![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xrlemin | Structured version Visualization version GIF version |
Description: Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
xrlemin | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrmin1 13196 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵) | |
2 | 1 | 3adant1 1127 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵) |
3 | simp1 1133 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐴 ∈ ℝ*) | |
4 | ifcl 4577 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ*) | |
5 | 4 | 3adant1 1127 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ*) |
6 | simp2 1134 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐵 ∈ ℝ*) | |
7 | xrletr 13177 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵) → 𝐴 ≤ 𝐵)) | |
8 | 3, 5, 6, 7 | syl3anc 1368 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵) → 𝐴 ≤ 𝐵)) |
9 | 2, 8 | mpan2d 692 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 ≤ 𝐵)) |
10 | xrmin2 13197 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶) | |
11 | 10 | 3adant1 1127 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶) |
12 | xrletr 13177 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶) → 𝐴 ≤ 𝐶)) | |
13 | 5, 12 | syld3an2 1408 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
14 | 11, 13 | mpan2d 692 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 ≤ 𝐶)) |
15 | 9, 14 | jcad 511 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) |
16 | breq2 5156 | . . 3 ⊢ (𝐵 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶))) | |
17 | breq2 5156 | . . 3 ⊢ (𝐶 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 ≤ 𝐶 ↔ 𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶))) | |
18 | 16, 17 | ifboth 4571 | . 2 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶) → 𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶)) |
19 | 15, 18 | impbid1 224 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 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: lemin 13211 stdbdxmet 24444 stdbdbl 24446 itgspliticc 25786 cvmliftlem10 34937 iccin 47993 |
Copyright terms: Public domain | W3C validator |