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Mirrors > Home > MPE Home > Th. List > xrmax2 | Structured version Visualization version GIF version |
Description: An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) |
Ref | Expression |
---|---|
xrmax2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrleid 13163 | . . . 4 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ 𝐵) | |
2 | 1 | ad2antlr 726 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵) |
3 | iftrue 4535 | . . . 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 5176 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
6 | xrletri 13165 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | |
7 | 6 | orcanai 1001 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐴) |
8 | iffalse 4538 | . . . 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 5176 | . 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 1534 ∈ wcel 2099 ifcif 4529 class class class wbr 5148 ℝ*cxr 11278 ≤ cle 11280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-pre-lttri 11213 ax-pre-lttrn 11214 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 |
This theorem is referenced by: xrmaxlt 13193 xrmaxle 13195 max2 13199 limsupgre 15458 pnfnei 23137 tgioo 24725 dvferm2lem 25931 mdegaddle 26023 plypf1 26159 |
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