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Theorem abssneg 28271
Description: Surreal absolute value of the negative. (Contributed by Scott Fenton, 16-Apr-2025.)
Assertion
Ref Expression
abssneg (𝐴 No → (abss‘( -us𝐴)) = (abss𝐴))

Proof of Theorem abssneg
StepHypRef Expression
1 negnegs 28076 . . . 4 (𝐴 No → ( -us ‘( -us𝐴)) = 𝐴)
21adantr 480 . . 3 ((𝐴 No ∧ 0s ≤s 𝐴) → ( -us ‘( -us𝐴)) = 𝐴)
3 negscl 28068 . . . 4 (𝐴 No → ( -us𝐴) ∈ No )
4 0sno 27871 . . . . . . . 8 0s No
54a1i 11 . . . . . . 7 (𝐴 No → 0s No )
6 id 22 . . . . . . 7 (𝐴 No 𝐴 No )
75, 6slenegd 28080 . . . . . 6 (𝐴 No → ( 0s ≤s 𝐴 ↔ ( -us𝐴) ≤s ( -us ‘ 0s )))
8 negs0s 28058 . . . . . . 7 ( -us ‘ 0s ) = 0s
98breq2i 5151 . . . . . 6 (( -us𝐴) ≤s ( -us ‘ 0s ) ↔ ( -us𝐴) ≤s 0s )
107, 9bitrdi 287 . . . . 5 (𝐴 No → ( 0s ≤s 𝐴 ↔ ( -us𝐴) ≤s 0s ))
1110biimpa 476 . . . 4 ((𝐴 No ∧ 0s ≤s 𝐴) → ( -us𝐴) ≤s 0s )
12 abssnid 28267 . . . 4 ((( -us𝐴) ∈ No ∧ ( -us𝐴) ≤s 0s ) → (abss‘( -us𝐴)) = ( -us ‘( -us𝐴)))
133, 11, 12syl2an2r 685 . . 3 ((𝐴 No ∧ 0s ≤s 𝐴) → (abss‘( -us𝐴)) = ( -us ‘( -us𝐴)))
14 abssid 28265 . . 3 ((𝐴 No ∧ 0s ≤s 𝐴) → (abss𝐴) = 𝐴)
152, 13, 143eqtr4d 2787 . 2 ((𝐴 No ∧ 0s ≤s 𝐴) → (abss‘( -us𝐴)) = (abss𝐴))
166, 5slenegd 28080 . . . . . 6 (𝐴 No → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐴)))
178breq1i 5150 . . . . . 6 (( -us ‘ 0s ) ≤s ( -us𝐴) ↔ 0s ≤s ( -us𝐴))
1816, 17bitrdi 287 . . . . 5 (𝐴 No → (𝐴 ≤s 0s ↔ 0s ≤s ( -us𝐴)))
1918biimpa 476 . . . 4 ((𝐴 No 𝐴 ≤s 0s ) → 0s ≤s ( -us𝐴))
20 abssid 28265 . . . 4 ((( -us𝐴) ∈ No ∧ 0s ≤s ( -us𝐴)) → (abss‘( -us𝐴)) = ( -us𝐴))
213, 19, 20syl2an2r 685 . . 3 ((𝐴 No 𝐴 ≤s 0s ) → (abss‘( -us𝐴)) = ( -us𝐴))
22 abssnid 28267 . . 3 ((𝐴 No 𝐴 ≤s 0s ) → (abss𝐴) = ( -us𝐴))
2321, 22eqtr4d 2780 . 2 ((𝐴 No 𝐴 ≤s 0s ) → (abss‘( -us𝐴)) = (abss𝐴))
24 sletric 27809 . . 3 (( 0s No 𝐴 No ) → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
254, 24mpan 690 . 2 (𝐴 No → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
2615, 23, 25mpjaodan 961 1 (𝐴 No → (abss‘( -us𝐴)) = (abss𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561   No csur 27684   ≤s csle 27789   0s c0s 27867   -us cnegs 28051  absscabss 28261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-abss 28262
This theorem is referenced by:  absslt  28273
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