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Theorem probun 26801
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 1027 . . . 4  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =  B )  /\  ( A  i^i  B )  =  (/) )  ->  P  e. Prob
)
2 simplr 754 . . . 4  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =  B )  /\  ( A  i^i  B )  =  (/) )  ->  A  =  B )
3 simpr 461 . . . 4  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =  B )  /\  ( A  i^i  B )  =  (/) )  ->  ( A  i^i  B )  =  (/) )
4 disj3 3722 . . . . . . . . . . 11  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
54biimpi 194 . . . . . . . . . 10  |-  ( ( A  i^i  B )  =  (/)  ->  A  =  ( A  \  B
) )
6 difeq1 3466 . . . . . . . . . . 11  |-  ( A  =  B  ->  ( A  \  B )  =  ( B  \  B
) )
7 difid 3746 . . . . . . . . . . 11  |-  ( B 
\  B )  =  (/)
86, 7syl6eq 2490 . . . . . . . . . 10  |-  ( A  =  B  ->  ( A  \  B )  =  (/) )
95, 8sylan9eqr 2496 . . . . . . . . 9  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  A  =  (/) )
10 eqtr2 2460 . . . . . . . . . 10  |-  ( ( A  =  B  /\  A  =  (/) )  ->  B  =  (/) )
119, 10syldan 470 . . . . . . . . 9  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  B  =  (/) )
129, 11uneq12d 3510 . . . . . . . 8  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  =  ( (/)  u.  (/) ) )
13 unidm 3498 . . . . . . . 8  |-  ( (/)  u.  (/) )  =  (/)
1412, 13syl6eq 2490 . . . . . . 7  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  =  (/) )
1514fveq2d 5694 . . . . . 6  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( P `  ( A  u.  B ) )  =  ( P `  (/) ) )
16 probnul 26796 . . . . . 6  |-  ( P  e. Prob  ->  ( P `  (/) )  =  0 )
1715, 16sylan9eqr 2496 . . . . 5  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( P `  ( A  u.  B )
)  =  0 )
189fveq2d 5694 . . . . . . . 8  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( P `  A )  =  ( P `  (/) ) )
1918, 16sylan9eqr 2496 . . . . . . 7  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( P `  A
)  =  0 )
2011fveq2d 5694 . . . . . . . 8  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( P `  B )  =  ( P `  (/) ) )
2120, 16sylan9eqr 2496 . . . . . . 7  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( P `  B
)  =  0 )
2219, 21oveq12d 6108 . . . . . 6  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( ( P `  A )  +  ( P `  B ) )  =  ( 0  +  0 ) )
23 00id 9543 . . . . . 6  |-  ( 0  +  0 )  =  0
2422, 23syl6eq 2490 . . . . 5  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( ( P `  A )  +  ( P `  B ) )  =  0 )
2517, 24eqtr4d 2477 . . . 4  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( P `  ( A  u.  B )
)  =  ( ( P `  A )  +  ( P `  B ) ) )
261, 2, 3, 25syl12anc 1216 . . 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 434 . 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 969 . . . . . . 7  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  <->  ( P  e. Prob  /\  ( A  e. 
dom  P  /\  B  e. 
dom  P ) ) )
2928anbi1i 695 . . . . . 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 967 . . . . . 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 252 . . . . 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 991 . . . . . . 7  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  P  e. Prob
)
33 prssi 4028 . . . . . . . . . 10  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P )  ->  { A ,  B }  C_  dom  P )
34333ad2ant2 1010 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  { A ,  B }  C_  dom  P )
3534adantr 465 . . . . . . . 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 4533 . . . . . . . . 9  |-  { A ,  B }  e.  _V
3736elpw 3865 . . . . . . . 8  |-  ( { A ,  B }  e.  ~P dom  P  <->  { A ,  B }  C_  dom  P )
3835, 37sylibr 212 . . . . . . 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 26011 . . . . . . . . 9  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P )  ->  { A ,  B }  ~<_  om )
40393ad2ant2 1010 . . . . . . . 8  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  { A ,  B }  ~<_  om )
4140adantr 465 . . . . . . 7  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  { A ,  B }  ~<_  om )
42 simp2l 1014 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  A  e.  dom  P )
43 simp2r 1015 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  B  e.  dom  P )
44 simp3 990 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  A  =/=  B )
45 id 22 . . . . . . . . . 10  |-  ( x  =  A  ->  x  =  A )
46 id 22 . . . . . . . . . 10  |-  ( x  =  B  ->  x  =  B )
4745, 46disjprg 4287 . . . . . . . . 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 1218 . . . . . . . 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 485 . . . . . . 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 26800 . . . . . . 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 1222 . . . . . 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 4104 . . . . . . . . . 10  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P )  ->  U. { A ,  B }  =  ( A  u.  B ) )
5352fveq2d 5694 . . . . . . . . 9  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P )  ->  ( P `  U. { A ,  B } )  =  ( P `  ( A  u.  B ) ) )
54533ad2ant2 1010 . . . . . . . 8  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  U. { A ,  B } )  =  ( P `  ( A  u.  B ) ) )
