dstrvval - Mathbox for Thierry Arnoux

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Theorem dstrvval 32337

Description: The value of the distribution of a random variable. (Contributed by Thierry Arnoux, 9-Feb-2017.)

**Hypotheses**
Ref	Expression
dstrvprob.1	⊢ (𝜑 → 𝑃 ∈ Prob)
dstrvprob.2	⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
dstrvprob.3	⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))))
dstrvval.1	⊢ (𝜑 → 𝐴 ∈ 𝔅_ℝ)

**Assertion**
Ref	Expression
dstrvval	⊢ (𝜑 → (𝐷‘𝐴) = (𝑃‘(^◡𝑋 “ 𝐴)))

Distinct variable groups: 𝑃,𝑎 𝑋,𝑎 𝐴,𝑎 Allowed substitution hints: 𝜑(𝑎) 𝐷(𝑎)

**Proof of Theorem dstrvval**
Step	Hyp	Ref	Expression
1		dstrvprob.3	. . 3 ⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))))
2	1	fveq1d 6758	. 2 ⊢ (𝜑 → (𝐷‘𝐴) = ((𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))‘𝐴))
3		dstrvval.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝔅_ℝ)
4		oveq2 7263	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑋∘_RV/𝑐 E 𝑎) = (𝑋∘_RV/𝑐 E 𝐴))
5	4	fveq2d 6760	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) = (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)))
6		eqid 2738	. . . 4 ⊢ (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))) = (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))
7		fvex 6769	. . . 4 ⊢ (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)) ∈ V
8	5, 6, 7	fvmpt 6857	. . 3 ⊢ (𝐴 ∈ 𝔅_ℝ → ((𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))‘𝐴) = (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)))
9	3, 8	syl 17	. 2 ⊢ (𝜑 → ((𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))‘𝐴) = (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)))
10		dstrvprob.1	. . . 4 ⊢ (𝜑 → 𝑃 ∈ Prob)
11		dstrvprob.2	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
12	10, 11, 3	orvcelval 32335	. . 3 ⊢ (𝜑 → (𝑋∘_RV/𝑐 E 𝐴) = (^◡𝑋 “ 𝐴))
13	12	fveq2d 6760	. 2 ⊢ (𝜑 → (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)) = (𝑃‘(^◡𝑋 “ 𝐴)))
14	2, 9, 13	3eqtrd 2782	1 ⊢ (𝜑 → (𝐷‘𝐴) = (𝑃‘(^◡𝑋 “ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ↦ cmpt 5153 E cep 5485 ^◡ccnv 5579 “ cima 5583 ‘cfv 6418 (class class class)co 7255 𝔅_ℝcbrsiga 32049 Probcprb 32274 rRndVarcrrv 32307 ∘_RV/𝑐corvc 32322

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-pre-lttri 10876 ax-pre-lttrn 10877

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-ioo 13012 df-topgen 17071 df-top 21951 df-bases 22004 df-esum 31896 df-siga 31977 df-sigagen 32007 df-brsiga 32050 df-meas 32064 df-mbfm 32118 df-prob 32275 df-rrv 32308 df-orvc 32323

This theorem is referenced by: dstrvprob 32338

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