xrmin2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xrmin2		Structured version Visualization version GIF version

Theorem xrmin2 12565

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 12538	. . . 4 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
2		iffalse 4475	. . . . 5 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐵)
3	2	breq1d 5068	. . . 4 ⊢ (¬ 𝐴 ≤ 𝐵 → (if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵 ↔ 𝐵 ≤ 𝐵))
4	1, 3	syl5ibrcom 249	. . 3 ⊢ (𝐵 ∈ ℝ^* → (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵))
5		iftrue 4472	. . . 4 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐴)
6		id 22	. . . 4 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵)
7	5, 6	eqbrtrd 5080	. . 3 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵)
8	4, 7	pm2.61d2 183	. 2 ⊢ (𝐵 ∈ ℝ^* → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵)
9	8	adantl 484	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2110 ifcif 4466 class class class wbr 5058 ℝ^*cxr 10668 ≤ cle 10670

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675

This theorem is referenced by: xrltmin 12569 xrlemin 12571 min2 12577 mnfnei 21823 stdbdxmet 23119 stdbdmet 23120 stdbdmopn 23122 tgioo 23398 metnrmlem1 23461 ismbfd 24234 dvferm1lem 24575 lhop1 24605 stoweid 42342