dstrvval - Mathbox for Thierry Arnoux

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Theorem dstrvval 34648

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 6844	. 2 ⊢ (𝜑 → (𝐷‘𝐴) = ((𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))‘𝐴))
3		dstrvval.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝔅_ℝ)
4		oveq2 7376	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑋∘_RV/𝑐 E 𝑎) = (𝑋∘_RV/𝑐 E 𝐴))
5	4	fveq2d 6846	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) = (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)))
6		eqid 2737	. . . 4 ⊢ (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))) = (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))
7		fvex 6855	. . . 4 ⊢ (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)) ∈ V
8	5, 6, 7	fvmpt 6949	. . 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 34646	. . 3 ⊢ (𝜑 → (𝑋∘_RV/𝑐 E 𝐴) = (^◡𝑋 “ 𝐴))
13	12	fveq2d 6846	. 2 ⊢ (𝜑 → (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)) = (𝑃‘(^◡𝑋 “ 𝐴)))
14	2, 9, 13	3eqtrd 2776	1 ⊢ (𝜑 → (𝐷‘𝐴) = (𝑃‘(^◡𝑋 “ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5181 E cep 5531 ^◡ccnv 5631 “ cima 5635 ‘cfv 6500 (class class class)co 7368 𝔅_ℝcbrsiga 34358 Probcprb 34584 rRndVarcrrv 34617 ∘_RV/𝑐corvc 34633

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-pre-lttri 11112 ax-pre-lttrn 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-ioo 13277 df-topgen 17375 df-top 22850 df-bases 22902 df-esum 34205 df-siga 34286 df-sigagen 34316 df-brsiga 34359 df-meas 34373 df-mbfm 34427 df-prob 34585 df-rrv 34618 df-orvc 34634

This theorem is referenced by: dstrvprob 34649

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