orvclteinc - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > orvclteinc		Structured version Visualization version GIF version

Theorem orvclteinc 29864

Description: Preimage maps produced by the "lower than or equal" relation are increasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)

**Hypotheses**
Ref	Expression
dstfrv.1	⊢ (𝜑 → 𝑃 ∈ Prob)
dstfrv.2	⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
orvclteinc.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
orvclteinc.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
orvclteinc.3	⊢ (𝜑 → 𝐴 ≤ 𝐵)

**Assertion**
Ref	Expression
orvclteinc	⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) ⊆ (𝑋∘_RV/𝑐 ≤ 𝐵))

Proof of Theorem orvclteinc Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dstfrv.1	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Prob)
2		dstfrv.2	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
3	1, 2	rrvf2 29837	. . . 4 ⊢ (𝜑 → 𝑋:dom 𝑋⟶ℝ)
4		ffun 5961	. . . 4 ⊢ (𝑋:dom 𝑋⟶ℝ → Fun 𝑋)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → Fun 𝑋)
6		simp2 1055	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ∈ ℝ)
7		orvclteinc.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
8	7	3ad2ant1 1075	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ∈ ℝ)
9		orvclteinc.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
10	9	3ad2ant1 1075	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐵 ∈ ℝ)
11		simp3 1056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐴)
12		orvclteinc.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
13	12	3ad2ant1 1075	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ≤ 𝐵)
14	6, 8, 10, 11, 13	letrd 10073	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐵)
15	14	3expia 1259	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝐴 → 𝑥 ≤ 𝐵))
16	15	ss2rabdv 3646	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ⊆ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵})
17		sspreima 28827	. . 3 ⊢ ((Fun 𝑋 ∧ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ⊆ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}) → (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) ⊆ (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}))
18	5, 16, 17	syl2anc 691	. 2 ⊢ (𝜑 → (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) ⊆ (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}))
19	1, 2, 7	orrvcval4 29853	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) = (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))
20	1, 2, 9	orrvcval4 29853	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐵) = (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}))
21	18, 19, 20	3sstr4d 3611	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) ⊆ (𝑋∘_RV/𝑐 ≤ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1031 ∈ wcel 1977 {crab 2900 ⊆ wss 3540 class class class wbr 4583 ^◡ccnv 5037 dom cdm 5038 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 ≤ cle 9954 Probcprb 29796 rRndVarcrrv 29829 ∘_RV/𝑐corvc 29844

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ioo 12050 df-topgen 15927 df-top 20521 df-bases 20522 df-esum 29417 df-siga 29498 df-sigagen 29529 df-brsiga 29572 df-meas 29586 df-mbfm 29640 df-prob 29797 df-rrv 29830 df-orvc 29845

This theorem is referenced by: dstfrvinc 29865 dstfrvclim1 29866

W3C validator