Theorem neg0s 28417

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 28285	. . . . 5 ⊢ (R‘ 0s ) = ∅
2	1	imaeq2i 6049	. . . 4 ⊢ ( -s “ (R‘ 0s )) = ( -s “ ∅)
3		ima0 6074	. . . 4 ⊢ ( -s “ ∅) = ∅
4	2, 3	eqtri 2783	. . 3 ⊢ ( -s “ (R‘ 0s )) = ∅
5		left0s 28284	. . . . 5 ⊢ (L‘ 0s ) = ∅
6	5	imaeq2i 6049	. . . 4 ⊢ ( -s “ (L‘ 0s )) = ( -s “ ∅)
7	6, 3	eqtri 2783	. . 3 ⊢ ( -s “ (L‘ 0s )) = ∅
8	4, 7	oveq12i 7432	. 2 ⊢ (( -s “ (R‘ 0s )) |s ( -s “ (L‘ 0s ))) = (∅ |s ∅)
9		0no 28200	. . 3 ⊢ 0s ∈ No
10		negsval 28416	. . 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 28198	. 2 ⊢ 0s = (∅ |s ∅)
13	8, 11, 12	3eqtr4i 2793	1 ⊢ ( -s‘ 0s ) = 0s

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  ∅c0 4278   “ cima 5653  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   |s ccuts 28150   0_s c0s 28196  Lcleft 28216  Rcright 28217   -_scnegs 28410

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-0s 28198  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-negs 28412

This theorem is used by:  neg1s  28418  lt0negs2d  28442  subsfo  28456  subsid1  28459  ltmulnegs1d  28567  mulscan2d  28570  recsex  28610  abssnid  28634  absmuls  28635  abssge0  28636  absnegs  28638  leabss  28639  elzs2  28790  elnnzs  28792  elznns  28793  z12bday  28876  bdayfin  28878  recut  28885