Theorem absnegs 28679

Description: Surreal absolute value of the negative. (Contributed by Scott Fenton, 16-Apr-2025.)

Assertion

Ref	Expression
absnegs	⊢ (𝐴 ∈ No → (abss‘( -s‘𝐴)) = (abss‘𝐴))

Proof of Theorem absnegs

Step	Hyp	Ref	Expression
1		negnegs 28476	. . . 4 ⊢ (𝐴 ∈ No → ( -s‘( -s‘𝐴)) = 𝐴)
2	1	adantr 486	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → ( -s‘( -s‘𝐴)) = 𝐴)
3		negscl 28468	. . . 4 ⊢ (𝐴 ∈ No → ( -s‘𝐴) ∈ No)
4		0no 28241	. . . . . . . 8 ⊢ 0s ∈ No
5	4	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ No → 0s ∈ No)
6		id 23	. . . . . . 7 ⊢ (𝐴 ∈ No → 𝐴 ∈ No)
7	5, 6	lenegsd 28480	. . . . . 6 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ↔ ( -s‘𝐴) ≤s ( -s‘ 0s )))
8		neg0s 28458	. . . . . . 7 ⊢ ( -s‘ 0s ) = 0s
9	8	breq2i 5114	. . . . . 6 ⊢ (( -s‘𝐴) ≤s ( -s‘ 0s ) ↔ ( -s‘𝐴) ≤s 0s )
10	7, 9	bitrdi 290	. . . . 5 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ↔ ( -s‘𝐴) ≤s 0s ))
11	10	biimpa 482	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → ( -s‘𝐴) ≤s 0s )
12		abssnid 28675	. . . 4 ⊢ ((( -s‘𝐴) ∈ No ∧ ( -s‘𝐴) ≤s 0s ) → (abss‘( -s‘𝐴)) = ( -s‘( -s‘𝐴)))
13	3, 11, 12	syl2an2r 698	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘( -s‘𝐴)) = ( -s‘( -s‘𝐴)))
14		abssid 28673	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
15	2, 13, 14	3eqtr4d 2808	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘( -s‘𝐴)) = (abss‘𝐴))
16	6, 5	lenegsd 28480	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ ( -s‘ 0s ) ≤s ( -s‘𝐴)))
17	8	breq1i 5113	. . . . . 6 ⊢ (( -s‘ 0s ) ≤s ( -s‘𝐴) ↔ 0s ≤s ( -s‘𝐴))
18	16, 17	bitrdi 290	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ 0s ≤s ( -s‘𝐴)))
19	18	biimpa 482	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 0s ≤s ( -s‘𝐴))
20		abssid 28673	. . . 4 ⊢ ((( -s‘𝐴) ∈ No ∧ 0s ≤s ( -s‘𝐴)) → (abss‘( -s‘𝐴)) = ( -s‘𝐴))
21	3, 19, 20	syl2an2r 698	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘( -s‘𝐴)) = ( -s‘𝐴))
22		abssnid 28675	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -s‘𝐴))
23	21, 22	eqtr4d 2801	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘( -s‘𝐴)) = (abss‘𝐴))
24		lestric 28171	. . 3 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
25	4, 24	mpan 703	. 2 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
26	15, 23, 25	mpjaodan 973	1 ⊢ (𝐴 ∈ No → (abss‘( -s‘𝐴)) = (abss‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   class class class wbr 5106  ‘cfv 6564  Nocsur 28043   ≤s cles 28147   0_s c0s 28237   -_scnegs 28451  abs_scabss 28669

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 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781

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 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-1o 8512  df-2o 8513  df-nadd 8711  df-no 28046  df-lts 28047  df-bday 28048  df-les 28148  df-slts 28190  df-cuts 28192  df-0s 28239  df-made 28259  df-old 28260  df-left 28262  df-right 28263  df-norec 28370  df-norec2 28381  df-adds 28392  df-negs 28453  df-abss 28670

This theorem is used by:  abslts  28681  abssubs  28682