Theorem negs1s 28077

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 27890	. . 3 ⊢ 1s ∈ No
2		negsval 28075	. . 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 27952	. . . . 5 ⊢ ( R ‘ 1s ) = ∅
5	4	imaeq2i 6087	. . . 4 ⊢ ( -us “ ( R ‘ 1s )) = ( -us “ ∅)
6		ima0 6106	. . . 4 ⊢ ( -us “ ∅) = ∅
7	5, 6	eqtri 2768	. . 3 ⊢ ( -us “ ( R ‘ 1s )) = ∅
8		left1s 27951	. . . . 5 ⊢ ( L ‘ 1s ) = { 0s }
9	8	imaeq2i 6087	. . . 4 ⊢ ( -us “ ( L ‘ 1s )) = ( -us “ { 0s })
10		negsfn 28073	. . . . . . 7 ⊢ -us Fn No
11		0sno 27889	. . . . . . 7 ⊢ 0s ∈ No
12		fnimapr 7005	. . . . . . 7 ⊢ (( -us Fn No ∧ 0s ∈ No ∧ 0s ∈ No ) → ( -us “ { 0s , 0s }) = {( -us ‘ 0s ), ( -us ‘ 0s )})
13	10, 11, 11, 12	mp3an 1461	. . . . . 6 ⊢ ( -us “ { 0s , 0s }) = {( -us ‘ 0s ), ( -us ‘ 0s )}
14		negs0s 28076	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
15	14, 14	preq12i 4763	. . . . . 6 ⊢ {( -us ‘ 0s ), ( -us ‘ 0s )} = { 0s , 0s }
16	13, 15	eqtri 2768	. . . . 5 ⊢ ( -us “ { 0s , 0s }) = { 0s , 0s }
17		dfsn2 4661	. . . . . 6 ⊢ { 0s } = { 0s , 0s }
18	17	imaeq2i 6087	. . . . 5 ⊢ ( -us “ { 0s }) = ( -us “ { 0s , 0s })
19	16, 18, 17	3eqtr4i 2778	. . . 4 ⊢ ( -us “ { 0s }) = { 0s }
20	9, 19	eqtri 2768	. . 3 ⊢ ( -us “ ( L ‘ 1s )) = { 0s }
21	7, 20	oveq12i 7460	. 2 ⊢ (( -us “ ( R ‘ 1s )) |s ( -us “ ( L ‘ 1s ))) = (∅ |s { 0s })
22	3, 21	eqtri 2768	1 ⊢ ( -us ‘ 1s ) = (∅ |s { 0s })

Syntax hints:   = wceq 1537   ∈ wcel 2108  ∅c0 4352  {csn 4648  {cpr 4650   “ cima 5703   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448   No csur 27702   |s cscut 27845   0_s c0s 27885   1_s c1s 27886   L cleft 27902   R cright 27903   -u_s cnegs 28069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-negs 28071

This theorem is referenced by: (None)