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Theorem xrmin2 13079

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 13052	. . . 4 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
2		iffalse 4483	. . . . 5 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐵)
3	2	breq1d 5103	. . . 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 4480	. . . 4 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐴)
6		id 22	. . . 4 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵)
7	5, 6	eqbrtrd 5115	. . 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 2113 ifcif 4474 class class class wbr 5093 ℝ^*cxr 11152 ≤ cle 11154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-pre-lttri 11087 ax-pre-lttrn 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 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 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159

This theorem is referenced by: xrltmin 13083 xrlemin 13085 min2 13091 mnfnei 23137 stdbdxmet 24431 stdbdmet 24432 stdbdmopn 24434 tgioo 24712 metnrmlem1 24776 ismbfd 25568 dvferm1lem 25916 lhop1 25947 stoweid 46185