Theorem abssnid 28235

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		0no 27801	. . . 4 ⊢ 0s ∈ No
2		lesloe 27718	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≤s 0s ↔ (𝐴 <s 0s ∨ 𝐴 = 0s )))
3	1, 2	mpan2 692	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ (𝐴 <s 0s ∨ 𝐴 = 0s )))
4		ltnles 27717	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 <s 0s ↔ ¬ 0s ≤s 𝐴))
5	1, 4	mpan2 692	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 <s 0s ↔ ¬ 0s ≤s 𝐴))
6		abssval 28231	. . . . . . 7 ⊢ (𝐴 ∈ No → (abss‘𝐴) = if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)))
7		iffalse 4475	. . . . . . 7 ⊢ (¬ 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)) = ( -us ‘𝐴))
8	6, 7	sylan9eq 2791	. . . . . 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 28234	. . . . . . 7 ⊢ (abss‘ 0s ) = 0s
12		neg0s 28018	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
13	11, 12	eqtr4i 2762	. . . . . 6 ⊢ (abss‘ 0s ) = ( -us ‘ 0s )
14		fveq2 6840	. . . . . 6 ⊢ (𝐴 = 0s → (abss‘𝐴) = (abss‘ 0s ))
15		fveq2 6840	. . . . . 6 ⊢ (𝐴 = 0s → ( -us ‘𝐴) = ( -us ‘ 0s ))
16	13, 14, 15	3eqtr4a 2797	. . . . 5 ⊢ (𝐴 = 0s → (abss‘𝐴) = ( -us ‘𝐴))
17	16	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → (𝐴 = 0s → (abss‘𝐴) = ( -us ‘𝐴)))
18	10, 17	jaod 860	. . 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 848   = wceq 1542   ∈ wcel 2114  ifcif 4466   class class class wbr 5085  ‘cfv 6498   No csur 27603   <s clts 27604   ≤s cles 27708   0_s c0s 27797   -u_s cnegs 28011  abs_scabss 28229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-negs 28013  df-abss 28230

This theorem is referenced by:  absmuls  28236  absnegs  28239  leabss  28240