Theorem neg0s 28442

Description: Negative surreal zero is surreal zero. (Contributed by Scott Fenton, 20-Aug-2024.)

Assertion

Ref	Expression
neg0s	⊢ ( -s‘ 0s ) = 0s

Proof of Theorem neg0s

Step	Hyp	Ref	Expression
1		right0s 28310	. . . . 5 ⊢ (R‘ 0s ) = ∅
2	1	imaeq2i 6062	. . . 4 ⊢ ( -s “ (R‘ 0s )) = ( -s “ ∅)
3		ima0 6087	. . . 4 ⊢ ( -s “ ∅) = ∅
4	2, 3	eqtri 2786	. . 3 ⊢ ( -s “ (R‘ 0s )) = ∅
5		left0s 28309	. . . . 5 ⊢ (L‘ 0s ) = ∅
6	5	imaeq2i 6062	. . . 4 ⊢ ( -s “ (L‘ 0s )) = ( -s “ ∅)
7	6, 3	eqtri 2786	. . 3 ⊢ ( -s “ (L‘ 0s )) = ∅
8	4, 7	oveq12i 7455	. 2 ⊢ (( -s “ (R‘ 0s )) |s ( -s “ (L‘ 0s ))) = (∅ |s ∅)
9		0no 28225	. . 3 ⊢ 0s ∈ No
10		negsval 28441	. . 3 ⊢ ( 0s ∈ No → ( -s‘ 0s ) = (( -s “ (R‘ 0s )) |s ( -s “ (L‘ 0s ))))
11	9, 10	ax-mp 5	. 2 ⊢ ( -s‘ 0s ) = (( -s “ (R‘ 0s )) |s ( -s “ (L‘ 0s )))
12		df-0s 28223	. 2 ⊢ 0s = (∅ |s ∅)
13	8, 11, 12	3eqtr4i 2796	1 ⊢ ( -s‘ 0s ) = 0s

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  ∅c0 4281   “ cima 5658  ‘cfv 6558  (class class class)co 7443  Nocsur 28027   |s ccuts 28175   0_s c0s 28221  Lcleft 28241  Rcright 28242   -_scnegs 28435

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-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-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-no 28030  df-lts 28031  df-bday 28032  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-negs 28437

This theorem is used by:  neg1s  28443  lt0negs2d  28467  subsfo  28481  subsid1  28484  ltmulnegs1d  28592  mulscan2d  28595  recsex  28635  abssnid  28659  absmuls  28660  abssge0  28661  absnegs  28663  leabss  28664  elzs2  28815  elnnzs  28817  elznns  28818  z12bday  28901  bdayfin  28903  recut  28910