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Theorem meadjun 38294
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 11714 . . . . . . 7  |-  ( 0 [,] +oo )  C_  RR*
2 meadjun.m . . . . . . . . 9  |-  ( ph  ->  M  e. Meas )
3 meadjun.x . . . . . . . . 9  |-  S  =  dom  M
42, 3meaf 38285 . . . . . . . 8  |-  ( ph  ->  M : S --> ( 0 [,] +oo ) )
5 meadjun.b . . . . . . . 8  |-  ( ph  ->  B  e.  S )
64, 5ffvelrnd 6021 . . . . . . 7  |-  ( ph  ->  ( M `  B
)  e.  ( 0 [,] +oo ) )
71, 6sseldi 3429 . . . . . 6  |-  ( ph  ->  ( M `  B
)  e.  RR* )
8 xaddid2 11530 . . . . . 6  |-  ( ( M `  B )  e.  RR*  ->  ( 0 +e ( M `
 B ) )  =  ( M `  B ) )
97, 8syl 17 . . . . 5  |-  ( ph  ->  ( 0 +e
( M `  B
) )  =  ( M `  B ) )
109eqcomd 2456 . . . 4  |-  ( ph  ->  ( M `  B
)  =  ( 0 +e ( M `
 B ) ) )
1110adantr 467 . . 3  |-  ( (
ph  /\  A  =  (/) )  ->  ( M `  B )  =  ( 0 +e ( M `  B ) ) )
12 uneq1 3580 . . . . . 6  |-  ( A  =  (/)  ->  ( A  u.  B )  =  ( (/)  u.  B
) )
13 0un 37380 . . . . . . 7  |-  ( (/)  u.  B )  =  B
1413a1i 11 . . . . . 6  |-  ( A  =  (/)  ->  ( (/)  u.  B )  =  B )
1512, 14eqtrd 2484 . . . . 5  |-  ( A  =  (/)  ->  ( A  u.  B )  =  B )
1615fveq2d 5867 . . . 4  |-  ( A  =  (/)  ->  ( M `
 ( A  u.  B ) )  =  ( M `  B
) )
1716adantl 468 . . 3  |-  ( (
ph  /\  A  =  (/) )  ->  ( M `  ( A  u.  B
) )  =  ( M `  B ) )
18 fveq2 5863 . . . . . 6  |-  ( A  =  (/)  ->  ( M `
 A )  =  ( M `  (/) ) )
1918adantl 468 . . . . 5  |-  ( (
ph  /\  A  =  (/) )  ->  ( M `  A )  =  ( M `  (/) ) )
202mea0 38286 . . . . . 6  |-  ( ph  ->  ( M `  (/) )  =  0 )
2120adantr 467 . . . . 5  |-  ( (
ph  /\  A  =  (/) )  ->  ( M `  (/) )  =  0 )
2219, 21eqtrd 2484 . . . 4  |-  ( (
ph  /\  A  =  (/) )  ->  ( M `  A )  =  0 )
2322oveq1d 6303 . . 3  |-  ( (
ph  /\  A  =  (/) )  ->  ( ( M `  A ) +e ( M `
 B ) )  =  ( 0 +e ( M `  B ) ) )
2411, 17, 233eqtr4d 2494 . 2  |-  ( (
ph  /\  A  =  (/) )  ->  ( M `  ( A  u.  B
) )  =  ( ( M `  A
) +e ( M `  B ) ) )
25 simpl 459 . . 3  |-  ( (
ph  /\  -.  A  =  (/) )  ->  ph )
26 meadjun.dj . . . . . 6  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
2726ad2antrr 731 . . . . 5  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  A  =  B )  ->  ( A  i^i  B
)  =  (/) )
28 inidm 3640 . . . . . . . . . . 11  |-  ( A  i^i  A )  =  A
2928eqcomi 2459 . . . . . . . . . 10  |-  A  =  ( A  i^i  A
)
30 ineq2 3627 . . . . . . . . . 10  |-  ( A  =  B  ->  ( A  i^i  A )  =  ( A  i^i  B
) )
3129, 30syl5req 2497 . . . . . . . . 9  |-  ( A  =  B  ->  ( A  i^i  B )  =  A )
3231adantl 468 . . . . . . . 8  |-  ( ( -.  A  =  (/)  /\  A  =  B )  ->  ( A  i^i  B )  =  A )
33 neqne 37368 . . . . . . . . 9  |-  ( -.  A  =  (/)  ->  A  =/=  (/) )
3433adantr 467 . . . . . . . 8  |-  ( ( -.  A  =  (/)  /\  A  =  B )  ->  A  =/=  (/) )
3532, 34eqnetrd 2690 . . . . . . 7  |-  ( ( -.  A  =  (/)  /\  A  =  B )  ->  ( A  i^i  B )  =/=  (/) )
3635neneqd 2628 . . . . . 6  |-  ( ( -.  A  =  (/)  /\  A  =  B )  ->  -.  ( A  i^i  B )  =  (/) )
