Theorem neg0s 28197

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 28065	. . . . 5 ⊢ ( R ‘ 0s ) = ∅
2	1	imaeq2i 6062	. . . 4 ⊢ ( -us “ ( R ‘ 0s )) = ( -us “ ∅)
3		ima0 6081	. . . 4 ⊢ ( -us “ ∅) = ∅
4	2, 3	eqtri 2786	. . 3 ⊢ ( -us “ ( R ‘ 0s )) = ∅
5		left0s 28064	. . . . 5 ⊢ ( L ‘ 0s ) = ∅
6	5	imaeq2i 6062	. . . 4 ⊢ ( -us “ ( L ‘ 0s )) = ( -us “ ∅)
7	6, 3	eqtri 2786	. . 3 ⊢ ( -us “ ( L ‘ 0s )) = ∅
8	4, 7	oveq12i 7424	. 2 ⊢ (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s ))) = (∅ |s ∅)
9		0no 27980	. . 3 ⊢ 0s ∈ No
10		negsval 28196	. . 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 27978	. 2 ⊢ 0s = (∅ |s ∅)
13	8, 11, 12	3eqtr4i 2796	1 ⊢ ( -us ‘ 0s ) = 0s

Syntax hints:   = wceq 1570   ∈ wcel 2143  ∅c0 4287   “ cima 5666  ‘cfv 6538  (class class class)co 7412   No csur 27782   |s ccuts 27930   0_s c0s 27976   L cleft 27996   R cright 27997   -u_s cnegs 28190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-1o 8454  df-2o 8455  df-no 27785  df-lts 27786  df-bday 27787  df-slts 27929  df-cuts 27931  df-0s 27978  df-made 27998  df-old 27999  df-left 28001  df-right 28002  df-norec 28109  df-negs 28192

This theorem is referenced by:  neg1s  28198  lt0negs2d  28222  subsfo  28236  subsid1  28239  ltmulnegs1d  28347  mulscan2d  28350  recsex  28390  abssnid  28414  absmuls  28415  abssge0  28416  absnegs  28418  leabss  28419  elzs2  28570  elnnzs  28572  elznns  28573  z12bday  28656  bdayfin  28658  recut  28665