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Theorem xrmin1 13337

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
xrmin1	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴)

**Proof of Theorem xrmin1**
Step	Hyp	Ref	Expression
1		iftrue 4491	. . . 4 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐴)
2	1	adantl 487	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐴)
3		xrleid 13310	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
4	3	ad2antrr 739	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴)
5	2, 4	eqbrtrd 5131	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴)
6		iffalse 4494	. . . 4 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐵)
7	6	adantl 487	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 ≤ 𝐵) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐵)
8		xrletri 13312	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
9	8	orcanai 1018	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐴)
10	7, 9	eqbrtrd 5131	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 ≤ 𝐵) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴)
11	5, 10	pm2.61dan 825	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ifcif 4485 class class class wbr 5106 ℝ^*cxr 11372 ≤ cle 11374

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-pre-lttri 11304 ax-pre-lttrn 11305

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-po 5564 df-so 5565 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8748 df-en 9005 df-dom 9006 df-sdom 9007 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379

This theorem is used by: xrltmin 13342 xrlemin 13344 min1 13349 mnfnei 23572 stdbdxmet 24867 stdbdmopn 24870 metnrmlem1a 25211 dvferm1lem 26337 lhop1 26367 stoweid 47288

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