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 31605 | . . . 4 ⊢ (𝜑 → 𝑋:dom 𝑋⟶ℝ) |
4 | 3 | ffund 6511 | . . 3 ⊢ (𝜑 → Fun 𝑋) |
5 | simp2 1129 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ∈ ℝ) | |
6 | orvclteinc.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | 6 | 3ad2ant1 1125 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ∈ ℝ) |
8 | orvclteinc.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | 8 | 3ad2ant1 1125 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐵 ∈ ℝ) |
10 | simp3 1130 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐴) | |
11 | orvclteinc.3 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
12 | 11 | 3ad2ant1 1125 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ≤ 𝐵) |
13 | 5, 7, 9, 10, 12 | letrd 10785 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐵) |
14 | 13 | 3expia 1113 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝐴 → 𝑥 ≤ 𝐵)) |
15 | 14 | ss2rabdv 4049 | . . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ⊆ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}) |
16 | sspreima 30320 | . . 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 584 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) ⊆ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵})) |
18 | 1, 2, 6 | orrvcval4 31621 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) |
19 | 1, 2, 8 | orrvcval4 31621 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐵) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵})) |
20 | 17, 18, 19 | 3sstr4d 4011 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) ⊆ (𝑋∘RV/𝑐 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 ∈ wcel 2105 {crab 3139 ⊆ wss 3933 class class class wbr 5057 ◡ccnv 5547 dom cdm 5548 “ cima 5551 Fun wfun 6342 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 ≤ cle 10664 Probcprb 31564 rRndVarcrrv 31597 ∘RV/𝑐corvc 31612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-ioo 12730 df-topgen 16705 df-top 21430 df-bases 21482 df-esum 31186 df-siga 31267 df-sigagen 31297 df-brsiga 31340 df-meas 31354 df-mbfm 31408 df-prob 31565 df-rrv 31598 df-orvc 31613 |
This theorem is referenced by: dstfrvinc 31633 dstfrvclim1 31634 |
Copyright terms: Public domain | W3C validator |