Theorem neg1s 28300

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 28083	. . 3 ⊢ 1s ∈ No
2		negsval 28298	. . 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 28169	. . . . 5 ⊢ ( R ‘ 1s ) = ∅
5	4	imaeq2i 6058	. . . 4 ⊢ ( -us “ ( R ‘ 1s )) = ( -us “ ∅)
6		ima0 6077	. . . 4 ⊢ ( -us “ ∅) = ∅
7	5, 6	eqtri 2785	. . 3 ⊢ ( -us “ ( R ‘ 1s )) = ∅
8		left1s 28168	. . . . 5 ⊢ ( L ‘ 1s ) = { 0s }
9	8	imaeq2i 6058	. . . 4 ⊢ ( -us “ ( L ‘ 1s )) = ( -us “ { 0s })
10		negsfn 28296	. . . . . . 7 ⊢ -us Fn No
11		0no 28082	. . . . . . 7 ⊢ 0s ∈ No
12		fnimapr 6965	. . . . . . 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 28299	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
15	14, 14	preq12i 4702	. . . . . 6 ⊢ {( -us ‘ 0s ), ( -us ‘ 0s )} = { 0s , 0s }
16	13, 15	eqtri 2785	. . . . 5 ⊢ ( -us “ { 0s , 0s }) = { 0s , 0s }
17		dfsn2 4600	. . . . . 6 ⊢ { 0s } = { 0s , 0s }
18	17	imaeq2i 6058	. . . . 5 ⊢ ( -us “ { 0s }) = ( -us “ { 0s , 0s })
19	16, 18, 17	3eqtr4i 2795	. . . 4 ⊢ ( -us “ { 0s }) = { 0s }
20	9, 19	eqtri 2785	. . 3 ⊢ ( -us “ ( L ‘ 1s )) = { 0s }
21	7, 20	oveq12i 7429	. 2 ⊢ (( -us “ ( R ‘ 1s )) |s ( -us “ ( L ‘ 1s ))) = (∅ |s { 0s })
22	3, 21	eqtri 2785	1 ⊢ ( -us ‘ 1s ) = (∅ |s { 0s })

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  ∅c0 4282  {csn 4587  {cpr 4589   “ cima 5662   Fn wfn 6532  ‘cfv 6537  (class class class)co 7417   No csur 27884   |s ccuts 28032   0_s c0s 28078   1_s c1s 28079   L cleft 28098   R cright 28099   -u_s cnegs 28292

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-1o 8459  df-2o 8460  df-no 27887  df-lts 27888  df-bday 27889  df-les 27989  df-slts 28031  df-cuts 28033  df-0s 28080  df-1s 28081  df-made 28100  df-old 28101  df-left 28103  df-right 28104  df-norec 28211  df-negs 28294

This theorem is used by: (None)