3736adantll 719 . . . . 5  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  A  =  B )  ->  -.  ( A  i^i  B )  =  (/) )
3827, 37pm2.65da 579 . . . 4  |-  ( (
ph  /\  -.  A  =  (/) )  ->  -.  A  =  B )
3938neqned 2630 . . 3  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A  =/=  B )
40 meadjun.a . . . . . . . 8  |-  ( ph  ->  A  e.  S )
41 uniprg 4211 . . . . . . . 8  |-  ( ( A  e.  S  /\  B  e.  S )  ->  U. { A ,  B }  =  ( A  u.  B )
)
4240, 5, 41syl2anc 666 . . . . . . 7  |-  ( ph  ->  U. { A ,  B }  =  ( A  u.  B )
)
4342eqcomd 2456 . . . . . 6  |-  ( ph  ->  ( A  u.  B
)  =  U. { A ,  B }
)
4443fveq2d 5867 . . . . 5  |-  ( ph  ->  ( M `  ( A  u.  B )
)  =  ( M `
 U. { A ,  B } ) )
4544adantr 467 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( M `  ( A  u.  B
) )  =  ( M `  U. { A ,  B }
) )
4640, 5prssd 37378 . . . . . 6  |-  ( ph  ->  { A ,  B }  C_  S )
47 prfi 7843 . . . . . . . 8  |-  { A ,  B }  e.  Fin
48 isfinite 8154 . . . . . . . . . 10  |-  ( { A ,  B }  e.  Fin  <->  { A ,  B }  ~<  om )
4948biimpi 198 . . . . . . . . 9  |-  ( { A ,  B }  e.  Fin  ->  { A ,  B }  ~<  om )
50 sdomdom 7594 . . . . . . . . 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 4395 . . . . . . . . . 10  |- Disj  x  e. 
{ B } x
5554a1i 11 . . . . . . . . 9  |-  ( A  =  B  -> Disj  x  e. 
{ B } x
)
56 preq1 4050 . . . . . . . . . . 11  |-  ( A  =  B  ->  { A ,  B }  =  { B ,  B }
)
57 dfsn2 3980 . . . . . . . . . . . . 13  |-  { B }  =  { B ,  B }
5857eqcomi 2459 . . . . . . . . . . . 12  |-  { B ,  B }  =  { B }
5958a1i 11 . . . . . . . . . . 11  |-  ( A  =  B  ->  { B ,  B }  =  { B } )
6056, 59eqtrd 2484 . . . . . . . . . 10  |-  ( A  =  B  ->  { A ,  B }  =  { B } )
6160disjeq1d 4380 . . . . . . . . 9  |-  ( A  =  B  ->  (Disj  x  e.  { A ,  B } x  <-> Disj  x  e.  { B } x ) )
6255, 61mpbird 236 . . . . . . . 8  |-  ( A  =  B  -> Disj  x  e. 
{ A ,  B } x )
6362adantl 468 . . . . . . 7  |-  ( (
ph  /\  A  =  B )  -> Disj  x  e. 
{ A ,  B } x )
64 simpl 459 . . . . . . . 8  |-  ( (
ph  /\  -.  A  =  B )  ->  ph )
65 neqne 37368 . . . . . . . . 9  |-  ( -.  A  =  B  ->  A  =/=  B )
6665adantl 468 . . . . . . . 8  |-  ( (
ph  /\  -.  A  =  B )  ->  A  =/=  B )
6726adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  B )  ->  ( A  i^i  B )  =  (/) )
6840adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  B )  ->  A  e.  S )
695adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  B )  ->  B  e.  S )
70 simpr 463 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  B )  ->  A  =/=  B )
71 id 22 . . . . . . . . . . 11  |-  ( x  =  A  ->  x  =  A )
72 id 22 . . . . . . . . . . 11  |-  ( x  =  B  ->  x  =  B )
7371, 72disjprg 4397 . . . . . . . . . 10  |-  ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  -> 
(Disj  x  e.  { A ,  B } x  <->  ( A  i^i  B )  =  (/) ) )
7468, 69, 70, 73syl3anc 1267 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  B )  ->  (Disj  x  e. 
