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Theorem probun 29086
Description: The probability of the union two incompatible events is the sum of their probabilities. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
probun  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  -> 
( ( A  i^i  B )  =  (/)  ->  ( P `  ( A  u.  B ) )  =  ( ( P `  A )  +  ( P `  B ) ) ) )

Proof of Theorem probun
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpll1 1044 . . . 4  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =  B )  /\  ( A  i^i  B )  =  (/) )  ->  P  e. Prob
)
2 simplr 760 . . . 4  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =  B )  /\  ( A  i^i  B )  =  (/) )  ->  A  =  B )
3 simpr 462 . . . 4  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =  B )  /\  ( A  i^i  B )  =  (/) )  ->  ( A  i^i  B )  =  (/) )
4 disj3 3843 . . . . . . . . . . 11  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
54biimpi 197 . . . . . . . . . 10  |-  ( ( A  i^i  B )  =  (/)  ->  A  =  ( A  \  B
) )
6 difeq1 3582 . . . . . . . . . . 11  |-  ( A  =  B  ->  ( A  \  B )  =  ( B  \  B
) )
7 difid 3869 . . . . . . . . . . 11  |-  ( B 
\  B )  =  (/)
86, 7syl6eq 2486 . . . . . . . . . 10  |-  ( A  =  B  ->  ( A  \  B )  =  (/) )
95, 8sylan9eqr 2492 . . . . . . . . 9  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  A  =  (/) )
10 eqtr2 2456 . . . . . . . . . 10  |-  ( ( A  =  B  /\  A  =  (/) )  ->  B  =  (/) )
119, 10syldan 472 . . . . . . . . 9  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  B  =  (/) )
129, 11uneq12d 3627 . . . . . . . 8  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  =  ( (/)  u.  (/) ) )
13 unidm 3615 . . . . . . . 8  |-  ( (/)  u.  (/) )  =  (/)
1412, 13syl6eq 2486 . . . . . . 7  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  =  (/) )
1514fveq2d 5885 . . . . . 6  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( P `  ( A  u.  B ) )  =  ( P `  (/) ) )
16 probnul 29081 . . . . . 6  |-  ( P  e. Prob  ->  ( P `  (/) )  =  0 )
1715, 16sylan9eqr 2492 . . . . 5  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( P `  ( A  u.  B )
)  =  0 )
189fveq2d 5885 . . . . . . . 8  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( P `  A )  =  ( P `  (/) ) )
1918, 16sylan9eqr 2492 . . . . . . 7  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( P `  A
)  =  0 )
2011fveq2d 5885 . . . . . . . 8  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( P `  B )  =  ( P `  (/) ) )
2120, 16sylan9eqr 2492 . . . . . . 7  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( P `  B
)  =  0 )
2219, 21oveq12d 6323 . . . . . 6  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( ( P `  A )  +  ( P `  B ) )  =  ( 0  +  0 ) )
23 00id 9807 . . . . . 6  |-  ( 0  +  0 )  =  0
2422, 23syl6eq 2486 . . . . 5  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( ( P `  A )  +  ( P `  B ) )  =  0 )
2517, 24eqtr4d 2473 . . . 4  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( P `  ( A  u.  B )
)  =  ( ( P `  A )  +  ( P `  B ) ) )
261, 2, 3, 25syl12anc 1262 . . 3  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =  B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 ( A  u.  B ) )  =  ( ( P `  A )  +  ( P `  B ) ) )
2726ex 435 . 2  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( P `  ( A  u.  B
) )  =  ( ( P `  A
)  +  ( P `
 B ) ) ) )
28 3anass 986 . . . . . . 7  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  <->  ( P  e. Prob  /\  ( A  e. 
dom  P  /\  B  e. 
dom  P ) ) )
2928anbi1i 699 . . . . . 6  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  <->  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P ) )  /\  A  =/=  B
) )
30 df-3an 984 . . . . . 6  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  <->  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P ) )  /\  A  =/=  B
) )
3129, 30bitr4i 255 . . . . 5  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  <->  ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B ) )
32 simpl1 1008 . . . . . . 7  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  P  e. Prob
)
33 prssi 4159 . . . . . . . . . 10  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P )  ->  { A ,  B }  C_  dom  P )
34333ad2ant2 1027 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  { A ,  B }  C_  dom  P )
3534adantr 466 . . . . . . . 8  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  { A ,  B }  C_  dom  P )
36 prex 4664 . . . . . . . . 9  |-  { A ,  B }  e.  _V
3736elpw 3991 . . . . . . . 8  |-  ( { A ,  B }  e.  ~P dom  P  <->  { A ,  B }  C_  dom  P )
3835, 37sylibr 215 . . . . . . 7  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  { A ,  B }  e.  ~P dom  P )
39 prct 28147 . . . . . . . . 9  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P )  ->  { A ,  B }  ~<_  om )
40393ad2ant2 1027 . . . . . . . 8  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  { A ,  B }  ~<_  om )
4140adantr 466 . . . . . . 7  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  { A ,  B }  ~<_  om )
42 simp2l 1031 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  A  e.  dom  P )
43 simp2r 1032 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  B  e.  dom  P )
44 simp3 1007 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  A  =/=  B )
45 id 23 . . . . . . . . . 10  |-  ( x  =  A  ->  x  =  A )
46 id 23 . . . . . . . . . 10  |-  ( x  =  B  ->  x  =  B )
4745, 46disjprg 4422 . . . . . . . . 9  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P  /\  A  =/=  B
)  ->  (Disj  x  e. 
