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Mirrors > Home > MPE Home > Th. List > xrmax1 | 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 |
---|---|
xrmax1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrleid 12867 | . . . 4 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) | |
2 | iffalse 4473 | . . . . 5 ⊢ (¬ 𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐴) | |
3 | 2 | breq2d 5090 | . . . 4 ⊢ (¬ 𝐴 ≤ 𝐵 → (𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ↔ 𝐴 ≤ 𝐴)) |
4 | 1, 3 | syl5ibrcom 246 | . . 3 ⊢ (𝐴 ∈ ℝ* → (¬ 𝐴 ≤ 𝐵 → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))) |
5 | id 22 | . . . 4 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵) | |
6 | iftrue 4470 | . . . 4 ⊢ (𝐴 ≤ 𝐵 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐵) | |
7 | 5, 6 | breqtrrd 5106 | . . 3 ⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
8 | 4, 7 | pm2.61d2 181 | . 2 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
9 | 8 | adantr 480 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2109 ifcif 4464 class class class wbr 5078 ℝ*cxr 10992 ≤ cle 10994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-pre-lttri 10929 ax-pre-lttrn 10930 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 |
This theorem is referenced by: xrmaxlt 12897 xrmaxle 12899 max1 12901 limsupgre 15171 pnfnei 22352 ismbfd 24784 dvferm2lem 25131 mdegaddle 25220 |
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