Theorem absnegs 28253

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

Assertion

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

Proof of Theorem absnegs

Step	Hyp	Ref	Expression
1		negnegs 28050	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
2	1	adantr 480	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → ( -us ‘( -us ‘𝐴)) = 𝐴)
3		negscl 28042	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
4		0no 27815	. . . . . . . 8 ⊢ 0s ∈ No
5	4	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ No → 0s ∈ No )
6		id 22	. . . . . . 7 ⊢ (𝐴 ∈ No → 𝐴 ∈ No )
7	5, 6	lenegsd 28054	. . . . . 6 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ↔ ( -us ‘𝐴) ≤s ( -us ‘ 0s )))
8		neg0s 28032	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
9	8	breq2i 5094	. . . . . 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 28249	. . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ ( -us ‘𝐴) ≤s 0s ) → (abss‘( -us ‘𝐴)) = ( -us ‘( -us ‘𝐴)))
13	3, 11, 12	syl2an2r 686	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘( -us ‘𝐴)) = ( -us ‘( -us ‘𝐴)))
14		abssid 28247	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
15	2, 13, 14	3eqtr4d 2782	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘( -us ‘𝐴)) = (abss‘𝐴))
16	6, 5	lenegsd 28054	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
17	8	breq1i 5093	. . . . . 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 28247	. . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ 0s ≤s ( -us ‘𝐴)) → (abss‘( -us ‘𝐴)) = ( -us ‘𝐴))
21	3, 19, 20	syl2an2r 686	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘( -us ‘𝐴)) = ( -us ‘𝐴))
22		abssnid 28249	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴))
23	21, 22	eqtr4d 2775	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘( -us ‘𝐴)) = (abss‘𝐴))
24		lestric 27746	. . 3 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
25	4, 24	mpan 691	. 2 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
26	15, 23, 25	mpjaodan 961	1 ⊢ (𝐴 ∈ No → (abss‘( -us ‘𝐴)) = (abss‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  ‘cfv 6492   No csur 27617   ≤s cles 27722   0_s c0s 27811   -u_s cnegs 28025  abs_scabss 28243

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-1o 8398  df-2o 8399  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-les 27723  df-slts 27764  df-cuts 27766  df-0s 27813  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec 27944  df-norec2 27955  df-adds 27966  df-negs 28027  df-abss 28244

This theorem is referenced by:  abslts  28255  abssubs  28256