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Theorem xrmax2 13081

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
xrmax2	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))

**Proof of Theorem xrmax2**
Step	Hyp	Ref	Expression
1		xrleid 13056	. . . 4 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
2	1	ad2antlr 727	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
3		iftrue 4480	. . . 4 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐵)
4	3	adantl 481	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐵)
5	2, 4	breqtrrd 5121	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
6		xrletri 13058	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
7	6	orcanai 1004	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐴)
8		iffalse 4483	. . . 4 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐴)
9	8	adantl 481	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 ≤ 𝐵) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐴)
10	7, 9	breqtrrd 5121	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
11	5, 10	pm2.61dan 812	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ifcif 4474 class class class wbr 5093 ℝ^*cxr 11151 ≤ cle 11153

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-pre-lttri 11086 ax-pre-lttrn 11087

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 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 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158

This theorem is referenced by: xrmaxlt 13086 xrmaxle 13088 max2 13092 limsupgre 15394 pnfnei 23141 tgioo 24717 dvferm2lem 25923 mdegaddle 26012 plypf1 26150