Theorem negs0s 28073

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 27947	. . . . 5 ⊢ ( R ‘ 0s ) = ∅
2	1	imaeq2i 6078	. . . 4 ⊢ ( -us “ ( R ‘ 0s )) = ( -us “ ∅)
3		ima0 6097	. . . 4 ⊢ ( -us “ ∅) = ∅
4	2, 3	eqtri 2763	. . 3 ⊢ ( -us “ ( R ‘ 0s )) = ∅
5		left0s 27946	. . . . 5 ⊢ ( L ‘ 0s ) = ∅
6	5	imaeq2i 6078	. . . 4 ⊢ ( -us “ ( L ‘ 0s )) = ( -us “ ∅)
7	6, 3	eqtri 2763	. . 3 ⊢ ( -us “ ( L ‘ 0s )) = ∅
8	4, 7	oveq12i 7443	. 2 ⊢ (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s ))) = (∅ |s ∅)
9		0sno 27886	. . 3 ⊢ 0s ∈ No
10		negsval 28072	. . 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 27884	. 2 ⊢ 0s = (∅ |s ∅)
13	8, 11, 12	3eqtr4i 2773	1 ⊢ ( -us ‘ 0s ) = 0s

Syntax hints:   = wceq 1537   ∈ wcel 2106  ∅c0 4339   “ cima 5692  ‘cfv 6563  (class class class)co 7431   No csur 27699   |s cscut 27842   0_s c0s 27882   L cleft 27899   R cright 27900   -u_s cnegs 28066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-negs 28068

This theorem is referenced by:  negs1s  28074  slt0neg2d  28098  subsfo  28110  subsid1  28113  sltmulneg1d  28217  mulscan2d  28220  recsex  28258  abssnid  28282  absmuls  28283  abssge0  28284  abssneg  28286  sleabs  28287  elzs2  28400  elnnzs  28402  elznns  28403  recut  28443  0reno  28444