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Mirrors > Home > MPE Home > Th. List > xrmineq | Structured version Visualization version GIF version |
Description: The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) |
Ref | Expression |
---|---|
xrmineq | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifid 4508 | . 2 ⊢ if(𝐴 ≤ 𝐵, 𝐵, 𝐵) = 𝐵 | |
2 | xrletri3 12550 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐵 = 𝐴 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) | |
3 | 2 | ancoms 461 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 = 𝐴 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) |
4 | 3 | biimpar 480 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐵 = 𝐴) |
5 | 4 | anassrs 470 | . . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐵 ≤ 𝐴) ∧ 𝐴 ≤ 𝐵) → 𝐵 = 𝐴) |
6 | 5 | ifeq1da 4499 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐵, 𝐵) = if(𝐴 ≤ 𝐵, 𝐴, 𝐵)) |
7 | 6 | 3impa 1106 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐵, 𝐵) = if(𝐴 ≤ 𝐵, 𝐴, 𝐵)) |
8 | 1, 7 | syl5reqr 2873 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ifcif 4469 class class class wbr 5068 ℝ*cxr 10676 ≤ cle 10678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 |
This theorem is referenced by: ssfzunsn 12956 setsstruct 16525 ioondisj2 41774 |
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