Theorem negs1s 27975

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 27777	. . 3 ⊢ 1s ∈ No
2		negsval 27973	. . 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 27847	. . . . 5 ⊢ ( R ‘ 1s ) = ∅
5	4	imaeq2i 6012	. . . 4 ⊢ ( -us “ ( R ‘ 1s )) = ( -us “ ∅)
6		ima0 6031	. . . 4 ⊢ ( -us “ ∅) = ∅
7	5, 6	eqtri 2754	. . 3 ⊢ ( -us “ ( R ‘ 1s )) = ∅
8		left1s 27846	. . . . 5 ⊢ ( L ‘ 1s ) = { 0s }
9	8	imaeq2i 6012	. . . 4 ⊢ ( -us “ ( L ‘ 1s )) = ( -us “ { 0s })
10		negsfn 27971	. . . . . . 7 ⊢ -us Fn No
11		0sno 27776	. . . . . . 7 ⊢ 0s ∈ No
12		fnimapr 6911	. . . . . . 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 27974	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
15	14, 14	preq12i 4690	. . . . . 6 ⊢ {( -us ‘ 0s ), ( -us ‘ 0s )} = { 0s , 0s }
16	13, 15	eqtri 2754	. . . . 5 ⊢ ( -us “ { 0s , 0s }) = { 0s , 0s }
17		dfsn2 4588	. . . . . 6 ⊢ { 0s } = { 0s , 0s }
18	17	imaeq2i 6012	. . . . 5 ⊢ ( -us “ { 0s }) = ( -us “ { 0s , 0s })
19	16, 18, 17	3eqtr4i 2764	. . . 4 ⊢ ( -us “ { 0s }) = { 0s }
20	9, 19	eqtri 2754	. . 3 ⊢ ( -us “ ( L ‘ 1s )) = { 0s }
21	7, 20	oveq12i 7364	. 2 ⊢ (( -us “ ( R ‘ 1s )) |s ( -us “ ( L ‘ 1s ))) = (∅ |s { 0s })
22	3, 21	eqtri 2754	1 ⊢ ( -us ‘ 1s ) = (∅ |s { 0s })

Syntax hints:   = wceq 1541   ∈ wcel 2111  ∅c0 4282  {csn 4575  {cpr 4577   “ cima 5622   Fn wfn 6482  ‘cfv 6487  (class class class)co 7352   No csur 27584   |s cscut 27728   0_s c0s 27772   1_s c1s 27773   L cleft 27792   R cright 27793   -u_s cnegs 27967

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-1o 8391  df-2o 8392  df-no 27587  df-slt 27588  df-bday 27589  df-sle 27690  df-sslt 27727  df-scut 27729  df-0s 27774  df-1s 27775  df-made 27794  df-old 27795  df-left 27797  df-right 27798  df-norec 27887  df-negs 27969

This theorem is referenced by: (None)