Theorem abssge0 28661

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 4491	. . . . 5 ⊢ ( 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)) = 𝐴)
3	1, 2	breqtrrd 5136	. . . 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 28442	. . . . 5 ⊢ ( -s‘ 0s ) = 0s
6		0no 28225	. . . . . . . . 9 ⊢ 0s ∈ No
7		lestric 28155	. . . . . . . . 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 28464	. . . . . 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 5144	. . . 4 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No) → 0s ≤s ( -s‘𝐴))
16		iffalse 4494	. . . . 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 5136	. . 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 28655	. 2 ⊢ (𝐴 ∈ No → (abss‘𝐴) = if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)))
21	19, 20	breqtrrd 5136	1 ⊢ (𝐴 ∈ No → 0s ≤s (abss‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ifcif 4485   class class class wbr 5106  ‘cfv 6558  Nocsur 28027   ≤s cles 28131   0_s c0s 28221   -_scnegs 28435  abs_scabss 28653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-abss 28654

This theorem is used by:  remulscllem2  28917