Theorem dstrvprob 30533

Description: The distribution of a random variable is a probability law. (TODO: could be shortened using dstrvval 30532) (Contributed by Thierry Arnoux, 10-Feb-2017.)

Hypotheses

Ref	Expression
dstrvprob.1	|- ( ph -> P e. Prob )
dstrvprob.2	|- ( ph -> X e. (rRndVar ` P ) )
dstrvprob.3	|- ( ph -> D = ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) ) )

Assertion

Ref	Expression
dstrvprob	|- ( ph -> D e. Prob )

Distinct variable groups:   P, a   X, a   D, a   ph, a

Proof of Theorem dstrvprob

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dstrvprob.3	. . . . . 6 |- ( ph -> D = ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) ) )
2		dstrvprob.1	. . . . . . . . 9 |- ( ph -> P e. Prob )
3	2	adantr 481	. . . . . . . 8 |- ( ( ph /\ a e. 𝔅ℝ ) -> P e. Prob )
4		dstrvprob.2	. . . . . . . . . 10 |- ( ph -> X e. (rRndVar ` P ) )
5	4	adantr 481	. . . . . . . . 9 |- ( ( ph /\ a e. 𝔅ℝ ) -> X e. (rRndVar ` P ) )
6		simpr 477	. . . . . . . . 9 |- ( ( ph /\ a e. 𝔅ℝ ) -> a e. 𝔅ℝ )
7	3, 5, 6	orvcelel 30531	. . . . . . . 8 |- ( ( ph /\ a e. 𝔅ℝ ) -> ( X∘RV/𝑐 _E a ) e. dom P )
8		prob01 30475	. . . . . . . 8 |- ( ( P e. Prob /\ ( X∘RV/𝑐 _E a ) e. dom P ) -> ( P ` ( X∘RV/𝑐 _E a ) ) e. ( 0 [,] 1 ) )
9	3, 7, 8	syl2anc 693	. . . . . . 7 |- ( ( ph /\ a e. 𝔅ℝ ) -> ( P ` ( X∘RV/𝑐 _E a ) ) e. ( 0 [,] 1 ) )
10		elunitrn 29943	. . . . . . . . 9 |- ( ( P ` ( X∘RV/𝑐 _E a ) ) e. ( 0 [,] 1 ) -> ( P ` ( X∘RV/𝑐 _E a ) ) e. RR )
11	10	rexrd 10089	. . . . . . . 8 |- ( ( P ` ( X∘RV/𝑐 _E a ) ) e. ( 0 [,] 1 ) -> ( P ` ( X∘RV/𝑐 _E a ) ) e. RR* )
12		elunitge0 29945	. . . . . . . 8 |- ( ( P ` ( X∘RV/𝑐 _E a ) ) e. ( 0 [,] 1 ) -> 0 <_ ( P ` ( X∘RV/𝑐 _E a ) ) )
13		elxrge0 12281	. . . . . . . 8 |- ( ( P ` ( X∘RV/𝑐 _E a ) ) e. ( 0 [,] +oo ) <-> ( ( P ` ( X∘RV/𝑐 _E a ) ) e. RR* /\ 0 <_ ( P ` ( X∘RV/𝑐 _E a ) ) ) )
14	11, 12, 13	sylanbrc 698	. . . . . . 7 |- ( ( P ` ( X∘RV/𝑐 _E a ) ) e. ( 0 [,] 1 ) -> ( P ` ( X∘RV/𝑐 _E a ) ) e. ( 0 [,] +oo ) )
15	9, 14	syl 17	. . . . . 6 |- ( ( ph /\ a e. 𝔅ℝ ) -> ( P ` ( X∘RV/𝑐 _E a ) ) e. ( 0 [,] +oo ) )
16	1, 15	fmpt3d 6386	. . . . 5 |- ( ph -> D :𝔅ℝ --> ( 0 [,] +oo ) )
17		simpr 477	. . . . . . . . 9 |- ( ( ph /\ a = (/) ) -> a = (/) )
18	17	oveq2d 6666	. . . . . . . 8 |- ( ( ph /\ a = (/) ) -> ( X∘RV/𝑐 _E a ) = ( X∘RV/𝑐 _E (/) ) )
19	18	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ a = (/) ) -> ( P ` ( X∘RV/𝑐 _E a ) ) = ( P ` ( X∘RV/𝑐 _E (/) ) ) )
20		brsigarn 30247	. . . . . . . . 9 |- 𝔅ℝ e. (sigAlgebra ` RR )
21		elrnsiga 30189	. . . . . . . . 9 |- (𝔅ℝ e. (sigAlgebra ` RR ) -> 𝔅ℝ e. U. ran sigAlgebra )
