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| 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 13193 | . . . 4 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ 𝐵) | |
| 2 | iffalse 4534 | . . . . 5 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐵) | |
| 3 | 2 | breq1d 5153 | . . . 4 ⊢ (¬ 𝐴 ≤ 𝐵 → (if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵 ↔ 𝐵 ≤ 𝐵)) | 
| 4 | 1, 3 | syl5ibrcom 247 | . . 3 ⊢ (𝐵 ∈ ℝ* → (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵)) | 
| 5 | iftrue 4531 | . . . 4 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐴) | |
| 6 | id 22 | . . . 4 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵) | |
| 7 | 5, 6 | eqbrtrd 5165 | . . 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 2108 ifcif 4525 class class class wbr 5143 ℝ*cxr 11294 ≤ cle 11296 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 | 
| This theorem is referenced by: xrltmin 13224 xrlemin 13226 min2 13232 mnfnei 23229 stdbdxmet 24528 stdbdmet 24529 stdbdmopn 24531 tgioo 24817 metnrmlem1 24881 ismbfd 25674 dvferm1lem 26022 lhop1 26053 stoweid 46078 | 
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