Theorem sleabs 28061

Description: A surreal is less than or equal to its absolute value. (Contributed by Scott Fenton, 16-Apr-2025.)

Assertion

Ref	Expression
sleabs	⊢ (𝐴 ∈ No → 𝐴 ≤s (abss‘𝐴))

Proof of Theorem sleabs

Step	Hyp	Ref	Expression
1		slerflex 27615	. . . 4 ⊢ (𝐴 ∈ No → 𝐴 ≤s 𝐴)
2	1	adantr 480	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → 𝐴 ≤s 𝐴)
3		abssid 28054	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
4	2, 3	breqtrrd 5167	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → 𝐴 ≤s (abss‘𝐴))
5		simpl 482	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 𝐴 ∈ No )
6		0sno 27678	. . . . 5 ⊢ 0s ∈ No
7	6	a1i 11	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 0s ∈ No )
8		negscl 27867	. . . . 5 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
9	8	adantr 480	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → ( -us ‘𝐴) ∈ No )
10		simpr 484	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 𝐴 ≤s 0s )
11		negs0s 27858	. . . . 5 ⊢ ( -us ‘ 0s ) = 0s
12	5, 7	slenegd 27879	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
13	10, 12	mpbid 231	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us ‘𝐴))
14	11, 13	eqbrtrrid 5175	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 0s ≤s ( -us ‘𝐴))
15	5, 7, 9, 10, 14	sletrd 27614	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 𝐴 ≤s ( -us ‘𝐴))
16		abssnid 28056	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴))
17	15, 16	breqtrrd 5167	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 𝐴 ≤s (abss‘𝐴))
18		sletric 27616	. . 3 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
19	6, 18	mpan 687	. 2 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
20	4, 17, 19	mpjaodan 955	1 ⊢ (𝐴 ∈ No → 𝐴 ≤s (abss‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844   ∈ wcel 2098   class class class wbr 5139  ‘cfv 6534   No csur 27492   ≤s csle 27596   0_s c0s 27674   -u_s cnegs 27851  abs_scabss 28050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-ot 4630  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-1o 8462  df-2o 8463  df-nadd 8662  df-no 27495  df-slt 27496  df-bday 27497  df-sle 27597  df-sslt 27633  df-scut 27635  df-0s 27676  df-made 27693  df-old 27694  df-left 27696  df-right 27697  df-norec 27774  df-norec2 27785  df-adds 27796  df-negs 27853  df-abss 28051

This theorem is referenced by:  absslt  28062