xrmax2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xrmax2		Structured version Visualization version GIF version

Theorem xrmax2 13119

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 13093	. . . 4 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
2	1	ad2antlr 728	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
3		iftrue 4473	. . . 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 5114	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
6		xrletri 13095	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
7	6	orcanai 1005	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐴)
8		iffalse 4476	. . . 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 5114	. 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 4467 class class class wbr 5086 ℝ^*cxr 11169 ≤ cle 11171

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 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-pre-lttri 11103 ax-pre-lttrn 11104

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176

This theorem is referenced by: xrmaxlt 13124 xrmaxle 13126 max2 13130 limsupgre 15434 pnfnei 23195 tgioo 24771 dvferm2lem 25963 mdegaddle 26049 plypf1 26187