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Theorem xrmin2 12202

Description: The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007.)

**Assertion**
Ref	Expression
xrmin2	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵)

**Proof of Theorem xrmin2**
Step	Hyp	Ref	Expression
1		xrleid 12176	. . . 4 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
2		iffalse 4239	. . . . 5 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐵)
3	2	breq1d 4814	. . . 4 ⊢ (¬ 𝐴 ≤ 𝐵 → (if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵 ↔ 𝐵 ≤ 𝐵))
4	1, 3	syl5ibrcom 237	. . 3 ⊢ (𝐵 ∈ ℝ^* → (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵))
5		iftrue 4236	. . . 4 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐴)
6		id 22	. . . 4 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵)
7	5, 6	eqbrtrd 4826	. . 3 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵)
8	4, 7	pm2.61d2 172	. 2 ⊢ (𝐵 ∈ ℝ^* → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵)
9	8	adantl 473	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∈ wcel 2139 ifcif 4230 class class class wbr 4804 ℝ^*cxr 10265 ≤ cle 10267

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-pre-lttri 10202 ax-pre-lttrn 10203

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272

This theorem is referenced by: xrltmin 12206 xrlemin 12208 min2 12214 mnfnei 21227 stdbdxmet 22521 stdbdmet 22522 stdbdmopn 22524 tgioo 22800 metnrmlem1 22863 ismbfd 23606 dvferm1lem 23946 lhop1 23976 stoweid 40783