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Theorem abssneg 28285
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 28090 . . . 4 (𝐴 No → ( -us ‘( -us𝐴)) = 𝐴)
21adantr 480 . . 3 ((𝐴 No ∧ 0s ≤s 𝐴) → ( -us ‘( -us𝐴)) = 𝐴)
3 negscl 28082 . . . 4 (𝐴 No → ( -us𝐴) ∈ No )
4 0sno 27885 . . . . . . . 8 0s No
54a1i 11 . . . . . . 7 (𝐴 No → 0s No )
6 id 22 . . . . . . 7 (𝐴 No 𝐴 No )
75, 6slenegd 28094 . . . . . 6 (𝐴 No → ( 0s ≤s 𝐴 ↔ ( -us𝐴) ≤s ( -us ‘ 0s )))
8 negs0s 28072 . . . . . . 7 ( -us ‘ 0s ) = 0s
98breq2i 5155 . . . . . 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 28281 . . . 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 28279 . . 3 ((𝐴 No ∧ 0s ≤s 𝐴) → (abss𝐴) = 𝐴)
152, 13, 143eqtr4d 2784 . 2 ((𝐴 No ∧ 0s ≤s 𝐴) → (abss‘( -us𝐴)) = (abss𝐴))
166, 5slenegd 28094 . . . . . 6 (𝐴 No → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐴)))
178breq1i 5154 . . . . . 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 28279 . . . 4 ((( -us𝐴) ∈ No ∧ 0s ≤s ( -us𝐴)) → (abss‘( -us𝐴)) = ( -us𝐴))
213, 19, 20syl2an2r 685 . . 3 ((𝐴 No 𝐴 ≤s 0s ) → (abss‘( -us𝐴)) = ( -us𝐴))
22 abssnid 28281 . . 3 ((𝐴 No 𝐴 ≤s 0s ) → (abss𝐴) = ( -us𝐴))
2321, 22eqtr4d 2777 . 2 ((𝐴 No 𝐴 ≤s 0s ) → (abss‘( -us𝐴)) = (abss𝐴))
24 sletric 27823 . . 3 (( 0s No 𝐴 No ) → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
254, 24mpan 690 . 2 (𝐴 No → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
2615, 23, 25mpjaodan 960 1 (𝐴 No → (abss‘( -us𝐴)) = (abss𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1536  wcel 2105   class class class wbr 5147  cfv 6562   No csur 27698   ≤s csle 27803   0s c0s 27881   -us cnegs 28065  absscabss 28275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-ot 4639  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-1o 8504  df-2o 8505  df-nadd 8702  df-no 27701  df-slt 27702  df-bday 27703  df-sle 27804  df-sslt 27840  df-scut 27842  df-0s 27883  df-made 27900  df-old 27901  df-left 27903  df-right 27904  df-norec 27985  df-norec2 27996  df-adds 28007  df-negs 28067  df-abss 28276
This theorem is referenced by:  absslt  28287
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