Theorem negs1s 27962

Description: An expression for negative surreal one. (Contributed by Scott Fenton, 24-Jul-2025.)

Assertion

Ref	Expression
negs1s	⊢ ( -us ‘ 1s ) = (∅ |s { 0s })

Proof of Theorem negs1s

Step	Hyp	Ref	Expression
1		1sno 27764	. . 3 ⊢ 1s ∈ No
2		negsval 27960	. . 3 ⊢ ( 1s ∈ No → ( -us ‘ 1s ) = (( -us “ ( R ‘ 1s )) |s ( -us “ ( L ‘ 1s ))))
3	1, 2	ax-mp 5	. 2 ⊢ ( -us ‘ 1s ) = (( -us “ ( R ‘ 1s )) |s ( -us “ ( L ‘ 1s )))
4		right1s 27834	. . . . 5 ⊢ ( R ‘ 1s ) = ∅
5	4	imaeq2i 6004	. . . 4 ⊢ ( -us “ ( R ‘ 1s )) = ( -us “ ∅)
6		ima0 6023	. . . 4 ⊢ ( -us “ ∅) = ∅
7	5, 6	eqtri 2753	. . 3 ⊢ ( -us “ ( R ‘ 1s )) = ∅
8		left1s 27833	. . . . 5 ⊢ ( L ‘ 1s ) = { 0s }
9	8	imaeq2i 6004	. . . 4 ⊢ ( -us “ ( L ‘ 1s )) = ( -us “ { 0s })
10		negsfn 27958	. . . . . . 7 ⊢ -us Fn No
11		0sno 27763	. . . . . . 7 ⊢ 0s ∈ No
12		fnimapr 6900	. . . . . . 7 ⊢ (( -us Fn No ∧ 0s ∈ No ∧ 0s ∈ No ) → ( -us “ { 0s , 0s }) = {( -us ‘ 0s ), ( -us ‘ 0s )})
13	10, 11, 11, 12	mp3an 1463	. . . . . 6 ⊢ ( -us “ { 0s , 0s }) = {( -us ‘ 0s ), ( -us ‘ 0s )}
14		negs0s 27961	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
15	14, 14	preq12i 4689	. . . . . 6 ⊢ {( -us ‘ 0s ), ( -us ‘ 0s )} = { 0s , 0s }
16	13, 15	eqtri 2753	. . . . 5 ⊢ ( -us “ { 0s , 0s }) = { 0s , 0s }
17		dfsn2 4587	. . . . . 6 ⊢ { 0s } = { 0s , 0s }
18	17	imaeq2i 6004	. . . . 5 ⊢ ( -us “ { 0s }) = ( -us “ { 0s , 0s })
19	16, 18, 17	3eqtr4i 2763	. . . 4 ⊢ ( -us “ { 0s }) = { 0s }
20	9, 19	eqtri 2753	. . 3 ⊢ ( -us “ ( L ‘ 1s )) = { 0s }
21	7, 20	oveq12i 7353	. 2 ⊢ (( -us “ ( R ‘ 1s )) |s ( -us “ ( L ‘ 1s ))) = (∅ |s { 0s })
22	3, 21	eqtri 2753	1 ⊢ ( -us ‘ 1s ) = (∅ |s { 0s })

Syntax hints:   = wceq 1541   ∈ wcel 2110  ∅c0 4281  {csn 4574  {cpr 4576   “ cima 5617   Fn wfn 6472  ‘cfv 6477  (class class class)co 7341   No csur 27571   |s cscut 27715   0_s c0s 27759   1_s c1s 27760   L cleft 27779   R cright 27780   -u_s cnegs 27954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-1o 8380  df-2o 8381  df-no 27574  df-slt 27575  df-bday 27576  df-sle 27677  df-sslt 27714  df-scut 27716  df-0s 27761  df-1s 27762  df-made 27781  df-old 27782  df-left 27784  df-right 27785  df-norec 27874  df-negs 27956

This theorem is referenced by: (None)