Theorem sleabs 28286

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 27822	. . . 4 ⊢ (𝐴 ∈ No → 𝐴 ≤s 𝐴)
2	1	adantr 480	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → 𝐴 ≤s 𝐴)
3		abssid 28279	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
4	2, 3	breqtrrd 5175	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → 𝐴 ≤s (abss‘𝐴))
5		simpl 482	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 𝐴 ∈ No )
6		0sno 27885	. . . . 5 ⊢ 0s ∈ No
7	6	a1i 11	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 0s ∈ No )
8		negscl 28082	. . . . 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 28072	. . . . 5 ⊢ ( -us ‘ 0s ) = 0s
12	5, 7	slenegd 28094	. . . . . 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 5183	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 0s ≤s ( -us ‘𝐴))
15	5, 7, 9, 10, 14	sletrd 27821	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 𝐴 ≤s ( -us ‘𝐴))
16		abssnid 28281	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴))
17	15, 16	breqtrrd 5175	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 𝐴 ≤s (abss‘𝐴))
18		sletric 27823	. . 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 960	1 ⊢ (𝐴 ∈ No → 𝐴 ≤s (abss‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∈ wcel 2105   class class class wbr 5147  ‘cfv 6562   No csur 27698   ≤s csle 27803   0_s c0s 27881   -u_s cnegs 28065  abs_scabss 28275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-ot 4639  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-1o 8504  df-2o 8505  df-nadd 8702  df-no 27701  df-slt 27702  df-bday 27703  df-sle 27804  df-sslt 27840  df-scut 27842  df-0s 27883  df-made 27900  df-old 27901  df-left 27903  df-right 27904  df-norec 27985  df-norec2 27996  df-adds 28007  df-negs 28067  df-abss 28276

This theorem is referenced by:  absslt  28287