Theorem abssnid 28508

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 28074	. . . 4 ⊢ 0s ∈ No
2		lesloe 27990	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≤s 0s ↔ (𝐴 <s 0s ∨ 𝐴 = 0s )))
3	1, 2	mpan2 704	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ (𝐴 <s 0s ∨ 𝐴 = 0s )))
4		ltnles 27989	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 <s 0s ↔ ¬ 0s ≤s 𝐴))
5	1, 4	mpan2 704	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 <s 0s ↔ ¬ 0s ≤s 𝐴))
6		abssval 28504	. . . . . . 7 ⊢ (𝐴 ∈ No → (abss‘𝐴) = if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)))
7		iffalse 4491	. . . . . . 7 ⊢ (¬ 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)) = ( -us ‘𝐴))
8	6, 7	sylan9eq 2815	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ¬ 0s ≤s 𝐴) → (abss‘𝐴) = ( -us ‘𝐴))
9	8	ex 418	. . . . 5 ⊢ (𝐴 ∈ No → (¬ 0s ≤s 𝐴 → (abss‘𝐴) = ( -us ‘𝐴)))
10	5, 9	sylbid 243	. . . 4 ⊢ (𝐴 ∈ No → (𝐴 <s 0s → (abss‘𝐴) = ( -us ‘𝐴)))
11		abs0s 28507	. . . . . . 7 ⊢ (abss‘ 0s ) = 0s
12		neg0s 28291	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
13	11, 12	eqtr4i 2786	. . . . . 6 ⊢ (abss‘ 0s ) = ( -us ‘ 0s )
14		fveq2 6878	. . . . . 6 ⊢ (𝐴 = 0s → (abss‘𝐴) = (abss‘ 0s ))
15		fveq2 6878	. . . . . 6 ⊢ (𝐴 = 0s → ( -us ‘𝐴) = ( -us ‘ 0s ))
16	13, 14, 15	3eqtr4a 2821	. . . . 5 ⊢ (𝐴 = 0s → (abss‘𝐴) = ( -us ‘𝐴))
17	16	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → (𝐴 = 0s → (abss‘𝐴) = ( -us ‘𝐴)))
18	10, 17	jaod 873	. . 3 ⊢ (𝐴 ∈ No → ((𝐴 <s 0s ∨ 𝐴 = 0s ) → (abss‘𝐴) = ( -us ‘𝐴)))
19	3, 18	sylbid 243	. 2 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s → (abss‘𝐴) = ( -us ‘𝐴)))
20	19	imp 412	1 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ifcif 4482   class class class wbr 5103  ‘cfv 6533   No csur 27876   <s clts 27877   ≤s cles 27980   0_s c0s 28070   -u_s cnegs 28284  abs_scabss 28502

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-1o 8455  df-2o 8456  df-no 27879  df-lts 27880  df-bday 27881  df-les 27981  df-slts 28023  df-cuts 28025  df-0s 28072  df-made 28092  df-old 28093  df-left 28095  df-right 28096  df-norec 28203  df-negs 28286  df-abss 28503

This theorem is used by:  absmuls  28509  absnegs  28512  leabss  28513