22		0elsiga 30177	. . . . . . . . 9 |- (𝔅ℝ e. U. ran sigAlgebra -> (/) e. 𝔅ℝ )
23	20, 21, 22	mp2b 10	. . . . . . . 8 |- (/) e. 𝔅ℝ
24	23	a1i 11	. . . . . . 7 |- ( ph -> (/) e. 𝔅ℝ )
25	2, 4, 24	orvcelel 30531	. . . . . . . 8 |- ( ph -> ( X∘RV/𝑐 _E (/) ) e. dom P )
26	2, 25	probvalrnd 30486	. . . . . . 7 |- ( ph -> ( P ` ( X∘RV/𝑐 _E (/) ) ) e. RR )
27	1, 19, 24, 26	fvmptd 6288	. . . . . 6 |- ( ph -> ( D ` (/) ) = ( P ` ( X∘RV/𝑐 _E (/) ) ) )
28	2, 4, 24	orvcelval 30530	. . . . . . 7 |- ( ph -> ( X∘RV/𝑐 _E (/) ) = ( `' X " (/) ) )
29	28	fveq2d 6195	. . . . . 6 |- ( ph -> ( P ` ( X∘RV/𝑐 _E (/) ) ) = ( P ` ( `' X " (/) ) ) )
30		ima0 5481	. . . . . . . 8 |- ( `' X " (/) ) = (/)
31	30	fveq2i 6194	. . . . . . 7 |- ( P ` ( `' X " (/) ) ) = ( P ` (/) )
32		probnul 30476	. . . . . . . 8 |- ( P e. Prob -> ( P ` (/) ) = 0 )
33	2, 32	syl 17	. . . . . . 7 |- ( ph -> ( P ` (/) ) = 0 )
34	31, 33	syl5eq 2668	. . . . . 6 |- ( ph -> ( P ` ( `' X " (/) ) ) = 0 )
35	27, 29, 34	3eqtrd 2660	. . . . 5 |- ( ph -> ( D ` (/) ) = 0 )
36	2, 4	rrvvf 30506	. . . . . . . . . . . 12 |- ( ph -> X : U. dom P --> RR )
37	36	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> X : U. dom P --> RR )
38		ffun 6048	. . . . . . . . . . 11 |- ( X : U. dom P --> RR -> Fun X )
39	37, 38	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> Fun X )
40		unipreima 29446	. . . . . . . . . . 11 |- ( Fun X -> ( `' X " U. x ) = U_ a e. x ( `' X " a ) )
41	40	fveq2d 6195	. . . . . . . . . 10 |- ( Fun X -> ( P ` ( `' X " U. x ) ) = ( P ` U_ a e. x ( `' X " a ) ) )
42	39, 41	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> ( P ` ( `' X " U. x ) ) = ( P ` U_ a e. x ( `' X " a ) ) )
43	2	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> P e. Prob )
44		domprobmeas 30472	. . . . . . . . . . 11 |- ( P e. Prob -> P e. (measures ` dom P ) )
45	43, 44	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> P e. (measures ` dom P ) )
46		nfv 1843	. . . . . . . . . . . 12 |- F/ a ( ph /\ x e. ~P𝔅ℝ )
47		nfv 1843	. . . . . . . . . . . . 13 |- F/ a x ~<_ om
48		nfdisj1 4633	. . . . . . . . . . . . 13 |- F/ aDisj a e. x a
49	47, 48	nfan 1828	. . . . . . . . . . . 12 |- F/ a ( x ~<_ om /\ Disj a e. x a )
50	46, 49	nfan 1828	. . . . . . . . . . 11 |- F/ a ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) )
51		simplll 798	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) /\ a e. x ) -> ph )
52		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) /\ a e. x ) -> a e. x )
53		simpllr 799	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) /\ a e. x ) -> x e. ~P𝔅ℝ )
