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Theorem xrmax2 13128

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 13102	. . . 4 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
2	1	ad2antlr 728	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
3		iftrue 4472	. . . 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 5113	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
6		xrletri 13104	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
7	6	orcanai 1005	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐴)
8		iffalse 4475	. . . 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 5113	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
11	5, 10	pm2.61dan 813	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 1542 ∈ wcel 2114 ifcif 4466 class class class wbr 5085 ℝ^*cxr 11178 ≤ cle 11180

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-pre-lttri 11112 ax-pre-lttrn 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185

This theorem is referenced by: xrmaxlt 13133 xrmaxle 13135 max2 13139 limsupgre 15443 pnfnei 23185 tgioo 24761 dvferm2lem 25953 mdegaddle 26039 plypf1 26177