Theorem neg0s 28394

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 28262	. . . . 5 ⊢ ( R ‘ 0s ) = ∅
2	1	imaeq2i 6052	. . . 4 ⊢ ( -us “ ( R ‘ 0s )) = ( -us “ ∅)
3		ima0 6071	. . . 4 ⊢ ( -us “ ∅) = ∅
4	2, 3	eqtri 2784	. . 3 ⊢ ( -us “ ( R ‘ 0s )) = ∅
5		left0s 28261	. . . . 5 ⊢ ( L ‘ 0s ) = ∅
6	5	imaeq2i 6052	. . . 4 ⊢ ( -us “ ( L ‘ 0s )) = ( -us “ ∅)
7	6, 3	eqtri 2784	. . 3 ⊢ ( -us “ ( L ‘ 0s )) = ∅
8	4, 7	oveq12i 7424	. 2 ⊢ (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s ))) = (∅ |s ∅)
9		0no 28177	. . 3 ⊢ 0s ∈ No
10		negsval 28393	. . 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 28175	. 2 ⊢ 0s = (∅ |s ∅)
13	8, 11, 12	3eqtr4i 2794	1 ⊢ ( -us ‘ 0s ) = 0s

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  ∅c0 4279   “ cima 5654  ‘cfv 6531  (class class class)co 7412   No csur 27979   |s ccuts 28127   0_s c0s 28173   L cleft 28193   R cright 28194   -u_s cnegs 28387

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8460  df-2o 8461  df-no 27982  df-lts 27983  df-bday 27984  df-slts 28126  df-cuts 28128  df-0s 28175  df-made 28195  df-old 28196  df-left 28198  df-right 28199  df-norec 28306  df-negs 28389

This theorem is used by:  neg1s  28395  lt0negs2d  28419  subsfo  28433  subsid1  28436  ltmulnegs1d  28544  mulscan2d  28547  recsex  28587  abssnid  28611  absmuls  28612  abssge0  28613  absnegs  28615  leabss  28616  elzs2  28767  elnnzs  28769  elznns  28770  z12bday  28853  bdayfin  28855  recut  28862