{ A ,  B } x  <->  ( A  i^i  B )  =  (/) ) )
4842, 43, 44, 47syl3anc 1264 . . . . . . . 8  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  (Disj  x  e. 
{ A ,  B } x  <->  ( A  i^i  B )  =  (/) ) )
4948biimpar 487 . . . . . . 7  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  -> Disj  x  e.  { A ,  B }
x )
50 probcun 29085 . . . . . . 7  |-  ( ( P  e. Prob  /\  { A ,  B }  e.  ~P dom  P  /\  ( { A ,  B }  ~<_  om  /\ Disj  x  e. 
{ A ,  B } x ) )  ->  ( P `  U. { A ,  B } )  = Σ* x  e. 
{ A ,  B }  ( P `  x ) )
5132, 38, 41, 49, 50syl112anc 1268 . . . . . 6  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 U. { A ,  B } )  = Σ* x  e.  { A ,  B }  ( P `  x ) )
52 uniprg 4236 . . . . . . . . . 10  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P )  ->  U. { A ,  B }  =  ( A  u.  B ) )
5352fveq2d 5885 . . . . . . . . 9  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P )  ->  ( P `  U. { A ,  B } )  =  ( P `  ( A  u.  B ) ) )
54533ad2ant2 1027 . . . . . . . 8  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  U. { A ,  B } )  =  ( P `  ( A  u.  B ) ) )
55 fveq2 5881 . . . . . . . . . 10  |-  ( x  =  A  ->  ( P `  x )  =  ( P `  A ) )
5655adantl 467 . . . . . . . . 9  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  x  =  A )  ->  ( P `  x )  =  ( P `  A ) )
57 fveq2 5881 . . . . . . . . . 10  |-  ( x  =  B  ->  ( P `  x )  =  ( P `  B ) )
5857adantl 467 . . . . . . . . 9  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  x  =  B )  ->  ( P `  x )  =  ( P `  B ) )
59 unitssxrge0 28553 . . . . . . . . . 10  |-  ( 0 [,] 1 )  C_  ( 0 [,] +oo )
60 simp1 1005 . . . . . . . . . . 11  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  P  e. Prob )
61 prob01 29080 . . . . . . . . . . 11  |-  ( ( P  e. Prob  /\  A  e.  dom  P )  -> 
( P `  A
)  e.  ( 0 [,] 1 ) )
6260, 42, 61syl2anc 665 . . . . . . . . . 10  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  A )  e.  ( 0 [,] 1 ) )
6359, 62sseldi 3468 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  A )  e.  ( 0 [,] +oo )
)
64 prob01 29080 . . . . . . . . . . 11  |-  ( ( P  e. Prob  /\  B  e.  dom  P )  -> 
( P `  B
)  e.  ( 0 [,] 1 ) )
6560, 43, 64syl2anc 665 . . . . . . . . . 10  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  B )  e.  ( 0 [,] 1 ) )
6659, 65sseldi 3468 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  B )  e.  ( 0 [,] +oo )
)
6756, 58, 42, 43, 63, 66, 44esumpr 28734 . . . . . . . 8  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  -> Σ* x  e.  { A ,  B }  ( P `
 x )  =  ( ( P `  A ) +e
( P `  B
) ) )
6854, 67eqeq12d 2451 . . . . . . 7  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( ( P `  U. { A ,  B } )  = Σ* x  e.  { A ,  B }  ( P `  x )  <->  ( P `  ( A  u.  B
) )  =  ( ( P `  A
) +e ( P `  B ) ) ) )
6968adantr 466 . . . . . 6  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( ( P `  U. { A ,  B }
)  = Σ* x  e.  { A ,  B }  ( P `
 x )  <->  ( P `  ( A  u.  B
) )  =  ( ( P `  A
) +e ( P `  B ) ) ) )
7051, 69mpbid 213 . . . . 5  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 ( A  u.  B ) )  =  ( ( P `  A ) +e
( P `  B
) ) )
7131, 70sylanb 474 . . . 4  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 ( A  u.  B ) )  =  ( ( P `  A ) +e
( P `  B
) ) )
72 unitssre 11777 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
73 simpll1 1044 . . . . . . 7  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  P  e. Prob
)
74 simpll2 1045 . . . . . . 7  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  A  e. 
