Theorem neg0s 28018

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 27886	. . . . 5 ⊢ ( R ‘ 0s ) = ∅
2	1	imaeq2i 6023	. . . 4 ⊢ ( -us “ ( R ‘ 0s )) = ( -us “ ∅)
3		ima0 6042	. . . 4 ⊢ ( -us “ ∅) = ∅
4	2, 3	eqtri 2759	. . 3 ⊢ ( -us “ ( R ‘ 0s )) = ∅
5		left0s 27885	. . . . 5 ⊢ ( L ‘ 0s ) = ∅
6	5	imaeq2i 6023	. . . 4 ⊢ ( -us “ ( L ‘ 0s )) = ( -us “ ∅)
7	6, 3	eqtri 2759	. . 3 ⊢ ( -us “ ( L ‘ 0s )) = ∅
8	4, 7	oveq12i 7379	. 2 ⊢ (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s ))) = (∅ |s ∅)
9		0no 27801	. . 3 ⊢ 0s ∈ No
10		negsval 28017	. . 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 27799	. 2 ⊢ 0s = (∅ |s ∅)
13	8, 11, 12	3eqtr4i 2769	1 ⊢ ( -us ‘ 0s ) = 0s

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∅c0 4273   “ cima 5634  ‘cfv 6498  (class class class)co 7367   No csur 27603   |s ccuts 27751   0_s c0s 27797   L cleft 27817   R cright 27818   -u_s cnegs 28011

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-slts 27750  df-cuts 27752  df-0s 27799  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-negs 28013

This theorem is referenced by:  neg1s  28019  lt0negs2d  28043  subsfo  28057  subsid1  28060  ltmulnegs1d  28168  mulscan2d  28171  recsex  28211  abssnid  28235  absmuls  28236  abssge0  28237  absnegs  28239  leabss  28240  elzs2  28391  elnnzs  28393  elznns  28394  z12bday  28477  bdayfin  28479  recut  28486