Theorem lemaxle 12580

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 12572	. . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))
2	1	ancoms 461	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))
3	2	adantr 483	. . 3 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐵 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))
4		simpr 487	. . . 4 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
5		simpll 765	. . . 4 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐵 ∈ ℝ)
6		ifcl 4509	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → if(𝐶 ≤ 𝐵, 𝐵, 𝐶) ∈ ℝ)
7	6	adantr 483	. . . 4 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → if(𝐶 ≤ 𝐵, 𝐵, 𝐶) ∈ ℝ)
8		letr 10726	. . . 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 1366	. . 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 692	. 2 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝐴 ≤ 𝐵 → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)))
11	10	3impia 1112	1 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1082   ∈ wcel 2108  ifcif 4465   class class class wbr 5057  ℝcr 10528   ≤ cle 10668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-pre-lttri 10603  ax-pre-lttrn 10604

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673

This theorem is referenced by: (None)