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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 13159 | . . . . 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 4568 | . . . . . 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 13140 | . . . . 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 691 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 ≤ 𝐵)) |
10 | xrmin2 13160 | . . . . 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 13140 | . . . . 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 691 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 ≤ 𝐶)) |
15 | 9, 14 | jcad 512 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) |
16 | breq2 5145 | . . 3 ⊢ (𝐵 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶))) | |
17 | breq2 5145 | . . 3 ⊢ (𝐶 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 ≤ 𝐶 ↔ 𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶))) | |
18 | 16, 17 | ifboth 4562 | . 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 395 ∧ w3a 1084 ∈ wcel 2098 ifcif 4523 class class class wbr 5141 ℝ*cxr 11248 ≤ cle 11250 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 |
This theorem is referenced by: lemin 13174 stdbdxmet 24374 stdbdbl 24376 itgspliticc 25716 cvmliftlem10 34812 iccin 47785 |
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