Theorem neg1s 28198

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

Assertion

Ref	Expression
neg1s	⊢ ( -us ‘ 1s ) = (∅ |s { 0s })

Proof of Theorem neg1s

Step	Hyp	Ref	Expression
1		1no 27981	. . 3 ⊢ 1s ∈ No
2		negsval 28196	. . 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 28067	. . . . 5 ⊢ ( R ‘ 1s ) = ∅
5	4	imaeq2i 6062	. . . 4 ⊢ ( -us “ ( R ‘ 1s )) = ( -us “ ∅)
6		ima0 6081	. . . 4 ⊢ ( -us “ ∅) = ∅
7	5, 6	eqtri 2786	. . 3 ⊢ ( -us “ ( R ‘ 1s )) = ∅
8		left1s 28066	. . . . 5 ⊢ ( L ‘ 1s ) = { 0s }
9	8	imaeq2i 6062	. . . 4 ⊢ ( -us “ ( L ‘ 1s )) = ( -us “ { 0s })
10		negsfn 28194	. . . . . . 7 ⊢ -us Fn No
11		0no 27980	. . . . . . 7 ⊢ 0s ∈ No
12		fnimapr 6966	. . . . . . 7 ⊢ (( -us Fn No ∧ 0s ∈ No ∧ 0s ∈ No ) → ( -us “ { 0s , 0s }) = {( -us ‘ 0s ), ( -us ‘ 0s )})
13	10, 11, 11, 12	mp3an 1490	. . . . . 6 ⊢ ( -us “ { 0s , 0s }) = {( -us ‘ 0s ), ( -us ‘ 0s )}
14		neg0s 28197	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
15	14, 14	preq12i 4705	. . . . . 6 ⊢ {( -us ‘ 0s ), ( -us ‘ 0s )} = { 0s , 0s }
16	13, 15	eqtri 2786	. . . . 5 ⊢ ( -us “ { 0s , 0s }) = { 0s , 0s }
17		dfsn2 4603	. . . . . 6 ⊢ { 0s } = { 0s , 0s }
18	17	imaeq2i 6062	. . . . 5 ⊢ ( -us “ { 0s }) = ( -us “ { 0s , 0s })
19	16, 18, 17	3eqtr4i 2796	. . . 4 ⊢ ( -us “ { 0s }) = { 0s }
20	9, 19	eqtri 2786	. . 3 ⊢ ( -us “ ( L ‘ 1s )) = { 0s }
21	7, 20	oveq12i 7424	. 2 ⊢ (( -us “ ( R ‘ 1s )) |s ( -us “ ( L ‘ 1s ))) = (∅ |s { 0s })
22	3, 21	eqtri 2786	1 ⊢ ( -us ‘ 1s ) = (∅ |s { 0s })

Syntax hints:   = wceq 1570   ∈ wcel 2143  ∅c0 4287  {csn 4590  {cpr 4592   “ cima 5666   Fn wfn 6533  ‘cfv 6538  (class class class)co 7412   No csur 27782   |s ccuts 27930   0_s c0s 27976   1_s c1s 27977   L cleft 27996   R cright 27997   -u_s cnegs 28190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-1o 8454  df-2o 8455  df-no 27785  df-lts 27786  df-bday 27787  df-les 27887  df-slts 27929  df-cuts 27931  df-0s 27978  df-1s 27979  df-made 27998  df-old 27999  df-left 28001  df-right 28002  df-norec 28109  df-negs 28192

This theorem is referenced by: (None)