Theorem sleabs 28272

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 27808	. . . 4 ⊢ (𝐴 ∈ No → 𝐴 ≤s 𝐴)
2	1	adantr 480	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → 𝐴 ≤s 𝐴)
3		abssid 28265	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
4	2, 3	breqtrrd 5171	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → 𝐴 ≤s (abss‘𝐴))
5		simpl 482	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 𝐴 ∈ No )
6		0sno 27871	. . . . 5 ⊢ 0s ∈ No
7	6	a1i 11	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 0s ∈ No )
8		negscl 28068	. . . . 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 28058	. . . . 5 ⊢ ( -us ‘ 0s ) = 0s
12	5, 7	slenegd 28080	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
13	10, 12	mpbid 232	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us ‘𝐴))
14	11, 13	eqbrtrrid 5179	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 0s ≤s ( -us ‘𝐴))
15	5, 7, 9, 10, 14	sletrd 27807	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 𝐴 ≤s ( -us ‘𝐴))
16		abssnid 28267	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴))
17	15, 16	breqtrrd 5171	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 𝐴 ≤s (abss‘𝐴))
18		sletric 27809	. . 3 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
19	6, 18	mpan 690	. 2 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
20	4, 17, 19	mpjaodan 961	1 ⊢ (𝐴 ∈ No → 𝐴 ≤s (abss‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∈ wcel 2108   class class class wbr 5143  ‘cfv 6561   No csur 27684   ≤s csle 27789   0_s c0s 27867   -u_s cnegs 28051  abs_scabss 28261

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-abss 28262

This theorem is referenced by:  absslt  28273