Theorem lemaxle 13098

Description: A real number which is less than or equal to a second real number is less than or equal to the maximum/supremum of the second real number and a third real number. (Contributed by AV, 8-Jun-2021.)

Assertion

Ref	Expression
lemaxle	⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))

Proof of Theorem lemaxle

Step	Hyp	Ref	Expression
1		max2 13090	. . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))
2	1	ancoms 458	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))
3	2	adantr 480	. . 3 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))
4		simpr 484	. . . 4 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
5		simpll 766	. . . 4 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐵 ∈ ℝ)
6		ifcl 4522	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → if(𝐶 ≤ 𝐵, 𝐵, 𝐶) ∈ ℝ)
7	6	adantr 480	. . . 4 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → if(𝐶 ≤ 𝐵, 𝐵, 𝐶) ∈ ℝ)
8		letr 11216	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ if(𝐶 ≤ 𝐵, 𝐵, 𝐶) ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)) → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)))
9	4, 5, 7, 8	syl3anc 1373	. . 3 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)) → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)))
10	3, 9	mpan2d 694	. 2 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝐴 ≤ 𝐵 → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)))
11	10	3impia 1117	1 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2113  ifcif 4476   class class class wbr 5095  ℝcr 11014   ≤ cle 11156

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676  ax-cnex 11071  ax-resscn 11072  ax-pre-lttri 11089  ax-pre-lttrn 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-po 5529  df-so 5530  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-er 8630  df-en 8878  df-dom 8879  df-sdom 8880  df-pnf 11157  df-mnf 11158  df-xr 11159  df-ltxr 11160  df-le 11161

This theorem is referenced by: (None)