![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > orvclteinc | Structured version Visualization version GIF version |
Description: Preimage maps produced by the "less than or equal to" relation are increasing. (Contributed by Thierry Arnoux, 11-Feb-2017.) |
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) |
Ref | Expression |
---|---|
orvclteinc | ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) ⊆ (𝑋∘RV/𝑐 ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dstfrv.1 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | dstfrv.2 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
3 | 1, 2 | rrvf2 31056 | . . . 4 ⊢ (𝜑 → 𝑋:dom 𝑋⟶ℝ) |
4 | 3 | ffund 6282 | . . 3 ⊢ (𝜑 → Fun 𝑋) |
5 | simp2 1173 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ∈ ℝ) | |
6 | orvclteinc.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | 6 | 3ad2ant1 1169 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ∈ ℝ) |
8 | orvclteinc.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | 8 | 3ad2ant1 1169 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐵 ∈ ℝ) |
10 | simp3 1174 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐴) | |
11 | orvclteinc.3 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
12 | 11 | 3ad2ant1 1169 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ≤ 𝐵) |
13 | 5, 7, 9, 10, 12 | letrd 10513 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐵) |
14 | 13 | 3expia 1156 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝐴 → 𝑥 ≤ 𝐵)) |
15 | 14 | ss2rabdv 3908 | . . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ⊆ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}) |
16 | sspreima 29996 | . . 3 ⊢ ((Fun 𝑋 ∧ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ⊆ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}) → (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) ⊆ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵})) | |
17 | 4, 15, 16 | syl2anc 581 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) ⊆ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵})) |
18 | 1, 2, 6 | orrvcval4 31072 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) |
19 | 1, 2, 8 | orrvcval4 31072 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐵) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵})) |
20 | 17, 18, 19 | 3sstr4d 3873 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) ⊆ (𝑋∘RV/𝑐 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 ∈ wcel 2166 {crab 3121 ⊆ wss 3798 class class class wbr 4873 ◡ccnv 5341 dom cdm 5342 “ cima 5345 Fun wfun 6117 ‘cfv 6123 (class class class)co 6905 ℝcr 10251 ≤ cle 10392 Probcprb 31015 rRndVarcrrv 31048 ∘RV/𝑐corvc 31063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-pre-lttri 10326 ax-pre-lttrn 10327 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-ioo 12467 df-topgen 16457 df-top 21069 df-bases 21121 df-esum 30635 df-siga 30716 df-sigagen 30747 df-brsiga 30790 df-meas 30804 df-mbfm 30858 df-prob 31016 df-rrv 31049 df-orvc 31064 |
This theorem is referenced by: dstfrvinc 31084 dstfrvclim1 31085 |
Copyright terms: Public domain | W3C validator |