Theorem xrmaxle 13222

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.)

Assertion

Ref	Expression
xrmaxle	⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶)))

Proof of Theorem xrmaxle

Step	Hyp	Ref	Expression
1		xrmax1 13214	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
2	1	3adant3 1131	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
3		ifcl 4576	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ*)
4	3	ancoms 458	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ*)
5	4	3adant3 1131	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ*)
6		xrletr 13197	. . . . 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 1410	. . . 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 695	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 → 𝐴 ≤ 𝐶))
9		xrmax2 13215	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
10	9	3adant3 1131	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
11		simp2 1136	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐵 ∈ ℝ*)
12		simp3 1137	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐶 ∈ ℝ*)
13		xrletr 13197	. . . . 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 1370	. . . 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 695	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 → 𝐵 ≤ 𝐶))
16	8, 15	jcad 512	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 → (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶)))
17		breq1 5151	. . . 4 ⊢ (𝐵 = if(𝐴 ≤ 𝐵, 𝐵, 𝐴) → (𝐵 ≤ 𝐶 ↔ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶))
18		breq1 5151	. . . 4 ⊢ (𝐴 = if(𝐴 ≤ 𝐵, 𝐵, 𝐴) → (𝐴 ≤ 𝐶 ↔ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶))
19	17, 18	ifboth 4570	. . 3 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐴 ≤ 𝐶) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶)
20	19	ancoms 458	. 2 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶)
21	16, 20	impbid1 225	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2106  ifcif 4531   class class class wbr 5148  ℝ^*cxr 11292   ≤ cle 11294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299

This theorem is referenced by:  maxle  13230  mbfmax  25698  itgspliticc  25887  deg1addle2  26156  deg1sublt  26164  cvmliftlem10  35279  iccin  48693