Theorem abssnid 28686

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) = ( -s‘𝐴))

Proof of Theorem abssnid

Step	Hyp	Ref	Expression
1		0no 28252	. . . 4 ⊢ 0s ∈ No
2		lesloe 28168	. . . 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 28167	. . . . . 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 28682	. . . . . . 7 ⊢ (𝐴 ∈ No → (abss‘𝐴) = if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)))
7		iffalse 4494	. . . . . . 7 ⊢ (¬ 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)) = ( -s‘𝐴))
8	6, 7	sylan9eq 2818	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ¬ 0s ≤s 𝐴) → (abss‘𝐴) = ( -s‘𝐴))
9	8	ex 418	. . . . 5 ⊢ (𝐴 ∈ No → (¬ 0s ≤s 𝐴 → (abss‘𝐴) = ( -s‘𝐴)))
10	5, 9	sylbid 243	. . . 4 ⊢ (𝐴 ∈ No → (𝐴 <s 0s → (abss‘𝐴) = ( -s‘𝐴)))
11		abs0s 28685	. . . . . . 7 ⊢ (abss‘ 0s ) = 0s
12		neg0s 28469	. . . . . . 7 ⊢ ( -s‘ 0s ) = 0s
13	11, 12	eqtr4i 2789	. . . . . 6 ⊢ (abss‘ 0s ) = ( -s‘ 0s )
14		fveq2 6916	. . . . . 6 ⊢ (𝐴 = 0s → (abss‘𝐴) = (abss‘ 0s ))
15		fveq2 6916	. . . . . 6 ⊢ (𝐴 = 0s → ( -s‘𝐴) = ( -s‘ 0s ))
16	13, 14, 15	3eqtr4a 2824	. . . . 5 ⊢ (𝐴 = 0s → (abss‘𝐴) = ( -s‘𝐴))
17	16	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → (𝐴 = 0s → (abss‘𝐴) = ( -s‘𝐴)))
18	10, 17	jaod 873	. . 3 ⊢ (𝐴 ∈ No → ((𝐴 <s 0s ∨ 𝐴 = 0s ) → (abss‘𝐴) = ( -s‘𝐴)))
19	3, 18	sylbid 243	. 2 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s → (abss‘𝐴) = ( -s‘𝐴)))
20	19	imp 412	1 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -s‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ifcif 4485   class class class wbr 5108  ‘cfv 6566  Nocsur 28054   <s clts 28055   ≤s cles 28158   0_s c0s 28248   -_scnegs 28462  abs_scabss 28680

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-les 28159  df-slts 28201  df-cuts 28203  df-0s 28250  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec 28381  df-negs 28464  df-abss 28681

This theorem is used by:  absmuls  28687  absnegs  28690  leabss  28691