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Theorem xrlemin 11889
 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.)
Assertion
Ref Expression
xrlemin ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶) ↔ (𝐴𝐵𝐴𝐶)))

Proof of Theorem xrlemin
StepHypRef Expression
1 xrmin1 11882 . . . . 5 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → if(𝐵𝐶, 𝐵, 𝐶) ≤ 𝐵)
213adant1 1072 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → if(𝐵𝐶, 𝐵, 𝐶) ≤ 𝐵)
3 simp1 1054 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → 𝐴 ∈ ℝ*)
4 ifcl 4080 . . . . . 6 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → if(𝐵𝐶, 𝐵, 𝐶) ∈ ℝ*)
543adant1 1072 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → if(𝐵𝐶, 𝐵, 𝐶) ∈ ℝ*)
6 simp2 1055 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → 𝐵 ∈ ℝ*)
7 xrletr 11865 . . . . 5 ((𝐴 ∈ ℝ* ∧ if(𝐵𝐶, 𝐵, 𝐶) ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶) ∧ if(𝐵𝐶, 𝐵, 𝐶) ≤ 𝐵) → 𝐴𝐵))
83, 5, 6, 7syl3anc 1318 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶) ∧ if(𝐵𝐶, 𝐵, 𝐶) ≤ 𝐵) → 𝐴𝐵))
92, 8mpan2d 706 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶) → 𝐴𝐵))
10 xrmin2 11883 . . . . 5 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → if(𝐵𝐶, 𝐵, 𝐶) ≤ 𝐶)
11103adant1 1072 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → if(𝐵𝐶, 𝐵, 𝐶) ≤ 𝐶)
12 xrletr 11865 . . . . 5 ((𝐴 ∈ ℝ* ∧ if(𝐵𝐶, 𝐵, 𝐶) ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶) ∧ if(𝐵𝐶, 𝐵, 𝐶) ≤ 𝐶) → 𝐴𝐶))
135, 12syld3an2 1365 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶) ∧ if(𝐵𝐶, 𝐵, 𝐶) ≤ 𝐶) → 𝐴𝐶))
1411, 13mpan2d 706 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶) → 𝐴𝐶))
159, 14jcad 554 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶) → (𝐴𝐵𝐴𝐶)))
16 breq2 4587 . . 3 (𝐵 = if(𝐵𝐶, 𝐵, 𝐶) → (𝐴𝐵𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶)))
17 breq2 4587 . . 3 (𝐶 = if(𝐵𝐶, 𝐵, 𝐶) → (𝐴𝐶𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶)))
1816, 17ifboth 4074 . 2 ((𝐴𝐵𝐴𝐶) → 𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶))
1915, 18impbid1 214 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶) ↔ (𝐴𝐵𝐴𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ifcif 4036   class class class wbr 4583  ℝ*cxr 9952   ≤ cle 9954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959 This theorem is referenced by:  lemin  11897  stdbdxmet  22130  stdbdbl  22132  itgspliticc  23409  cvmliftlem10  30530
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