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Theorem negs0s 27328
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
StepHypRef Expression
1 right0s 27226 . . . . 5 ( R ‘ 0s ) = ∅
21imaeq2i 6012 . . . 4 ( -us “ ( R ‘ 0s )) = ( -us “ ∅)
3 ima0 6030 . . . 4 ( -us “ ∅) = ∅
42, 3eqtri 2765 . . 3 ( -us “ ( R ‘ 0s )) = ∅
5 left0s 27225 . . . . 5 ( L ‘ 0s ) = ∅
65imaeq2i 6012 . . . 4 ( -us “ ( L ‘ 0s )) = ( -us “ ∅)
76, 3eqtri 2765 . . 3 ( -us “ ( L ‘ 0s )) = ∅
84, 7oveq12i 7370 . 2 (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s ))) = (∅ |s ∅)
9 0sno 27168 . . 3 0s No
10 negsval 27327 . . 3 ( 0s No → ( -us ‘ 0s ) = (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s ))))
119, 10ax-mp 5 . 2 ( -us ‘ 0s ) = (( -us “ ( R ‘ 0s )) |s ( -us “ ( L ‘ 0s )))
12 df-0s 27166 . 2 0s = (∅ |s ∅)
138, 11, 123eqtr4i 2775 1 ( -us ‘ 0s ) = 0s
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  c0 4283  cima 5637  cfv 6497  (class class class)co 7358   No csur 26991   |s cscut 27125   0s c0s 27164   L cleft 27178   R cright 27179   -us cnegs 27321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-1o 8413  df-2o 8414  df-no 26994  df-slt 26995  df-bday 26996  df-sslt 27124  df-scut 27126  df-0s 27166  df-made 27180  df-old 27181  df-left 27183  df-right 27184  df-norec 27253  df-negs 27323
This theorem is referenced by:  subsid1  27358
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