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 13098

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 13071	. . . 4 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
2		iffalse 4487	. . . . 5 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐵)
3	2	breq1d 5105	. . . 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 4484	. . . 4 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐴)
6		id 22	. . . 4 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵)
7	5, 6	eqbrtrd 5117	. . 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 2109 ifcif 4478 class class class wbr 5095 ℝ^*cxr 11167 ≤ cle 11169

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-pre-lttri 11102 ax-pre-lttrn 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-po 5531 df-so 5532 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174

This theorem is referenced by: xrltmin 13102 xrlemin 13104 min2 13110 mnfnei 23124 stdbdxmet 24419 stdbdmet 24420 stdbdmopn 24422 tgioo 24700 metnrmlem1 24764 ismbfd 25556 dvferm1lem 25904 lhop1 25935 stoweid 46045