MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abssneg Structured version   Visualization version   GIF version

Theorem abssneg 28154
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 27969 . . . 4 (𝐴 No → ( -us ‘( -us𝐴)) = 𝐴)
21adantr 480 . . 3 ((𝐴 No ∧ 0s ≤s 𝐴) → ( -us ‘( -us𝐴)) = 𝐴)
3 negscl 27961 . . . 4 (𝐴 No → ( -us𝐴) ∈ No )
4 0sno 27772 . . . . . . . 8 0s No
54a1i 11 . . . . . . 7 (𝐴 No → 0s No )
6 id 22 . . . . . . 7 (𝐴 No 𝐴 No )
75, 6slenegd 27973 . . . . . 6 (𝐴 No → ( 0s ≤s 𝐴 ↔ ( -us𝐴) ≤s ( -us ‘ 0s )))
8 negs0s 27952 . . . . . . 7 ( -us ‘ 0s ) = 0s
98breq2i 5156 . . . . . 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 28150 . . . 4 ((( -us𝐴) ∈ No ∧ ( -us𝐴) ≤s 0s ) → (abss‘( -us𝐴)) = ( -us ‘( -us𝐴)))
133, 11, 12syl2an2r 684 . . 3 ((𝐴 No ∧ 0s ≤s 𝐴) → (abss‘( -us𝐴)) = ( -us ‘( -us𝐴)))
14 abssid 28148 . . 3 ((𝐴 No ∧ 0s ≤s 𝐴) → (abss𝐴) = 𝐴)
152, 13, 143eqtr4d 2778 . 2 ((𝐴 No ∧ 0s ≤s 𝐴) → (abss‘( -us𝐴)) = (abss𝐴))
166, 5slenegd 27973 . . . . . 6 (𝐴 No → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐴)))
178breq1i 5155 . . . . . 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 28148 . . . 4 ((( -us𝐴) ∈ No ∧ 0s ≤s ( -us𝐴)) → (abss‘( -us𝐴)) = ( -us𝐴))
213, 19, 20syl2an2r 684 . . 3 ((𝐴 No 𝐴 ≤s 0s ) → (abss‘( -us𝐴)) = ( -us𝐴))
22 abssnid 28150 . . 3 ((𝐴 No 𝐴 ≤s 0s ) → (abss𝐴) = ( -us𝐴))
2321, 22eqtr4d 2771 . 2 ((𝐴 No 𝐴 ≤s 0s ) → (abss‘( -us𝐴)) = (abss𝐴))
24 sletric 27710 . . 3 (( 0s No 𝐴 No ) → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
254, 24mpan 689 . 2 (𝐴 No → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
2615, 23, 25mpjaodan 957 1 (𝐴 No → (abss‘( -us𝐴)) = (abss𝐴))
Colors of variables: wff setvar class
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   0s c0s 27768   -us cnegs 27945  absscabss 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
  Copyright terms: Public domain W3C validator