55 fveq2 5690 . . . . . . . . . 10  |-  ( x  =  A  ->  ( P `  x )  =  ( P `  A ) )
5655adantl 466 . . . . . . . . 9  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  x  =  A )  ->  ( P `  x )  =  ( P `  A ) )
57 fveq2 5690 . . . . . . . . . 10  |-  ( x  =  B  ->  ( P `  x )  =  ( P `  B ) )
5857adantl 466 . . . . . . . . 9  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  x  =  B )  ->  ( P `  x )  =  ( P `  B ) )
59 unitssxrge0 26329 . . . . . . . . . 10  |-  ( 0 [,] 1 )  C_  ( 0 [,] +oo )
60 simp1 988 . . . . . . . . . . 11  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  P  e. Prob )
61 prob01 26795 . . . . . . . . . . 11  |-  ( ( P  e. Prob  /\  A  e.  dom  P )  -> 
( P `  A
)  e.  ( 0 [,] 1 ) )
6260, 42, 61syl2anc 661 . . . . . . . . . 10  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  A )  e.  ( 0 [,] 1 ) )
6359, 62sseldi 3353 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  A )  e.  ( 0 [,] +oo )
)
64 prob01 26795 . . . . . . . . . . 11  |-  ( ( P  e. Prob  /\  B  e.  dom  P )  -> 
( P `  B
)  e.  ( 0 [,] 1 ) )
6560, 43, 64syl2anc 661 . . . . . . . . . 10  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  B )  e.  ( 0 [,] 1 ) )
6659, 65sseldi 3353 . . . . . . . . 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 26515 . . . . . . . 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 2456 . . . . . . 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 465 . . . . . 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 210 . . . . 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 472 . . . 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 11431 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
73 simpll1 1027 . . . . . . 7  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  P  e. Prob
)
74 simpll2 1028 . . . . . . 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 661 . . . . . 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 3353 . . . . 5  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 A )  e.  RR )
77 simpll3 1029 . . . . . . 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 661 . . . . . 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 3353 . . . . 5  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 B )  e.  RR )
80 rexadd 11201 . . . . 5  |-  ( ( ( P `  A
)  e.  RR  /\  ( P `  B )  e.  RR )  -> 
( ( P `  A ) +e
( P `  B
) )  =  ( ( P `  A
)  +  ( P `
 B ) ) )
8176, 79, 80syl2anc 661 . . . 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 2474 . . 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 434 . 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 2688 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2605    \ cdif 3324    u. cun 3325    i^i cin 3326    C_ wss 3327   (/)c0 3636   ~Pcpw 3859   {cpr 3878   U.cuni 4090  Disj wdisj 4261   class class class wbr 4291   dom cdm 4839   ` cfv 5417  (class class class)co 6090   omcom 6475    ~<_ cdom 7307   RRcr 9280   0cc0 9281   1c1 9282    + caddc 9284   +oocpnf 9414   +ecxad 11086   [,]cicc 11302  Σ*cesum 26482  Probcprb 26789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-ac2 8631  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-disj 4262  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  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-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-fi 7660  df-sup 7690  df-oi 7723  df-card 8108  df-acn 8111  df-ac 8285  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ioo 11303  df-ioc 11304  df-ico 11305  df-icc 11306  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-fac 12051  df-bc 12078  df-hash 12103  df-shft 12555  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-limsup 12948  df-clim 12965  df-rlim 12966  df-sum 13163  df-ef 13352  df-sin 13354  df-cos 13355  df-pi 13357  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-hom 14261  df-cco 14262  df-rest 14360  df-topn 14361  df-0g 14379  df-gsum 14380  df-topgen 14381  df-pt 14382  df-prds 14385  df-ordt 14438  df-xrs 14439  df-qtop 14444  df-imas 14445  df-xps 14447  df-mre 14523  df-mrc 14524  df-acs 14526  df-ps 15369  df-tsr 15370  df-mnd 15414  df-plusf 15415  df-mhm 15463  df-submnd 15464  df-grp 15544  df-minusg 15545  df-sbg 15546  df-mulg 15547  df-subg 15677  df-cntz 15834  df-cmn 16278  df-abl 16279  df-mgp 16591  df-ur 16603  df-rng 16646  df-cring 16647  df-subrg 16862  df-abv 16901  df-lmod 16949  df-scaf 16950  df-sra 17252  df-rgmod 17253  df-psmet 17808  df-xmet 17809  df-met 17810  df-bl 17811  df-mopn 17812  df-fbas 17813  df-fg 17814  df-cnfld 17818  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-cld 18622  df-ntr 18623  df-cls 18624  df-nei 18701  df-lp 18739  df-perf 18740  df-cn 18830  df-cnp 18831  df-haus 18918  df-tx 19134  df-hmeo 19327  df-fil 19418  df-fm 19510  df-flim 19511  df-flf 19512  df-tmd 19642  df-tgp 19643  df-tsms 19696  df-trg 19733  df-xms 19894  df-ms 19895  df-tms 19896  df-nm 20174  df-ngp 20175  df-nrg 20177  df-nlm 20178  df-ii 20452  df-cncf 20453  df-limc 21340  df-dv 21341  df-log 22007  df-esum 26483  df-siga 26550  df-meas 26609  df-prob 26790
This theorem is referenced by:  probdif  26802
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