Theorem negs0s 28076

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 27950	. . . . 5 ⊢ ( R ‘ 0s ) = ∅
2	1	imaeq2i 6087	. . . 4 ⊢ ( -us “ ( R ‘ 0s )) = ( -us “ ∅)
3		ima0 6106	. . . 4 ⊢ ( -us “ ∅) = ∅
4	2, 3	eqtri 2768	. . 3 ⊢ ( -us “ ( R ‘ 0s )) = ∅
5		left0s 27949	. . . . 5 ⊢ ( L ‘ 0s ) = ∅
6	5	imaeq2i 6087	. . . 4 ⊢ ( -us “ ( L ‘ 0s )) = ( -us “ ∅)
7	6, 3	eqtri 2768	. . 3 ⊢ ( -us “ ( L ‘ 0s )) = ∅
8	4, 7	oveq12i 7460	. 2 ⊢ (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s ))) = (∅ |s ∅)
9		0sno 27889	. . 3 ⊢ 0s ∈ No
10		negsval 28075	. . 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 27887	. 2 ⊢ 0s = (∅ |s ∅)
13	8, 11, 12	3eqtr4i 2778	1 ⊢ ( -us ‘ 0s ) = 0s

Syntax hints:   = wceq 1537   ∈ wcel 2108  ∅c0 4352   “ cima 5703  ‘cfv 6573  (class class class)co 7448   No csur 27702   |s cscut 27845   0_s c0s 27885   L cleft 27902   R cright 27903   -u_s cnegs 28069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-negs 28071

This theorem is referenced by:  negs1s  28077  slt0neg2d  28101  subsfo  28113  subsid1  28116  sltmulneg1d  28220  mulscan2d  28223  recsex  28261  abssnid  28285  absmuls  28286  abssge0  28287  abssneg  28289  sleabs  28290  elzs2  28403  elnnzs  28405  elznns  28406  recut  28446  0reno  28447