{ A ,  B } x  <->  ( A  i^i  B )  =  (/) ) )
7567, 74mpbird 236 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  B )  -> Disj  x  e.  { A ,  B }
x )
7664, 66, 75syl2anc 666 . . . . . . 7  |-  ( (
ph  /\  -.  A  =  B )  -> Disj  x  e. 
{ A ,  B } x )
7763, 76pm2.61dan 799 . . . . . 6  |-  ( ph  -> Disj  x  e.  { A ,  B } x )
782, 3, 46, 53, 77meadjuni 38289 . . . . 5  |-  ( ph  ->  ( M `  U. { A ,  B }
)  =  (Σ^ `  ( M  |`  { A ,  B } ) ) )
7978adantr 467 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( M `  U. { A ,  B } )  =  (Σ^ `  ( M  |`  { A ,  B } ) ) )
804, 40ffvelrnd 6021 . . . . . . 7  |-  ( ph  ->  ( M `  A
)  e.  ( 0 [,] +oo ) )
8180adantr 467 . . . . . 6  |-  ( (
ph  /\  A  =/=  B )  ->  ( M `  A )  e.  ( 0 [,] +oo )
)
826adantr 467 . . . . . 6  |-  ( (
ph  /\  A  =/=  B )  ->  ( M `  B )  e.  ( 0 [,] +oo )
)
83 fveq2 5863 . . . . . 6  |-  ( x  =  A  ->  ( M `  x )  =  ( M `  A ) )
84 fveq2 5863 . . . . . 6  |-  ( x  =  B  ->  ( M `  x )  =  ( M `  B ) )
8568, 69, 81, 82, 83, 84, 70sge0pr 38230 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  (Σ^ `  ( x  e.  { A ,  B }  |->  ( M `  x
) ) )  =  ( ( M `  A ) +e
( M `  B
) ) )
864, 46fssresd 5748 . . . . . . . . 9  |-  ( ph  ->  ( M  |`  { A ,  B } ) : { A ,  B }
--> ( 0 [,] +oo ) )
8786feqmptd 5916 . . . . . . . 8  |-  ( ph  ->  ( M  |`  { A ,  B } )  =  ( x  e.  { A ,  B }  |->  ( ( M  |`  { A ,  B }
) `  x )
) )
88 fvres 5877 . . . . . . . . . 10  |-  ( x  e.  { A ,  B }  ->  ( ( M  |`  { A ,  B } ) `  x )  =  ( M `  x ) )
8988mpteq2ia 4484 . . . . . . . . 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 2484 . . . . . . 7  |-  ( ph  ->  ( M  |`  { A ,  B } )  =  ( x  e.  { A ,  B }  |->  ( M `  x
) ) )
9291fveq2d 5867 . . . . . 6  |-  ( ph  ->  (Σ^ `  ( M  |`  { A ,  B } ) )  =  (Σ^ `  ( x  e.  { A ,  B }  |->  ( M `  x
) ) ) )
9392adantr 467 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  (Σ^ `  ( M  |`  { A ,  B } ) )  =  (Σ^ `  ( x  e.  { A ,  B }  |->  ( M `  x
) ) ) )
94 eqidd 2451 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  ( ( M `  A ) +e ( M `
 B ) )  =  ( ( M `
 A ) +e ( M `  B ) ) )
9585, 93, 943eqtr4d 2494 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  (Σ^ `  ( M  |`  { A ,  B } ) )  =  ( ( M `
 A ) +e ( M `  B ) ) )
9645, 79, 953eqtrd 2488 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  ( M `  ( A  u.  B
) )  =  ( ( M `  A
) +e ( M `  B ) ) )
9725, 39, 96syl2anc 666 . 2  |-  ( (
ph  /\  -.  A  =  (/) )  ->  ( M `  ( A  u.  B ) )  =  ( ( M `  A ) +e
( M `  B
) ) )
9824, 97pm2.61dan 799 1  |-  ( ph  ->  ( M `  ( A  u.  B )
)  =  ( ( M `  A ) +e ( M `
 B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621    u. cun 3401    i^i cin 3402   (/)c0 3730   {csn 3967   {cpr 3969   U.cuni 4197  Disj wdisj 4372   class class class wbr 4401    |-> cmpt 4460   dom cdm 4833    |` cres 4835   ` cfv 5581  (class class class)co 6288   omcom 6689    ~<_ cdom 7564    ~< csdm 7565   Fincfn 7566   0cc0 9536   +oocpnf 9669   RR*cxr 9671   +ecxad 11404   [,]cicc 11635  Σ^csumge0 38198  Meascmea 38281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-disj 4373  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-xadd 11407  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746  df-sumge0 38199  df-mea 38282
This theorem is referenced by:  meassle  38295  meaunle  38296
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