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Mirrors > Home > MPE Home > Th. List > xrmin2 | Structured version Visualization version GIF version |
Description: The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007.) |
Ref | Expression |
---|---|
xrmin2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrleid 13133 | . . . 4 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ 𝐵) | |
2 | iffalse 4532 | . . . . 5 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐵) | |
3 | 2 | breq1d 5151 | . . . 4 ⊢ (¬ 𝐴 ≤ 𝐵 → (if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵 ↔ 𝐵 ≤ 𝐵)) |
4 | 1, 3 | syl5ibrcom 246 | . . 3 ⊢ (𝐵 ∈ ℝ* → (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵)) |
5 | iftrue 4529 | . . . 4 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐴) | |
6 | id 22 | . . . 4 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵) | |
7 | 5, 6 | eqbrtrd 5163 | . . 3 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) |
8 | 4, 7 | pm2.61d2 181 | . 2 ⊢ (𝐵 ∈ ℝ* → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) |
9 | 8 | adantl 481 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2098 ifcif 4523 class class class wbr 5141 ℝ*cxr 11248 ≤ cle 11250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 |
This theorem is referenced by: xrltmin 13164 xrlemin 13166 min2 13172 mnfnei 23076 stdbdxmet 24375 stdbdmet 24376 stdbdmopn 24378 tgioo 24663 metnrmlem1 24726 ismbfd 25519 dvferm1lem 25867 lhop1 25898 stoweid 45332 |
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