Theorem abssge0 28636

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 4487	. . . . 5 ⊢ ( 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)) = 𝐴)
3	1, 2	breqtrrd 5132	. . . 4 ⊢ ( 0s ≤s 𝐴 → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)))
4	3	adantr 486	. . 3 ⊢ (( 0s ≤s 𝐴 ∧ 𝐴 ∈ No) → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)))
5		neg0s 28417	. . . . 5 ⊢ ( -s‘ 0s ) = 0s
6		0no 28200	. . . . . . . . 9 ⊢ 0s ∈ No
7		lestric 28130	. . . . . . . . 9 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
8	6, 7	mpan 703	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
9	8	ord 878	. . . . . . 7 ⊢ (𝐴 ∈ No → (¬ 0s ≤s 𝐴 → 𝐴 ≤s 0s ))
10	9	impcom 413	. . . . . 6 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No) → 𝐴 ≤s 0s )
11		simpr 490	. . . . . . 7 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No) → 𝐴 ∈ No)
12	6	a1i 11	. . . . . . 7 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No) → 0s ∈ No)
13	11, 12	lenegsd 28439	. . . . . 6 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No) → (𝐴 ≤s 0s ↔ ( -s‘ 0s ) ≤s ( -s‘𝐴)))
14	10, 13	mpbid 235	. . . . 5 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No) → ( -s‘ 0s ) ≤s ( -s‘𝐴))
15	5, 14	eqbrtrrid 5140	. . . 4 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No) → 0s ≤s ( -s‘𝐴))
16		iffalse 4490	. . . . 5 ⊢ (¬ 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)) = ( -s‘𝐴))
17	16	adantr 486	. . . 4 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No) → if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)) = ( -s‘𝐴))
18	15, 17	breqtrrd 5132	. . 3 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No) → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)))
19	4, 18	pm2.61ian 824	. 2 ⊢ (𝐴 ∈ No → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)))
20		abssval 28630	. 2 ⊢ (𝐴 ∈ No → (abss‘𝐴) = if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)))
21	19, 20	breqtrrd 5132	1 ⊢ (𝐴 ∈ No → 0s ≤s (abss‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ifcif 4481   class class class wbr 5102  ‘cfv 6538  Nocsur 28002   ≤s cles 28106   0_s c0s 28196   -_scnegs 28410  abs_scabss 28628

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-norec2 28340  df-adds 28351  df-negs 28412  df-abss 28629

This theorem is used by:  remulscllem2  28892