dstrvval - Mathbox for Thierry Arnoux

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Theorem dstrvval 34479

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 6824	. 2 ⊢ (𝜑 → (𝐷‘𝐴) = ((𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))‘𝐴))
3		dstrvval.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝔅_ℝ)
4		oveq2 7354	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑋∘_RV/𝑐 E 𝑎) = (𝑋∘_RV/𝑐 E 𝐴))
5	4	fveq2d 6826	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) = (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)))
6		eqid 2731	. . . 4 ⊢ (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))) = (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))
7		fvex 6835	. . . 4 ⊢ (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)) ∈ V
8	5, 6, 7	fvmpt 6929	. . 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 34477	. . 3 ⊢ (𝜑 → (𝑋∘_RV/𝑐 E 𝐴) = (^◡𝑋 “ 𝐴))
13	12	fveq2d 6826	. 2 ⊢ (𝜑 → (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)) = (𝑃‘(^◡𝑋 “ 𝐴)))
14	2, 9, 13	3eqtrd 2770	1 ⊢ (𝜑 → (𝐷‘𝐴) = (𝑃‘(^◡𝑋 “ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 ↦ cmpt 5172 E cep 5515 ^◡ccnv 5615 “ cima 5619 ‘cfv 6481 (class class class)co 7346 𝔅_ℝcbrsiga 34189 Probcprb 34415 rRndVarcrrv 34448 ∘_RV/𝑐corvc 34464

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-pre-lttri 11077 ax-pre-lttrn 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-ioo 13246 df-topgen 17344 df-top 22807 df-bases 22859 df-esum 34036 df-siga 34117 df-sigagen 34147 df-brsiga 34190 df-meas 34204 df-mbfm 34258 df-prob 34416 df-rrv 34449 df-orvc 34465

This theorem is referenced by: dstrvprob 34480

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