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Mathbox for Thierry Arnoux |
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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 34430 | . . . 4 ⊢ (𝜑 → 𝑋:dom 𝑋⟶ℝ) |
4 | 3 | ffund 6741 | . . 3 ⊢ (𝜑 → Fun 𝑋) |
5 | simp2 1136 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ∈ ℝ) | |
6 | orvclteinc.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | 6 | 3ad2ant1 1132 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ∈ ℝ) |
8 | orvclteinc.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | 8 | 3ad2ant1 1132 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐵 ∈ ℝ) |
10 | simp3 1137 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐴) | |
11 | orvclteinc.3 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
12 | 11 | 3ad2ant1 1132 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ≤ 𝐵) |
13 | 5, 7, 9, 10, 12 | letrd 11416 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐵) |
14 | 13 | 3expia 1120 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝐴 → 𝑥 ≤ 𝐵)) |
15 | 14 | ss2rabdv 4086 | . . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ⊆ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}) |
16 | sspreima 7088 | . . 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 34446 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) |
19 | 1, 2, 8 | orrvcval4 34446 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐵) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵})) |
20 | 17, 18, 19 | 3sstr4d 4043 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) ⊆ (𝑋∘RV/𝑐 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2106 {crab 3433 ⊆ wss 3963 class class class wbr 5148 ◡ccnv 5688 dom cdm 5689 “ cima 5692 Fun wfun 6557 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 ≤ cle 11294 Probcprb 34389 rRndVarcrrv 34422 ∘RV/𝑐corvc 34437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ioo 13388 df-topgen 17490 df-top 22916 df-bases 22969 df-esum 34009 df-siga 34090 df-sigagen 34120 df-brsiga 34163 df-meas 34177 df-mbfm 34231 df-prob 34390 df-rrv 34423 df-orvc 34438 |
This theorem is referenced by: dstfrvinc 34458 dstfrvclim1 34459 |
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