54		elelpwi 4171	. . . . . . . . . . . . . 14 |- ( ( a e. x /\ x e. ~P𝔅ℝ ) -> a e. 𝔅ℝ )
55	52, 53, 54	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) /\ a e. x ) -> a e. 𝔅ℝ )
56	2, 4	rrvfinvima 30512	. . . . . . . . . . . . . 14 |- ( ph -> A. a e. 𝔅ℝ ( `' X " a ) e. dom P )
57	56	r19.21bi 2932	. . . . . . . . . . . . 13 |- ( ( ph /\ a e. 𝔅ℝ ) -> ( `' X " a ) e. dom P )
58	51, 55, 57	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) /\ a e. x ) -> ( `' X " a ) e. dom P )
59	58	ex 450	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> ( a e. x -> ( `' X " a ) e. dom P ) )
60	50, 59	ralrimi 2957	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> A. a e. x ( `' X " a ) e. dom P )
61		simprl 794	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> x ~<_ om )
62		simprr 796	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> Disj a e. x a )
63		disjpreima 29397	. . . . . . . . . . 11 |- ( ( Fun X /\ Disj a e. x a ) -> Disj a e. x ( `' X " a ) )
64	39, 62, 63	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> Disj a e. x ( `' X " a ) )
65		measvuni 30277	. . . . . . . . . 10 |- ( ( P e. (measures ` dom P ) /\ A. a e. x ( `' X " a ) e. dom P /\ ( x ~<_ om /\ Disj a e. x ( `' X " a ) ) ) -> ( P ` U_ a e. x ( `' X " a ) ) = Σ* a e. x ( P ` ( `' X " a ) ) )
66	45, 60, 61, 64, 65	syl112anc 1330	. . . . . . . . 9 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> ( P ` U_ a e. x ( `' X " a ) ) = Σ* a e. x ( P ` ( `' X " a ) ) )
67	42, 66	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> ( P ` ( `' X " U. x ) ) = Σ* a e. x ( P ` ( `' X " a ) ) )
68	4	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> X e. (rRndVar ` P ) )
69	1	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> D = ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) ) )
70	20, 21	mp1i 13	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> 𝔅ℝ e. U. ran sigAlgebra )
71		simplr 792	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> x e. ~P𝔅ℝ )
72		sigaclcu 30180	. . . . . . . . . 10 |- ( (𝔅ℝ e. U. ran sigAlgebra /\ x e. ~P𝔅ℝ /\ x ~<_ om ) -> U. x e. 𝔅ℝ )
73	70, 71, 61, 72	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> U. x e. 𝔅ℝ )
74	43, 68, 69, 73	dstrvval 30532	. . . . . . . 8 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> ( D ` U. x ) = ( P ` ( `' X " U. x ) ) )
75	1, 9	fvmpt2d 6293	. . . . . . . . . . . . 13 |- ( ( ph /\ a e. 𝔅ℝ ) -> ( D ` a ) = ( P ` ( X∘RV/𝑐 _E a ) ) )
76	51, 55, 75	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) /\ a e. x ) -> ( D ` a ) = ( P ` ( X∘RV/𝑐 _E a ) ) )
77	43	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) /\ a e. x ) -> P e. Prob )
