Theorem abssnid 28145

Description: For a negative surreal, its absolute value is its negation. (Contributed by Scott Fenton, 16-Apr-2025.)

Assertion

Ref	Expression
abssnid	⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴))

Proof of Theorem abssnid

Step	Hyp	Ref	Expression
1		0sno 27738	. . . 4 ⊢ 0s ∈ No
2		sleloe 27666	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≤s 0s ↔ (𝐴 <s 0s ∨ 𝐴 = 0s )))
3	1, 2	mpan2 691	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ (𝐴 <s 0s ∨ 𝐴 = 0s )))
4		sltnle 27665	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 <s 0s ↔ ¬ 0s ≤s 𝐴))
5	1, 4	mpan2 691	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 <s 0s ↔ ¬ 0s ≤s 𝐴))
6		abssval 28141	. . . . . . 7 ⊢ (𝐴 ∈ No → (abss‘𝐴) = if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)))
7		iffalse 4497	. . . . . . 7 ⊢ (¬ 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)) = ( -us ‘𝐴))
8	6, 7	sylan9eq 2784	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ¬ 0s ≤s 𝐴) → (abss‘𝐴) = ( -us ‘𝐴))
9	8	ex 412	. . . . 5 ⊢ (𝐴 ∈ No → (¬ 0s ≤s 𝐴 → (abss‘𝐴) = ( -us ‘𝐴)))
10	5, 9	sylbid 240	. . . 4 ⊢ (𝐴 ∈ No → (𝐴 <s 0s → (abss‘𝐴) = ( -us ‘𝐴)))
11		abs0s 28144	. . . . . . 7 ⊢ (abss‘ 0s ) = 0s
12		negs0s 27932	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
13	11, 12	eqtr4i 2755	. . . . . 6 ⊢ (abss‘ 0s ) = ( -us ‘ 0s )
14		fveq2 6858	. . . . . 6 ⊢ (𝐴 = 0s → (abss‘𝐴) = (abss‘ 0s ))
15		fveq2 6858	. . . . . 6 ⊢ (𝐴 = 0s → ( -us ‘𝐴) = ( -us ‘ 0s ))
16	13, 14, 15	3eqtr4a 2790	. . . . 5 ⊢ (𝐴 = 0s → (abss‘𝐴) = ( -us ‘𝐴))
17	16	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → (𝐴 = 0s → (abss‘𝐴) = ( -us ‘𝐴)))
18	10, 17	jaod 859	. . 3 ⊢ (𝐴 ∈ No → ((𝐴 <s 0s ∨ 𝐴 = 0s ) → (abss‘𝐴) = ( -us ‘𝐴)))
19	3, 18	sylbid 240	. 2 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s → (abss‘𝐴) = ( -us ‘𝐴)))
20	19	imp 406	1 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ifcif 4488   class class class wbr 5107  ‘cfv 6511   No csur 27551   <s cslt 27552   ≤s csle 27656   0_s c0s 27734   -u_s cnegs 27925  abs_scabss 28139

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-1o 8434  df-2o 8435  df-no 27554  df-slt 27555  df-bday 27556  df-sle 27657  df-sslt 27693  df-scut 27695  df-0s 27736  df-made 27755  df-old 27756  df-left 27758  df-right 27759  df-norec 27845  df-negs 27927  df-abss 28140

This theorem is referenced by:  absmuls  28146  abssneg  28149  sleabs  28150