Theorem negs0s 33708

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 33668	. . . . 5 ⊢ ( R ‘ 0s ) = ∅
2	1	imaeq2i 5904	. . . 4 ⊢ ( -us “ ( R ‘ 0s )) = ( -us “ ∅)
3		ima0 5922	. . . 4 ⊢ ( -us “ ∅) = ∅
4	2, 3	eqtri 2781	. . 3 ⊢ ( -us “ ( R ‘ 0s )) = ∅
5		left0s 33667	. . . . 5 ⊢ ( L ‘ 0s ) = ∅
6	5	imaeq2i 5904	. . . 4 ⊢ ( -us “ ( L ‘ 0s )) = ( -us “ ∅)
7	6, 3	eqtri 2781	. . 3 ⊢ ( -us “ ( L ‘ 0s )) = ∅
8	4, 7	oveq12i 7168	. 2 ⊢ (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s ))) = (∅ |s ∅)
9		0sno 33615	. . 3 ⊢ 0s ∈ No
10		negsfv 33707	. . 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 33613	. 2 ⊢ 0s = (∅ |s ∅)
13	8, 11, 12	3eqtr4i 2791	1 ⊢ ( -us ‘ 0s ) = 0s

Syntax hints:   = wceq 1538   ∈ wcel 2111  ∅c0 4227   “ cima 5531  ‘cfv 6340  (class class class)co 7156   No csur 33441   |s cscut 33575   0s c0s 33611   L cleft 33624   R cright 33625   -us cnegs 33701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-wrecs 7963  df-recs 8024  df-1o 8118  df-2o 8119  df-frecs 33393  df-no 33444  df-slt 33445  df-bday 33446  df-sslt 33574  df-scut 33576  df-0s 33613  df-made 33626  df-old 33627  df-left 33629  df-right 33630  df-norec 33678  df-negs 33704

This theorem is referenced by: (None)