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Theorem abssge0 28284
Description: The absolute value of a surreal number is non-negative. (Contributed by Scott Fenton, 16-Apr-2025.)
Assertion
Ref Expression
abssge0 (𝐴 No → 0s ≤s (abss𝐴))

Proof of Theorem abssge0
StepHypRef Expression
1 id 22 . . . . 5 ( 0s ≤s 𝐴 → 0s ≤s 𝐴)
2 iftrue 4537 . . . . 5 ( 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -us𝐴)) = 𝐴)
31, 2breqtrrd 5176 . . . 4 ( 0s ≤s 𝐴 → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us𝐴)))
43adantr 480 . . 3 (( 0s ≤s 𝐴𝐴 No ) → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us𝐴)))
5 negs0s 28073 . . . . 5 ( -us ‘ 0s ) = 0s
6 0sno 27886 . . . . . . . . 9 0s No
7 sletric 27824 . . . . . . . . 9 (( 0s No 𝐴 No ) → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
86, 7mpan 690 . . . . . . . 8 (𝐴 No → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
98ord 864 . . . . . . 7 (𝐴 No → (¬ 0s ≤s 𝐴𝐴 ≤s 0s ))
109impcom 407 . . . . . 6 ((¬ 0s ≤s 𝐴𝐴 No ) → 𝐴 ≤s 0s )
11 simpr 484 . . . . . . 7 ((¬ 0s ≤s 𝐴𝐴 No ) → 𝐴 No )
126a1i 11 . . . . . . 7 ((¬ 0s ≤s 𝐴𝐴 No ) → 0s No )
1311, 12slenegd 28095 . . . . . 6 ((¬ 0s ≤s 𝐴𝐴 No ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐴)))
1410, 13mpbid 232 . . . . 5 ((¬ 0s ≤s 𝐴𝐴 No ) → ( -us ‘ 0s ) ≤s ( -us𝐴))
155, 14eqbrtrrid 5184 . . . 4 ((¬ 0s ≤s 𝐴𝐴 No ) → 0s ≤s ( -us𝐴))
16 iffalse 4540 . . . . 5 (¬ 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -us𝐴)) = ( -us𝐴))
1716adantr 480 . . . 4 ((¬ 0s ≤s 𝐴𝐴 No ) → if( 0s ≤s 𝐴, 𝐴, ( -us𝐴)) = ( -us𝐴))
1815, 17breqtrrd 5176 . . 3 ((¬ 0s ≤s 𝐴𝐴 No ) → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us𝐴)))
194, 18pm2.61ian 812 . 2 (𝐴 No → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us𝐴)))
20 abssval 28278 . 2 (𝐴 No → (abss𝐴) = if( 0s ≤s 𝐴, 𝐴, ( -us𝐴)))
2119, 20breqtrrd 5176 1 (𝐴 No → 0s ≤s (abss𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  ifcif 4531   class class class wbr 5148  cfv 6563   No csur 27699   ≤s csle 27804   0s c0s 27882   -us cnegs 28066  absscabss 28276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-abss 28277
This theorem is referenced by:  remulscllem2  28448
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