xrmax1 - Metamath Proof Explorer

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Theorem xrmax1 12220

Description: An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.)

**Assertion**
Ref	Expression
xrmax1	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))

**Proof of Theorem xrmax1**
Step	Hyp	Ref	Expression
1		xrleid 12197	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
2		iffalse 4240	. . . . 5 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐴)
3	2	breq2d 4817	. . . 4 ⊢ (¬ 𝐴 ≤ 𝐵 → (𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ↔ 𝐴 ≤ 𝐴))
4	1, 3	syl5ibrcom 237	. . 3 ⊢ (𝐴 ∈ ℝ^* → (¬ 𝐴 ≤ 𝐵 → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)))
5		id 22	. . . 4 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵)
6		iftrue 4237	. . . 4 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐵)
7	5, 6	breqtrrd 4833	. . 3 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
8	4, 7	pm2.61d2 172	. 2 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
9	8	adantr 472	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∈ wcel 2140 ifcif 4231 class class class wbr 4805 ℝ^*cxr 10286 ≤ cle 10288

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 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-cnex 10205 ax-resscn 10206 ax-pre-lttri 10223 ax-pre-lttrn 10224

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 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-op 4329 df-uni 4590 df-br 4806 df-opab 4866 df-mpt 4883 df-id 5175 df-po 5188 df-so 5189 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-er 7914 df-en 8125 df-dom 8126 df-sdom 8127 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293

This theorem is referenced by: xrmaxlt 12226 xrmaxle 12228 max1 12230 limsupgre 14432 pnfnei 21247 ismbfd 23627 dvferm2lem 23969 mdegaddle 24054