78	68	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) /\ a e. x ) -> X e. (rRndVar ` P ) )
79	77, 78, 55	orvcelval 30530	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) /\ a e. x ) -> ( X∘RV/𝑐 _E a ) = ( `' X " a ) )
80	79	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) /\ a e. x ) -> ( P ` ( X∘RV/𝑐 _E a ) ) = ( P ` ( `' X " a ) ) )
81	76, 80	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) /\ a e. x ) -> ( D ` a ) = ( P ` ( `' X " a ) ) )
82	81	ex 450	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> ( a e. x -> ( D ` a ) = ( P ` ( `' X " a ) ) ) )
83	50, 82	ralrimi 2957	. . . . . . . . 9 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> A. a e. x ( D ` a ) = ( P ` ( `' X " a ) ) )
84	50, 83	esumeq2d 30099	. . . . . . . 8 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> Σ* a e. x ( D ` a ) = Σ* a e. x ( P ` ( `' X " a ) ) )
85	67, 74, 84	3eqtr4d 2666	. . . . . . 7 |- ( ( ( ph /\ x e. ~P𝔅ℝ ) /\ ( x ~<_ om /\ Disj a e. x a ) ) -> ( D ` U. x ) = Σ* a e. x ( D ` a ) )
86	85	ex 450	. . . . . 6 |- ( ( ph /\ x e. ~P𝔅ℝ ) -> ( ( x ~<_ om /\ Disj a e. x a ) -> ( D ` U. x ) = Σ* a e. x ( D ` a ) ) )
87	86	ralrimiva 2966	. . . . 5 |- ( ph -> A. x e. ~P 𝔅ℝ ( ( x ~<_ om /\ Disj a e. x a ) -> ( D ` U. x ) = Σ* a e. x ( D ` a ) ) )
88		ismeas 30262	. . . . . 6 |- (𝔅ℝ e. U. ran sigAlgebra -> ( D e. (measures ` 𝔅ℝ ) <-> ( D :𝔅ℝ --> ( 0 [,] +oo ) /\ ( D ` (/) ) = 0 /\ A. x e. ~P 𝔅ℝ ( ( x ~<_ om /\ Disj a e. x a ) -> ( D ` U. x ) = Σ* a e. x ( D ` a ) ) ) ) )
89	20, 21, 88	mp2b 10	. . . . 5 |- ( D e. (measures ` 𝔅ℝ ) <-> ( D :𝔅ℝ --> ( 0 [,] +oo ) /\ ( D ` (/) ) = 0 /\ A. x e. ~P 𝔅ℝ ( ( x ~<_ om /\ Disj a e. x a ) -> ( D ` U. x ) = Σ* a e. x ( D ` a ) ) ) )
90	16, 35, 87, 89	syl3anbrc 1246	. . . 4 |- ( ph -> D e. (measures ` 𝔅ℝ ) )
91	1	dmeqd 5326	. . . . . 6 |- ( ph -> dom D = dom ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) ) )
92	15	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. a e. 𝔅ℝ ( P ` ( X∘RV/𝑐 _E a ) ) e. ( 0 [,] +oo ) )
93		dmmptg 5632	. . . . . . 7 |- ( A. a e. 𝔅ℝ ( P ` ( X∘RV/𝑐 _E a ) ) e. ( 0 [,] +oo ) -> dom ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) ) = 𝔅ℝ )
94	92, 93	syl 17	. . . . . 6 |- ( ph -> dom ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) ) = 𝔅ℝ )
95	91, 94	eqtrd 2656	. . . . 5 |- ( ph -> dom D = 𝔅ℝ )
96	95	fveq2d 6195	. . . 4 |- ( ph -> (measures ` dom D ) = (measures ` 𝔅ℝ ) )
97	90, 96	eleqtrrd 2704	. . 3 |- ( ph -> D e. (measures ` dom D ) )
98		measbasedom 30265	. . 3 |- ( D e. U. ran measures <-> D e. (measures ` dom D ) )
