Theorem abssge0 28447

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 (abss‘𝐴))

Proof of Theorem abssge0

Step	Hyp	Ref	Expression
1		id 23	. . . . 5 ⊢ ( 0s ≤s 𝐴 → 0s ≤s 𝐴)
2		iftrue 4493	. . . . 5 ⊢ ( 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)) = 𝐴)
3	1, 2	breqtrrd 5139	. . . 4 ⊢ ( 0s ≤s 𝐴 → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)))
4	3	adantr 485	. . 3 ⊢ (( 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)))
5		neg0s 28228	. . . . 5 ⊢ ( -us ‘ 0s ) = 0s
6		0no 28011	. . . . . . . . 9 ⊢ 0s ∈ No
7		lestric 27941	. . . . . . . . 9 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
8	6, 7	mpan 702	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
9	8	ord 877	. . . . . . 7 ⊢ (𝐴 ∈ No → (¬ 0s ≤s 𝐴 → 𝐴 ≤s 0s ))
10	9	impcom 412	. . . . . 6 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → 𝐴 ≤s 0s )
11		simpr 489	. . . . . . 7 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → 𝐴 ∈ No )
12	6	a1i 11	. . . . . . 7 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → 0s ∈ No )
13	11, 12	lenegsd 28250	. . . . . 6 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
14	10, 13	mpbid 235	. . . . 5 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → ( -us ‘ 0s ) ≤s ( -us ‘𝐴))
15	5, 14	eqbrtrrid 5147	. . . 4 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → 0s ≤s ( -us ‘𝐴))
16		iffalse 4496	. . . . 5 ⊢ (¬ 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)) = ( -us ‘𝐴))
17	16	adantr 485	. . . 4 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)) = ( -us ‘𝐴))
18	15, 17	breqtrrd 5139	. . 3 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)))
19	4, 18	pm2.61ian 823	. 2 ⊢ (𝐴 ∈ No → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)))
20		abssval 28441	. 2 ⊢ (𝐴 ∈ No → (abss‘𝐴) = if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)))
21	19, 20	breqtrrd 5139	1 ⊢ (𝐴 ∈ No → 0s ≤s (abss‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ifcif 4487   class class class wbr 5109  ‘cfv 6536   No csur 27813   ≤s cles 27917   0_s c0s 28007   -u_s cnegs 28221  abs_scabss 28439

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-1o 8449  df-2o 8450  df-nadd 8648  df-no 27816  df-lts 27817  df-bday 27818  df-les 27918  df-slts 27960  df-cuts 27962  df-0s 28009  df-made 28029  df-old 28030  df-left 28032  df-right 28033  df-norec 28140  df-norec2 28151  df-adds 28162  df-negs 28223  df-abss 28440

This theorem is used by:  remulscllem2  28703