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Theorem probun 24630
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 996 . . . 4  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =  B )  /\  ( A  i^i  B )  =  (/) )  ->  P  e. Prob
)
2 simplr 732 . . . 4  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =  B )  /\  ( A  i^i  B )  =  (/) )  ->  A  =  B )
3 simpr 448 . . . 4  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =  B )  /\  ( A  i^i  B )  =  (/) )  ->  ( A  i^i  B )  =  (/) )
4 disj3 3632 . . . . . . . . . . 11  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
54biimpi 187 . . . . . . . . . 10  |-  ( ( A  i^i  B )  =  (/)  ->  A  =  ( A  \  B
) )
6 difeq1 3418 . . . . . . . . . . 11  |-  ( A  =  B  ->  ( A  \  B )  =  ( B  \  B
) )
7 difid 3656 . . . . . . . . . . 11  |-  ( B 
\  B )  =  (/)
86, 7syl6eq 2452 . . . . . . . . . 10  |-  ( A  =  B  ->  ( A  \  B )  =  (/) )
95, 8sylan9eqr 2458 . . . . . . . . 9  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  A  =  (/) )
10 eqtr2 2422 . . . . . . . . . 10  |-  ( ( A  =  B  /\  A  =  (/) )  ->  B  =  (/) )
119, 10syldan 457 . . . . . . . . 9  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  B  =  (/) )
129, 11uneq12d 3462 . . . . . . . 8  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  =  ( (/)  u.  (/) ) )
13 unidm 3450 . . . . . . . 8  |-  ( (/)  u.  (/) )  =  (/)
1412, 13syl6eq 2452 . . . . . . 7  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  =  (/) )
1514fveq2d 5691 . . . . . 6  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( P `  ( A  u.  B ) )  =  ( P `  (/) ) )
16 probnul 24625 . . . . . 6  |-  ( P  e. Prob  ->  ( P `  (/) )  =  0 )
1715, 16sylan9eqr 2458 . . . . 5  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( P `  ( A  u.  B )
)  =  0 )
189fveq2d 5691 . . . . . . . 8  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( P `  A )  =  ( P `  (/) ) )
1918, 16sylan9eqr 2458 . . . . . . 7  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( P `  A
)  =  0 )
2011fveq2d 5691 . . . . . . . 8  |-  ( ( A  =  B  /\  ( A  i^i  B )  =  (/) )  ->  ( P `  B )  =  ( P `  (/) ) )
2120, 16sylan9eqr 2458 . . . . . . 7  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( P `  B
)  =  0 )
2219, 21oveq12d 6058 . . . . . 6  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( ( P `  A )  +  ( P `  B ) )  =  ( 0  +  0 ) )
23 00id 9197 . . . . . 6  |-  ( 0  +  0 )  =  0
2422, 23syl6eq 2452 . . . . 5  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( ( P `  A )  +  ( P `  B ) )  =  0 )
2517, 24eqtr4d 2439 . . . 4  |-  ( ( P  e. Prob  /\  ( A  =  B  /\  ( A  i^i  B )  =  (/) ) )  -> 
( P `  ( A  u.  B )
)  =  ( ( P `  A )  +  ( P `  B ) ) )
261, 2, 3, 25syl12anc 1182 . . 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 424 . 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 940 . . . . . . 7  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  <->  ( P  e. Prob  /\  ( A  e. 
dom  P  /\  B  e. 
