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Theorem probun 30481
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 1100 . . . 4 |- ( ( ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) /\ A = B ) /\ ( A i^i B ) = (/) ) -> P e. Prob )
2 simplr 792 . . . 4 |- ( ( ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) /\ A = B ) /\ ( A i^i B ) = (/) ) -> A = B )
3 simpr 477 . . . 4 |- ( ( ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) /\ A = B ) /\ ( A i^i B ) = (/) ) -> ( A i^i B ) = (/) )
4 disj3 4021 . . . . . . . . . . 11 |- ( ( A i^i B ) = (/) <-> A = ( A \ B ) )
54biimpi 206 . . . . . . . . . 10 |- ( ( A i^i B ) = (/) -> A = ( A \ B ) )
6 difeq1 3721 . . . . . . . . . . 11 |- ( A = B -> ( A \ B ) = ( B \ B ) )
7 difid 3948 . . . . . . . . . . 11 |- ( B \ B ) = (/)
86, 7syl6eq 2672 . . . . . . . . . 10 |- ( A = B -> ( A \ B ) = (/) )
95, 8sylan9eqr 2678 . . . . . . . . 9 |- ( ( A = B /\ ( A i^i B ) = (/) ) -> A = (/) )
10 eqtr2 2642 . . . . . . . . . 10 |- ( ( A = B /\ A = (/) ) -> B = (/) )
119, 10syldan 487 . . . . . . . . 9 |- ( ( A = B /\ ( A i^i B ) = (/) ) -> B = (/) )
129, 11uneq12d 3768 . . . . . . . 8 |- ( ( A = B /\ ( A i^i B ) = (/) ) -> ( A u. B ) = ( (/) u. (/) ) )
13 unidm 3756 . . . . . . . 8 |- ( (/) u. (/) ) = (/)
1412, 13syl6eq 2672 . . . . . . 7 |- ( ( A = B /\ ( A i^i B ) = (/) ) -> ( A u. B ) = (/) )
1514fveq2d 6195 . . . . . 6 |- ( ( A = B /\ ( A i^i B ) = (/) ) -> ( P ` ( A u. B ) ) = ( P ` (/) ) )
16 probnul 30476 . . . . . 6 |- ( P e. Prob -> ( P ` (/) ) = 0 )
1715, 16sylan9eqr 2678 . . . . 5 |- ( ( P e. Prob /\ ( A = B /\ ( A i^i B ) = (/) ) ) -> ( P ` ( A u. B ) ) = 0 )
189fveq2d 6195 . . . . . . . 8 |- ( ( A = B /\ ( A i^i B ) = (/) ) -> ( P ` A ) = ( P ` (/) ) )
1918, 16sylan9eqr 2678 . . . . . . 7 |- ( ( P e. Prob /\ ( A = B /\ ( A i^i B ) = (/) ) ) -> ( P ` A ) = 0 )
2011fveq2d 6195 . . . . . . . 8 |- ( ( A = B /\ ( A i^i B ) = (/) ) -> ( P ` B ) = ( P ` (/) ) )
2120, 16sylan9eqr 2678 . . . . . . 7 |- ( ( P e. Prob /\ ( A = B /\ ( A i^i B ) = (/) ) ) -> ( P ` B ) = 0 )
2219, 21oveq12d 6668 . . . . . 6 |- ( ( P e. Prob /\ ( A = B /\ ( A i^i B ) = (/) ) ) -> ( ( P ` A ) + ( P ` B ) ) = ( 0 + 0 ) )
23 00id 10211 . . . . . 6 |- ( 0 + 0 ) = 0
2422, 23syl6eq 2672 . . . . 5 |- ( ( P e. Prob /\ ( A = B /\ ( A i^i B ) = (/) ) ) -> ( ( P ` A ) + ( P ` B ) ) = 0 )
2517, 24eqtr4d 2659 . . . 4 |- ( ( P e. Prob /\ ( A = B /\ ( A i^i B ) = (/) ) ) -> ( P ` ( A u. B ) ) = ( ( P ` A ) + ( P ` B ) ) )
261, 2, 3, 25syl12anc 1324 . . 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 450 . 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 1042 . . . . . . 7 |- ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) <-> ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) ) )
2928anbi1i 731 . . . . . 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 1039 . . . . . 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 267 . . . . 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 1064 . . . . . . 7 |- ( ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) /\ ( A i^i B ) = (/) ) -> P e. Prob )
33 prssi 4353 . . . . . . . . . 10 |- ( ( A e. dom P /\ B e. dom P ) -> { A , B } C_ dom P )
34333ad2ant2 1083 . . . . . . . . 9 |- ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) -> { A , B } C_ dom P )
3534adantr 481 . . . . . . . 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 4909 . . . . . . . . 9 |- { A , B } e. _V
3736elpw 4164 . . . . . . . 8 |- ( { A , B } e. ~P dom P <-> { A , B } C_ dom P )
3835, 37sylibr 224 . . . . . . 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 29492 . . . . . . . . 9 |- ( ( A e. dom P /\ B e. dom P ) -> { A , B } ~<_ om )
40393ad2ant2 1083 . . . . . . . 8 |- ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) -> { A , B } ~<_ om )
4140adantr 481 . . . . . . 7 |- ( ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) /\ ( A i^i B ) = (/) ) -> { A , B } ~<_ om )
42 simp2l 1087 . . . . . . . . 9 |- ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) -> A e. dom P )
43 simp2r 1088 . . . . . . . . 9 |- ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) -> B e. dom P )
