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Mirrors > Home > MPE Home > Th. List > xrmaxle | Structured version Visualization version GIF version |
Description: Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
xrmaxle | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrmax1 13194 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | |
2 | 1 | 3adant3 1129 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
3 | ifcl 4577 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ*) | |
4 | 3 | ancoms 457 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ*) |
5 | 4 | 3adant3 1129 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ*) |
6 | xrletr 13177 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∧ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶) → 𝐴 ≤ 𝐶)) | |
7 | 5, 6 | syld3an2 1408 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∧ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
8 | 2, 7 | mpand 693 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 → 𝐴 ≤ 𝐶)) |
9 | xrmax2 13195 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | |
10 | 9 | 3adant3 1129 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
11 | simp2 1134 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐵 ∈ ℝ*) | |
12 | simp3 1135 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐶 ∈ ℝ*) | |
13 | xrletr 13177 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∧ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶) → 𝐵 ≤ 𝐶)) | |
14 | 11, 5, 12, 13 | syl3anc 1368 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∧ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶) → 𝐵 ≤ 𝐶)) |
15 | 10, 14 | mpand 693 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 → 𝐵 ≤ 𝐶)) |
16 | 8, 15 | jcad 511 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 → (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶))) |
17 | breq1 5155 | . . . 4 ⊢ (𝐵 = if(𝐴 ≤ 𝐵, 𝐵, 𝐴) → (𝐵 ≤ 𝐶 ↔ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶)) | |
18 | breq1 5155 | . . . 4 ⊢ (𝐴 = if(𝐴 ≤ 𝐵, 𝐵, 𝐴) → (𝐴 ≤ 𝐶 ↔ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶)) | |
19 | 17, 18 | ifboth 4571 | . . 3 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐴 ≤ 𝐶) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶) |
20 | 19 | ancoms 457 | . 2 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶) |
21 | 16, 20 | impbid1 224 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ifcif 4532 class class class wbr 5152 ℝ*cxr 11285 ≤ cle 11287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-pre-lttri 11220 ax-pre-lttrn 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 |
This theorem is referenced by: maxle 13210 mbfmax 25598 itgspliticc 25786 deg1addle2 26058 deg1sublt 26066 cvmliftlem10 34937 iccin 47993 |
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