Theorem abssneg 28154

Description: Surreal absolute value of the negative. (Contributed by Scott Fenton, 16-Apr-2025.)

Assertion

Ref	Expression
abssneg	⊢ (𝐴 ∈ No → (abss‘( -us ‘𝐴)) = (abss‘𝐴))

Proof of Theorem abssneg

Step	Hyp	Ref	Expression
1		negnegs 27969	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
2	1	adantr 480	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → ( -us ‘( -us ‘𝐴)) = 𝐴)
3		negscl 27961	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
4		0sno 27772	. . . . . . . 8 ⊢ 0s ∈ No
5	4	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ No → 0s ∈ No )
6		id 22	. . . . . . 7 ⊢ (𝐴 ∈ No → 𝐴 ∈ No )
7	5, 6	slenegd 27973	. . . . . 6 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ↔ ( -us ‘𝐴) ≤s ( -us ‘ 0s )))
8		negs0s 27952	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
9	8	breq2i 5156	. . . . . 6 ⊢ (( -us ‘𝐴) ≤s ( -us ‘ 0s ) ↔ ( -us ‘𝐴) ≤s 0s )
10	7, 9	bitrdi 287	. . . . 5 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ↔ ( -us ‘𝐴) ≤s 0s ))
11	10	biimpa 476	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → ( -us ‘𝐴) ≤s 0s )
12		abssnid 28150	. . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ ( -us ‘𝐴) ≤s 0s ) → (abss‘( -us ‘𝐴)) = ( -us ‘( -us ‘𝐴)))
13	3, 11, 12	syl2an2r 684	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘( -us ‘𝐴)) = ( -us ‘( -us ‘𝐴)))
14		abssid 28148	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
15	2, 13, 14	3eqtr4d 2778	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘( -us ‘𝐴)) = (abss‘𝐴))
16	6, 5	slenegd 27973	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
17	8	breq1i 5155	. . . . . 6 ⊢ (( -us ‘ 0s ) ≤s ( -us ‘𝐴) ↔ 0s ≤s ( -us ‘𝐴))
18	16, 17	bitrdi 287	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ 0s ≤s ( -us ‘𝐴)))
19	18	biimpa 476	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 0s ≤s ( -us ‘𝐴))
20		abssid 28148	. . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ 0s ≤s ( -us ‘𝐴)) → (abss‘( -us ‘𝐴)) = ( -us ‘𝐴))
21	3, 19, 20	syl2an2r 684	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘( -us ‘𝐴)) = ( -us ‘𝐴))
22		abssnid 28150	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴))
23	21, 22	eqtr4d 2771	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘( -us ‘𝐴)) = (abss‘𝐴))
24		sletric 27710	. . 3 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
25	4, 24	mpan 689	. 2 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
26	15, 23, 25	mpjaodan 957	1 ⊢ (𝐴 ∈ No → (abss‘( -us ‘𝐴)) = (abss‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099   class class class wbr 5148  ‘cfv 6548   No csur 27586   ≤s csle 27690   0_s c0s 27768   -u_s cnegs 27945  abs_scabss 28144

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-ot 4638  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-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-1o 8487  df-2o 8488  df-nadd 8687  df-no 27589  df-slt 27590  df-bday 27591  df-sle 27691  df-sslt 27727  df-scut 27729  df-0s 27770  df-made 27787  df-old 27788  df-left 27790  df-right 27791  df-norec 27868  df-norec2 27879  df-adds 27890  df-negs 27947  df-abss 28145

This theorem is referenced by:  absslt  28156