Disj  x  e.  { A ,  B } x )
782, 3, 46, 53, 77meadjuni 38411 . . . . 5  |-  ( ph  ->  ( M `  U. { A ,  B }
)  =  (Σ^ `  ( M  |`  { A ,  B } ) ) )
7978adantr 472 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( M `  U. { A ,  B } )  =  (Σ^ `  ( M  |`  { A ,  B } ) ) )
804, 40ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( M `  A
)  e.  ( 0 [,] +oo ) )
8180adantr 472 . . . . . 6  |-  ( (
ph  /\  A  =/=  B )  ->  ( M `  A )  e.  ( 0 [,] +oo )
)
826adantr 472 . . . . . 6  |-  ( (
ph  /\  A  =/=  B )  ->  ( M `  B )  e.  ( 0 [,] +oo )
)
83 fveq2 5879 . . . . . 6  |-  ( x  =  A  ->  ( M `  x )  =  ( M `  A ) )
84 fveq2 5879 . . . . . 6  |-  ( x  =  B  ->  ( M `  x )  =  ( M `  B ) )
8568, 69, 81, 82, 83, 84, 70sge0pr 38350 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  (Σ^ `  ( x  e.  { A ,  B }  |->  ( M `  x
) ) )  =  ( ( M `  A ) +e
( M `  B
) ) )
864, 46fssresd 5762 . . . . . . . . 9  |-  ( ph  ->  ( M  |`  { A ,  B } ) : { A ,  B }
--> ( 0 [,] +oo ) )
8786feqmptd 5932 . . . . . . . 8  |-  ( ph  ->  ( M  |`  { A ,  B } )  =  ( x  e.  { A ,  B }  |->  ( ( M  |`  { A ,  B }
) `  x )
) )
88 fvres 5893 . . . . . . . . . 10  |-  ( x  e.  { A ,  B }  ->  ( ( M  |`  { A ,  B } ) `  x )  =  ( M `  x ) )
8988mpteq2ia 4478 . . . . . . . . 9  |-  ( x  e.  { A ,  B }  |->  ( ( M  |`  { A ,  B } ) `  x ) )  =  ( x  e.  { A ,  B }  |->  ( M `  x
) )
9089a1i 11 . . . . . . . 8  |-  ( ph  ->  ( x  e.  { A ,  B }  |->  ( ( M  |`  { A ,  B }
) `  x )
)  =  ( x  e.  { A ,  B }  |->  ( M `
 x ) ) )
9187, 90eqtrd 2505 . . . . . . 7  |-  ( ph  ->  ( M  |`  { A ,  B } )  =  ( x  e.  { A ,  B }  |->  ( M `  x
) ) )
9291fveq2d 5883 . . . . . 6  |-  ( ph  ->  (Σ^ `  ( M  |`  { A ,  B } ) )  =  (Σ^ `  ( x  e.  { A ,  B }  |->  ( M `  x
) ) ) )
9392adantr 472 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  (Σ^ `  ( M  |`  { A ,  B } ) )  =  (Σ^ `  ( x  e.  { A ,  B }  |->  ( M `  x
) ) ) )
94 eqidd 2472 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  ( ( M `  A ) +e ( M `
 B ) )  =  ( ( M `
 A ) +e ( M `  B ) ) )
9585, 93, 943eqtr4d 2515 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  (Σ^ `  ( M  |`  { A ,  B } ) )  =  ( ( M `
 A ) +e ( M `  B ) ) )
9645, 79, 953eqtrd 2509 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  ( M `  ( A  u.  B
) )  =  ( ( M `  A
) +e ( M `  B ) ) )
9725, 39, 96syl2anc 673 . 2  |-  ( (
ph  /\  -.  A  =  (/) )  ->  ( M `  ( A  u.  B ) )  =  ( ( M `  A ) +e
( M `  B
) ) )
9824, 97pm2.61dan 808 1  |-  ( ph  ->  ( M `  ( A  u.  B )
)  =  ( ( M `  A ) +e ( M `
 B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641    u. cun 3388    i^i cin 3389   (/)c0 3722   {csn 3959   {cpr 3961   U.cuni 4190  Disj wdisj 4366   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839    |` cres 4841   ` cfv 5589  (class class class)co 6308   omcom 6711    ~<_ cdom 7585    ~< csdm 7586   Fincfn 7587   0cc0 9557   +oocpnf 9690   RR*cxr 9692   +ecxad 11430   [,]cicc 11663  Σ^csumge0 38318  Meascmea 38403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-xadd 11433  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-sumge0 38319  df-mea 38404
This theorem is referenced by:  meassle  38417  meaunle  38418
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