Theorem abssnid 28136

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 27758	. . . 4 ⊢ 0s ∈ No
2		sleloe 27686	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≤s 0s ↔ (𝐴 <s 0s ∨ 𝐴 = 0s )))
3	1, 2	mpan2 690	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ (𝐴 <s 0s ∨ 𝐴 = 0s )))
4		sltnle 27685	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 <s 0s ↔ ¬ 0s ≤s 𝐴))
5	1, 4	mpan2 690	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 <s 0s ↔ ¬ 0s ≤s 𝐴))
6		abssval 28132	. . . . . . 7 ⊢ (𝐴 ∈ No → (abss‘𝐴) = if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)))
7		iffalse 4538	. . . . . . 7 ⊢ (¬ 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)) = ( -us ‘𝐴))
8	6, 7	sylan9eq 2788	. . . . . 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 239	. . . 4 ⊢ (𝐴 ∈ No → (𝐴 <s 0s → (abss‘𝐴) = ( -us ‘𝐴)))
11		abs0s 28135	. . . . . . 7 ⊢ (abss‘ 0s ) = 0s
12		negs0s 27938	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
13	11, 12	eqtr4i 2759	. . . . . 6 ⊢ (abss‘ 0s ) = ( -us ‘ 0s )
14		fveq2 6897	. . . . . 6 ⊢ (𝐴 = 0s → (abss‘𝐴) = (abss‘ 0s ))
15		fveq2 6897	. . . . . 6 ⊢ (𝐴 = 0s → ( -us ‘𝐴) = ( -us ‘ 0s ))
16	13, 14, 15	3eqtr4a 2794	. . . . 5 ⊢ (𝐴 = 0s → (abss‘𝐴) = ( -us ‘𝐴))
17	16	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → (𝐴 = 0s → (abss‘𝐴) = ( -us ‘𝐴)))
18	10, 17	jaod 858	. . 3 ⊢ (𝐴 ∈ No → ((𝐴 <s 0s ∨ 𝐴 = 0s ) → (abss‘𝐴) = ( -us ‘𝐴)))
19	3, 18	sylbid 239	. 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 205   ∧ wa 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099  ifcif 4529   class class class wbr 5148  ‘cfv 6548   No csur 27572   <s cslt 27573   ≤s csle 27676   0_s c0s 27754   -u_s cnegs 27931  abs_scabss 28130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-1o 8486  df-2o 8487  df-no 27575  df-slt 27576  df-bday 27577  df-sle 27677  df-sslt 27713  df-scut 27715  df-0s 27756  df-made 27773  df-old 27774  df-left 27776  df-right 27777  df-norec 27854  df-negs 27933  df-abss 28131

This theorem is referenced by:  absmuls  28137  abssneg  28140  sleabs  28141

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