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Theorem abssge0 28132
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 4530 . . . . 5 ( 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -us𝐴)) = 𝐴)
31, 2breqtrrd 5170 . . . 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 27932 . . . . 5 ( -us ‘ 0s ) = 0s
6 0sno 27752 . . . . . . . . 9 0s No
7 sletric 27690 . . . . . . . . 9 (( 0s No 𝐴 No ) → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
86, 7mpan 689 . . . . . . . 8 (𝐴 No → ( 0s ≤s 𝐴𝐴 ≤s 0s ))
98ord 863 . . . . . . 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 27953 . . . . . 6 ((¬ 0s ≤s 𝐴𝐴 No ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us𝐴)))
1410, 13mpbid 231 . . . . 5 ((¬ 0s ≤s 𝐴𝐴 No ) → ( -us ‘ 0s ) ≤s ( -us𝐴))
155, 14eqbrtrrid 5178 . . . 4 ((¬ 0s ≤s 𝐴𝐴 No ) → 0s ≤s ( -us𝐴))
16 iffalse 4533 . . . . 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 5170 . . 3 ((¬ 0s ≤s 𝐴𝐴 No ) → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us𝐴)))
194, 18pm2.61ian 811 . 2 (𝐴 No → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us𝐴)))
20 abssval 28126 . 2 (𝐴 No → (abss𝐴) = if( 0s ≤s 𝐴, 𝐴, ( -us𝐴)))
2119, 20breqtrrd 5170 1 (𝐴 No → 0s ≤s (abss𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 846   = wceq 1534  wcel 2099  ifcif 4524   class class class wbr 5142  cfv 6542   No csur 27566   ≤s csle 27670   0s c0s 27748   -us cnegs 27925  absscabss 28124
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 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
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 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-ot 4633  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-1o 8480  df-2o 8481  df-nadd 8680  df-no 27569  df-slt 27570  df-bday 27571  df-sle 27671  df-sslt 27707  df-scut 27709  df-0s 27750  df-made 27767  df-old 27768  df-left 27770  df-right 27771  df-norec 27848  df-norec2 27859  df-adds 27870  df-negs 27927  df-abss 28125
This theorem is referenced by:  remulscllem2  28222
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