dstrvval - Mathbox for Thierry Arnoux

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Theorem dstrvval 34435

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 6922	. 2 ⊢ (𝜑 → (𝐷‘𝐴) = ((𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))‘𝐴))
3		dstrvval.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝔅_ℝ)
4		oveq2 7456	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑋∘_RV/𝑐 E 𝑎) = (𝑋∘_RV/𝑐 E 𝐴))
5	4	fveq2d 6924	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) = (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)))
6		eqid 2740	. . . 4 ⊢ (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))) = (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))
7		fvex 6933	. . . 4 ⊢ (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)) ∈ V
8	5, 6, 7	fvmpt 7029	. . 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 34433	. . 3 ⊢ (𝜑 → (𝑋∘_RV/𝑐 E 𝐴) = (^◡𝑋 “ 𝐴))
13	12	fveq2d 6924	. 2 ⊢ (𝜑 → (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)) = (𝑃‘(^◡𝑋 “ 𝐴)))
14	2, 9, 13	3eqtrd 2784	1 ⊢ (𝜑 → (𝐷‘𝐴) = (𝑃‘(^◡𝑋 “ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ↦ cmpt 5249 E cep 5598 ^◡ccnv 5699 “ cima 5703 ‘cfv 6573 (class class class)co 7448 𝔅_ℝcbrsiga 34145 Probcprb 34372 rRndVarcrrv 34405 ∘_RV/𝑐corvc 34420

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-pre-lttri 11258 ax-pre-lttrn 11259

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-ioo 13411 df-topgen 17503 df-top 22921 df-bases 22974 df-esum 33992 df-siga 34073 df-sigagen 34103 df-brsiga 34146 df-meas 34160 df-mbfm 34214 df-prob 34373 df-rrv 34406 df-orvc 34421

This theorem is referenced by: dstrvprob 34436

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