99	97, 98	sylibr 224	. 2 |- ( ph -> D e. U. ran measures )
100	95	unieqd 4446	. . . . 5 |- ( ph -> U. dom D = U.𝔅ℝ )
101		unibrsiga 30249	. . . . 5 |- U.𝔅ℝ = RR
102	100, 101	syl6eq 2672	. . . 4 |- ( ph -> U. dom D = RR )
103	102	fveq2d 6195	. . 3 |- ( ph -> ( D ` U. dom D ) = ( D ` RR ) )
104		simpr 477	. . . . . . . 8 |- ( ( ph /\ a = RR ) -> a = RR )
105	104	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ a = RR ) -> ( X∘RV/𝑐 _E a ) = ( X∘RV/𝑐 _E RR ) )
106		baselsiga 30178	. . . . . . . . . 10 |- (𝔅ℝ e. (sigAlgebra ` RR ) -> RR e. 𝔅ℝ )
107	20, 106	mp1i 13	. . . . . . . . 9 |- ( ph -> RR e. 𝔅ℝ )
108	2, 4, 107	orvcelval 30530	. . . . . . . 8 |- ( ph -> ( X∘RV/𝑐 _E RR ) = ( `' X " RR ) )
109	108	adantr 481	. . . . . . 7 |- ( ( ph /\ a = RR ) -> ( X∘RV/𝑐 _E RR ) = ( `' X " RR ) )
110	105, 109	eqtrd 2656	. . . . . 6 |- ( ( ph /\ a = RR ) -> ( X∘RV/𝑐 _E a ) = ( `' X " RR ) )
111	110	fveq2d 6195	. . . . 5 |- ( ( ph /\ a = RR ) -> ( P ` ( X∘RV/𝑐 _E a ) ) = ( P ` ( `' X " RR ) ) )
112		fimacnv 6347	. . . . . . . . 9 |- ( X : U. dom P --> RR -> ( `' X " RR ) = U. dom P )
113	36, 112	syl 17	. . . . . . . 8 |- ( ph -> ( `' X " RR ) = U. dom P )
114	113	fveq2d 6195	. . . . . . 7 |- ( ph -> ( P ` ( `' X " RR ) ) = ( P ` U. dom P ) )
115		probtot 30474	. . . . . . . 8 |- ( P e. Prob -> ( P ` U. dom P ) = 1 )
116	2, 115	syl 17	. . . . . . 7 |- ( ph -> ( P ` U. dom P ) = 1 )
117	114, 116	eqtrd 2656	. . . . . 6 |- ( ph -> ( P ` ( `' X " RR ) ) = 1 )
118	117	adantr 481	. . . . 5 |- ( ( ph /\ a = RR ) -> ( P ` ( `' X " RR ) ) = 1 )
119	111, 118	eqtrd 2656	. . . 4 |- ( ( ph /\ a = RR ) -> ( P ` ( X∘RV/𝑐 _E a ) ) = 1 )
120		1red 10055	. . . 4 |- ( ph -> 1 e. RR )
121	1, 119, 107, 120	fvmptd 6288	. . 3 |- ( ph -> ( D ` RR ) = 1 )
122	103, 121	eqtrd 2656	. 2 |- ( ph -> ( D ` U. dom D ) = 1 )
123		elprob 30471	. 2 |- ( D e. Prob <-> ( D e. U. ran measures /\ ( D ` U. dom D ) = 1 ) )
124	99, 122, 123	sylanbrc 698	1 |- ( ph -> D e. Prob )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   U_ciun 4520  Disj wdisj 4620   class class class wbr 4653   |-> cmpt 4729   _E cep 5028   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   RRcr 9935   0cc0 9936   1c1 9937   +oocpnf 10071   RR*cxr 10073   <_ cle 10075   [,]cicc 12178  Σ^*cesum 30089  sigAlgebracsiga 30170  𝔅_ℝcbrsiga 30244  measurescmeas 30258  Probcprb 30469  rRndVarcrrv 30502  ∘_RV/𝑐corvc 30517

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-sigagen 30202  df-brsiga 30245  df-meas 30259  df-mbfm 30313  df-prob 30470  df-rrv 30503  df-orvc 30518

This theorem is referenced by: (None)