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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 13179 | . . . 4 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ 𝐵) | |
2 | iffalse 4541 | . . . . 5 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐵) | |
3 | 2 | breq1d 5162 | . . . 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 4538 | . . . 4 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐴) | |
6 | id 22 | . . . 4 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵) | |
7 | 5, 6 | eqbrtrd 5174 | . . 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 480 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∈ wcel 2098 ifcif 4532 class class class wbr 5152 ℝ*cxr 11293 ≤ cle 11295 |
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 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-pre-lttri 11228 ax-pre-lttrn 11229 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-po 5593 df-so 5594 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 |
This theorem is referenced by: xrltmin 13210 xrlemin 13212 min2 13218 mnfnei 23208 stdbdxmet 24507 stdbdmet 24508 stdbdmopn 24510 tgioo 24795 metnrmlem1 24858 ismbfd 25651 dvferm1lem 25999 lhop1 26030 stoweid 45621 |
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