Theorem negs1s 27940

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 27746	. . 3 ⊢ 1s ∈ No
2		negsval 27938	. . 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 27814	. . . . 5 ⊢ ( R ‘ 1s ) = ∅
5	4	imaeq2i 6032	. . . 4 ⊢ ( -us “ ( R ‘ 1s )) = ( -us “ ∅)
6		ima0 6051	. . . 4 ⊢ ( -us “ ∅) = ∅
7	5, 6	eqtri 2753	. . 3 ⊢ ( -us “ ( R ‘ 1s )) = ∅
8		left1s 27813	. . . . 5 ⊢ ( L ‘ 1s ) = { 0s }
9	8	imaeq2i 6032	. . . 4 ⊢ ( -us “ ( L ‘ 1s )) = ( -us “ { 0s })
10		negsfn 27936	. . . . . . 7 ⊢ -us Fn No
11		0sno 27745	. . . . . . 7 ⊢ 0s ∈ No
12		fnimapr 6947	. . . . . . 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 27939	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
15	14, 14	preq12i 4705	. . . . . 6 ⊢ {( -us ‘ 0s ), ( -us ‘ 0s )} = { 0s , 0s }
16	13, 15	eqtri 2753	. . . . 5 ⊢ ( -us “ { 0s , 0s }) = { 0s , 0s }
17		dfsn2 4605	. . . . . 6 ⊢ { 0s } = { 0s , 0s }
18	17	imaeq2i 6032	. . . . 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 7402	. 2 ⊢ (( -us “ ( R ‘ 1s )) |s ( -us “ ( L ‘ 1s ))) = (∅ |s { 0s })
22	3, 21	eqtri 2753	1 ⊢ ( -us ‘ 1s ) = (∅ |s { 0s })

Syntax hints:   = wceq 1540   ∈ wcel 2109  ∅c0 4299  {csn 4592  {cpr 4594   “ cima 5644   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390   No csur 27558   |s cscut 27701   0_s c0s 27741   1_s c1s 27742   L cleft 27760   R cright 27761   -u_s cnegs 27932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  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 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562  df-bday 27563  df-sle 27664  df-sslt 27700  df-scut 27702  df-0s 27743  df-1s 27744  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec 27852  df-negs 27934

This theorem is referenced by: (None)