Theorem xrmaxle 13150

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 13142	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
2	1	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
3		ifcl 4537	. . . . . . 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 1132	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ*)
6		xrletr 13125	. . . . 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 1413	. . . 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 13143	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
10	9	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
11		simp2 1137	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐵 ∈ ℝ*)
12		simp3 1138	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐶 ∈ ℝ*)
13		xrletr 13125	. . . . 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 1373	. . . 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 5113	. . . 4 ⊢ (𝐵 = if(𝐴 ≤ 𝐵, 𝐵, 𝐴) → (𝐵 ≤ 𝐶 ↔ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶))
18		breq1 5113	. . . 4 ⊢ (𝐴 = if(𝐴 ≤ 𝐵, 𝐵, 𝐴) → (𝐴 ≤ 𝐶 ↔ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶))
19	17, 18	ifboth 4531	. . 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 2109  ifcif 4491   class class class wbr 5110  ℝ^*cxr 11214   ≤ cle 11216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-pre-lttri 11149  ax-pre-lttrn 11150

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-po 5549  df-so 5550  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221

This theorem is referenced by:  maxle  13158  mbfmax  25557  itgspliticc  25745  deg1addle2  26014  deg1sublt  26022  cvmliftlem10  35288  iccin  48888