Theorem absnegs 28255

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

Assertion

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

Proof of Theorem absnegs

Step	Hyp	Ref	Expression
1		negnegs 28052	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
2	1	adantr 480	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → ( -us ‘( -us ‘𝐴)) = 𝐴)
3		negscl 28044	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
4		0no 27817	. . . . . . . 8 ⊢ 0s ∈ No
5	4	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ No → 0s ∈ No )
6		id 22	. . . . . . 7 ⊢ (𝐴 ∈ No → 𝐴 ∈ No )
7	5, 6	lenegsd 28056	. . . . . 6 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ↔ ( -us ‘𝐴) ≤s ( -us ‘ 0s )))
8		neg0s 28034	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
9	8	breq2i 5108	. . . . . 6 ⊢ (( -us ‘𝐴) ≤s ( -us ‘ 0s ) ↔ ( -us ‘𝐴) ≤s 0s )
10	7, 9	bitrdi 287	. . . . 5 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ↔ ( -us ‘𝐴) ≤s 0s ))
11	10	biimpa 476	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → ( -us ‘𝐴) ≤s 0s )
12		abssnid 28251	. . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ ( -us ‘𝐴) ≤s 0s ) → (abss‘( -us ‘𝐴)) = ( -us ‘( -us ‘𝐴)))
13	3, 11, 12	syl2an2r 686	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘( -us ‘𝐴)) = ( -us ‘( -us ‘𝐴)))
14		abssid 28249	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
15	2, 13, 14	3eqtr4d 2782	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘( -us ‘𝐴)) = (abss‘𝐴))
16	6, 5	lenegsd 28056	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
17	8	breq1i 5107	. . . . . 6 ⊢ (( -us ‘ 0s ) ≤s ( -us ‘𝐴) ↔ 0s ≤s ( -us ‘𝐴))
18	16, 17	bitrdi 287	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ 0s ≤s ( -us ‘𝐴)))
19	18	biimpa 476	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 0s ≤s ( -us ‘𝐴))
20		abssid 28249	. . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ 0s ≤s ( -us ‘𝐴)) → (abss‘( -us ‘𝐴)) = ( -us ‘𝐴))
21	3, 19, 20	syl2an2r 686	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘( -us ‘𝐴)) = ( -us ‘𝐴))
22		abssnid 28251	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴))
23	21, 22	eqtr4d 2775	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘( -us ‘𝐴)) = (abss‘𝐴))
24		lestric 27748	. . 3 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
25	4, 24	mpan 691	. 2 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
26	15, 23, 25	mpjaodan 961	1 ⊢ (𝐴 ∈ No → (abss‘( -us ‘𝐴)) = (abss‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   class class class wbr 5100  ‘cfv 6500   No csur 27619   ≤s cles 27724   0_s c0s 27813   -u_s cnegs 28027  abs_scabss 28245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-1o 8407  df-2o 8408  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029  df-abss 28246

This theorem is referenced by:  abslts  28257  abssubs  28258