dom  P ) ) )
2928anbi1i 677 . . . . . 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 938 . . . . . 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 244 . . . . 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 960 . . . . . . 7  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  P  e. Prob
)
33 prssi 3914 . . . . . . . . . 10  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P )  ->  { A ,  B }  C_  dom  P )
34333ad2ant2 979 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  { A ,  B }  C_  dom  P )
3534adantr 452 . . . . . . . 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 4366 . . . . . . . . 9  |-  { A ,  B }  e.  _V
3736elpw 3765 . . . . . . . 8  |-  ( { A ,  B }  e.  ~P dom  P  <->  { A ,  B }  C_  dom  P )
3835, 37sylibr 204 . . . . . . 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 24057 . . . . . . . . 9  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P )  ->  { A ,  B }  ~<_  om )
40393ad2ant2 979 . . . . . . . 8  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  { A ,  B }  ~<_  om )
4140adantr 452 . . . . . . 7  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  { A ,  B }  ~<_  om )
42 simp2l 983 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  A  e.  dom  P )
43 simp2r 984 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  B  e.  dom  P )
44 simp3 959 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  A  =/=  B )
45 id 20 . . . . . . . . . 10  |-  ( x  =  A  ->  x  =  A )
46 id 20 . . . . . . . . . 10  |-  ( x  =  B  ->  x  =  B )
4745, 46disjprg 4168 . . . . . . . . 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 1184 . . . . . . . 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 472 . . . . . . 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 24629 . . . . . . 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 1188 . . . . . 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 3990 . . . . . . . . . 10  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P )  ->  U. { A ,  B }  =  ( A  u.  B ) )
5352fveq2d 5691 . . . . . . . . 9  |-  ( ( A  e.  dom  P  /\  B  e.  dom  P )  ->  ( P `  U. { A ,  B } )  =  ( P `  ( A  u.  B ) ) )
54533ad2ant2 979 . . . . . . . 8  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  U. { A ,  B } )  =  ( P `  ( A  u.  B ) ) )
55 fveq2 5687 . . . . . . . . . 10  |-  ( x  =  A  ->  ( P `  x )  =  ( P `  A ) )
5655adantl 453 . . . . . . . . 9  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  x  =  A )  ->  ( P `  x )  =  ( P `  A ) )
57 fveq2 5687 . . . . . . . . . 10  |-  ( x  =  B  ->  ( P `  x )  =  ( P `  B ) )
5857adantl 453 . . . . . . . . 9  |-  ( ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  x  =  B )  ->  ( P `  x )  =  ( P `  B ) )
59 unitssxrge0 24251 . . . . . . . . . 10  |-  ( 0 [,] 1 )  C_  ( 0 [,]  +oo )
60 simp1 957 . . . . . . . . . . 11  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  P  e. Prob )
61 prob01 24624 . . . . . . . . . . 11  |-  ( ( P  e. Prob  /\  A  e.  dom  P )  -> 
( P `  A
)  e.  ( 0 [,] 1 ) )
6260, 42, 61syl2anc 643 . . . . . . . . . 10  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  A )  e.  ( 0 [,] 1 ) )
6359, 62sseldi 3306 . . . . . . . . 9  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  A )  e.  ( 0 [,]  +oo )
)
64 prob01 24624 . . . . . . . . . . 11  |-  ( ( P  e. Prob  /\  B  e.  dom  P )  -> 
( P `  B
)  e.  ( 0 [,] 1 ) )
6560, 43, 64syl2anc 643 . . . . . . . . . 10  |-  ( ( P  e. Prob  /\  ( A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/=  B
)  ->  ( P `  B )  e.  ( 0 [,] 1 ) )
6659, 65sseldi 3306 . . . . . . . . 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 24410 . . . . . . . 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 2418 . . . . . . 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 452 . . . . . 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 202 . . . . 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 459 . . . 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 10998 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
73 simpll1 996 . . . . . . 7  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  P  e. Prob
)
74 simpll2 997 . . . . . . 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 643 . . . . . 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 3306 . . . . 5  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 A )  e.  RR )
77 simpll3 998 . . . . . . 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 643 . . . . . 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 3306 . . . . 5  |-  ( ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  =/= 
B )  /\  ( A  i^i  B )  =  (/) )  ->  ( P `
 B )  e.  RR )
80 rexadd 10774 . . . . 5  |-  ( ( ( P `  A
)  e.  RR  /\  ( P `  B )  e.  RR )  -> 
( ( P `  A ) + e
( P `  B
) )  =  ( ( P `  A
)  +  ( P `
 B ) ) )
8176, 79, 80syl2anc 643 . . . 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 2436 . . 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 424 . 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 2645 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {cpr 3775   U.cuni 3975  Disj wdisj 4142   class class class wbr 4172   omcom 4804   dom cdm 4837   ` cfv 5413  (class class class)co 6040    ~<_ cdom 7066   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    +oocpnf 9073   + ecxad 10664   [,]cicc 10875  Σ*cesum 24377  Probcprb 24618
This theorem is referenced by:  probdif  24631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-ac2 8299  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-ac 7953  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-ordt 13680  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-ps 14584  df-tsr 14585  df-mnd 14645  df-plusf 14646  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-subrg 15821  df-abv 15860  df-lmod 15907  df-scaf 15908  df-sra 16199  df-rgmod 16200  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-tmd 18055  df-tgp 18056  df-tsms 18109  df-trg 18142  df-xms 18303  df-ms 18304  df-tms 18305  df-nm 18583  df-ngp 18584  df-nrg 18586  df-nlm 18587  df-ii 18860  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-esum 24378  df-siga 24444  df-meas 24503  df-prob 24619
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