Theorem neg0s 28299

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

Assertion

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

Proof of Theorem neg0s

Step	Hyp	Ref	Expression
1		right0s 28167	. . . . 5 ⊢ ( R ‘ 0s ) = ∅
2	1	imaeq2i 6058	. . . 4 ⊢ ( -us “ ( R ‘ 0s )) = ( -us “ ∅)
3		ima0 6077	. . . 4 ⊢ ( -us “ ∅) = ∅
4	2, 3	eqtri 2785	. . 3 ⊢ ( -us “ ( R ‘ 0s )) = ∅
5		left0s 28166	. . . . 5 ⊢ ( L ‘ 0s ) = ∅
6	5	imaeq2i 6058	. . . 4 ⊢ ( -us “ ( L ‘ 0s )) = ( -us “ ∅)
7	6, 3	eqtri 2785	. . 3 ⊢ ( -us “ ( L ‘ 0s )) = ∅
8	4, 7	oveq12i 7429	. 2 ⊢ (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s ))) = (∅ |s ∅)
9		0no 28082	. . 3 ⊢ 0s ∈ No
10		negsval 28298	. . 3 ⊢ ( 0s ∈ No → ( -us ‘ 0s ) = (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s ))))
11	9, 10	ax-mp 5	. 2 ⊢ ( -us ‘ 0s ) = (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s )))
12		df-0s 28080	. 2 ⊢ 0s = (∅ |s ∅)
13	8, 11, 12	3eqtr4i 2795	1 ⊢ ( -us ‘ 0s ) = 0s

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  ∅c0 4282   “ cima 5662  ‘cfv 6537  (class class class)co 7417   No csur 27884   |s ccuts 28032   0_s c0s 28078   L cleft 28098   R cright 28099   -u_s cnegs 28292

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-1o 8459  df-2o 8460  df-no 27887  df-lts 27888  df-bday 27889  df-slts 28031  df-cuts 28033  df-0s 28080  df-made 28100  df-old 28101  df-left 28103  df-right 28104  df-norec 28211  df-negs 28294

This theorem is used by:  neg1s  28300  lt0negs2d  28324  subsfo  28338  subsid1  28341  ltmulnegs1d  28449  mulscan2d  28452  recsex  28492  abssnid  28516  absmuls  28517  abssge0  28518  absnegs  28520  leabss  28521  elzs2  28672  elnnzs  28674  elznns  28675  z12bday  28758  bdayfin  28760  recut  28767