dom  P )
7573, 74, 61syl2anc 665 . . . . . 6  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 A )  e.  ( 0 [,] 1
) )
7672, 75sseldi 3468 . . . . 5  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 A )  e.  RR )
77 simpll3 1046 . . . . . . 7  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  B  e. 
dom  P )
7873, 77, 64syl2anc 665 . . . . . 6  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 B )  e.  ( 0 [,] 1
) )
7972, 78sseldi 3468 . . . . 5  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 B )  e.  RR )
80 rexadd 11525 . . . . 5  |-  ( ( ( P `  A
)  e.  RR  /\  ( P `  B )  e.  RR )  -> 
( ( P `  A ) +e
( P `  B
) )  =  ( ( P `  A
)  +  ( P `
 B ) ) )
8176, 79, 80syl2anc 665 . . . 4  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( ( P `  A ) +e ( P `
 B ) )  =  ( ( P `
 A )  +  ( P `  B
) ) )
8271, 81eqtrd 2470 . . 3  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 ( A  u.  B ) )  =  ( ( P `  A )  +  ( P `  B ) ) )
8382ex 435 . 2  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( ( A  i^i  B )  =  (/)  ->  ( P `  ( A  u.  B
) )  =  ( ( P `  A
)  +  ( P `
 B ) ) ) )
8427, 83pm2.61dane 2749 1  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  -> 
( ( A  i^i  B )  =  (/)  ->  ( P `  ( A  u.  B ) )  =  ( ( P `  A )  +  ( P `  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625    \ cdif 3439    u. cun 3440    i^i cin 3441    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   {cpr 4004   U.cuni 4222  Disj wdisj 4397   class class class wbr 4426   dom cdm 4854   ` cfv 5601  (class class class)co 6305   omcom 6706    ~<_ cdom 7575   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541   +oocpnf 9671   +ecxad 11407   [,]cicc 11638  Σ*cesum 28695  Probcprb 29074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-ac2 8891  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-acn 8375  df-ac 8545  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-ef 14099  df-sin 14101  df-cos 14102  df-pi 14104  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15166  df-mulr 15167  df-starv 15168  df-sca 15169  df-vsca 15170  df-ip 15171  df-tset 15172  df-ple 15173  df-ds 15175  df-unif 15176  df-hom 15177  df-cco 15178  df-rest 15284  df-topn 15285  df-0g 15303  df-gsum 15304  df-topgen 15305  df-pt 15306  df-prds 15309  df-ordt 15362  df-xrs 15363  df-qtop 15368  df-imas 15369  df-xps 15371  df-mre 15447  df-mrc 15448  df-acs 15450  df-ps 16401  df-tsr 16402  df-plusf 16442  df-mgm 16443  df-sgrp 16482  df-mnd 16492  df-mhm 16537  df-submnd 16538  df-grp 16628  df-minusg 16629  df-sbg 16630  df-mulg 16631  df-subg 16769  df-cntz 16926  df-cmn 17371  df-abl 17372  df-mgp 17663  df-ur 17675  df-ring 17721  df-cring 17722  df-subrg 17945  df-abv 17984  df-lmod 18032  df-scaf 18033  df-sra 18334  df-rgmod 18335  df-psmet 18901  df-xmet 18902  df-met 18903  df-bl 18904  df-mopn 18905  df-fbas 18906  df-fg 18907  df-cnfld 18910  df-top 19856  df-bases 19857  df-topon 19858  df-topsp 19859  df-cld 19969  df-ntr 19970  df-cls 19971  df-nei 20049  df-lp 20087  df-perf 20088  df-cn 20178  df-cnp 20179  df-haus 20266  df-tx 20512  df-hmeo 20705  df-fil 20796  df-fm 20888  df-flim 20889  df-flf 20890  df-tmd 21022  df-tgp 21023  df-tsms 21076  df-trg 21109  df-xms 21270  df-ms 21271  df-tms 21272  df-nm 21532  df-ngp 21533  df-nrg 21535  df-nlm 21536  df-ii 21809  df-cncf 21810  df-limc 22706  df-dv 22707  df-log 23379  df-esum 28696  df-siga 28777  df-meas 28865  df-prob 29075
This theorem is referenced by:  probdif  29087
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