Theorem negs0s 28058

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

Assertion

Ref	Expression
negs0s	⊢ ( -us ‘ 0s ) = 0s

Proof of Theorem negs0s

Step	Hyp	Ref	Expression
1		right0s 27932	. . . . 5 ⊢ ( R ‘ 0s ) = ∅
2	1	imaeq2i 6076	. . . 4 ⊢ ( -us “ ( R ‘ 0s )) = ( -us “ ∅)
3		ima0 6095	. . . 4 ⊢ ( -us “ ∅) = ∅
4	2, 3	eqtri 2765	. . 3 ⊢ ( -us “ ( R ‘ 0s )) = ∅
5		left0s 27931	. . . . 5 ⊢ ( L ‘ 0s ) = ∅
6	5	imaeq2i 6076	. . . 4 ⊢ ( -us “ ( L ‘ 0s )) = ( -us “ ∅)
7	6, 3	eqtri 2765	. . 3 ⊢ ( -us “ ( L ‘ 0s )) = ∅
8	4, 7	oveq12i 7443	. 2 ⊢ (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s ))) = (∅ |s ∅)
9		0sno 27871	. . 3 ⊢ 0s ∈ No
10		negsval 28057	. . 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 27869	. 2 ⊢ 0s = (∅ |s ∅)
13	8, 11, 12	3eqtr4i 2775	1 ⊢ ( -us ‘ 0s ) = 0s

Syntax hints:   = wceq 1540   ∈ wcel 2108  ∅c0 4333   “ cima 5688  ‘cfv 6561  (class class class)co 7431   No csur 27684   |s cscut 27827   0_s c0s 27867   L cleft 27884   R cright 27885   -u_s cnegs 28051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-negs 28053

This theorem is referenced by:  negs1s  28059  slt0neg2d  28083  subsfo  28095  subsid1  28098  sltmulneg1d  28202  mulscan2d  28205  recsex  28243  abssnid  28267  absmuls  28268  abssge0  28269  abssneg  28271  sleabs  28272  elzs2  28385  elnnzs  28387  elznns  28388  recut  28428  0reno  28429