44 simp3 1063 . . . . . . . . 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 4648 . . . . . . . . 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 1326 . . . . . . . 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 502 . . . . . . 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 30480 . . . . . . 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 1330 . . . . . 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 4450 . . . . . . . . . 10 |- ( ( A e. dom P /\ B e. dom P ) -> U. { A , B } = ( A u. B ) )
5352fveq2d 6195 . . . . . . . . 9 |- ( ( A e. dom P /\ B e. dom P ) -> ( P ` U. { A , B } ) = ( P ` ( A u. B ) ) )
54533ad2ant2 1083 . . . . . . . 8 |- ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) -> ( P ` U. { A , B } ) = ( P ` ( A u. B ) ) )
55 fveq2 6191 . . . . . . . . . 10 |- ( x = A -> ( P ` x ) = ( P ` A ) )
5655adantl 482 . . . . . . . . 9 |- ( ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) /\ x = A ) -> ( P ` x ) = ( P ` A ) )
57 fveq2 6191 . . . . . . . . . 10 |- ( x = B -> ( P ` x ) = ( P ` B ) )
5857adantl 482 . . . . . . . . 9 |- ( ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) /\ x = B ) -> ( P ` x ) = ( P ` B ) )
59 unitssxrge0 29946 . . . . . . . . . 10 |- ( 0 [,] 1 ) C_ ( 0 [,] +oo )
60 simp1 1061 . . . . . . . . . . 11 |- ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) -> P e. Prob )
61 prob01 30475 . . . . . . . . . . 11 |- ( ( P e. Prob /\ A e. dom P ) -> ( P ` A ) e. ( 0 [,] 1 ) )
6260, 42, 61syl2anc 693 . . . . . . . . . 10 |- ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) -> ( P ` A ) e. ( 0 [,] 1 ) )
6359, 62sseldi 3601 . . . . . . . . 9 |- ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) -> ( P ` A ) e. ( 0 [,] +oo ) )
64 prob01 30475 . . . . . . . . . . 11 |- ( ( P e. Prob /\ B e. dom P ) -> ( P ` B ) e. ( 0 [,] 1 ) )
6560, 43, 64syl2anc 693 . . . . . . . . . 10 |- ( ( P e. Prob /\ ( A e. dom P /\ B e. dom P ) /\ A =/= B ) -> ( P ` B ) e. ( 0 [,] 1 ) )
6659, 65sseldi 3601 . . . . . . . . 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 30128 . . . . . . . 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 2637 . . . . . . 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 481 . . . . . 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 222 . . . . 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 489 . . . 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 12319 . . . . . 6 |- ( 0 [,] 1 ) C_ RR
73 simpll1 1100 . . . . . . 7 |- ( ( ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) /\ A =/= B ) /\ ( A i^i B ) = (/) ) -> P e. Prob )
74 simpll2 1101 . . . . . . 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 693 . . . . . 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 3601 . . . . 5 |- ( ( ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) /\ A =/= B ) /\ ( A i^i B ) = (/) ) -> ( P ` A ) e. RR )
77 simpll3 1102 . . . . . . 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 693 . . . . . 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 3601 . . . . 5 |- ( ( ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) /\ A =/= B ) /\ ( A i^i B ) = (/) ) -> ( P ` B ) e. RR )
80 rexadd 12063 . . . . 5 |- ( ( ( P ` A ) e. RR /\ ( P ` B ) e. RR ) -> ( ( P ` A ) +e ( P ` B ) ) = ( ( P ` A ) + ( P ` B ) ) )
8176, 79, 80syl2anc 693 . . . 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 2656 . . 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 450 . 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 2881 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {cpr 4179   U.cuni 4436  Disj wdisj 4620   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   +ecxad 11944   [,]cicc 12178  Σ*cesum 30089  Probcprb 30469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-ordt 16161  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-abv 18817  df-lmod 18865  df-scaf 18866  df-sra 19172  df-rgmod 19173  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tmd 21876  df-tgp 21877  df-tsms 21930  df-trg 21963  df-xms 22125  df-ms 22126  df-tms 22127  df-nm 22387  df-ngp 22388  df-nrg 22390  df-nlm 22391  df-ii 22680  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-esum 30090  df-siga 30171  df-meas 30259  df-prob 30470
This theorem is referenced by:  probdif  30482
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