Theorem neg1s 28443

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

Assertion

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

Proof of Theorem neg1s

Step	Hyp	Ref	Expression
1		1no 28226	. . 3 ⊢ 1s ∈ No
2		negsval 28441	. . 3 ⊢ ( 1s ∈ No → ( -s‘ 1s ) = (( -s “ (R‘ 1s )) |s ( -s “ (L‘ 1s ))))
3	1, 2	ax-mp 5	. 2 ⊢ ( -s‘ 1s ) = (( -s “ (R‘ 1s )) |s ( -s “ (L‘ 1s )))
4		right1s 28312	. . . . 5 ⊢ (R‘ 1s ) = ∅
5	4	imaeq2i 6062	. . . 4 ⊢ ( -s “ (R‘ 1s )) = ( -s “ ∅)
6		ima0 6087	. . . 4 ⊢ ( -s “ ∅) = ∅
7	5, 6	eqtri 2786	. . 3 ⊢ ( -s “ (R‘ 1s )) = ∅
8		left1s 28311	. . . . 5 ⊢ (L‘ 1s ) = { 0s }
9	8	imaeq2i 6062	. . . 4 ⊢ ( -s “ (L‘ 1s )) = ( -s “ { 0s })
10		negsfn 28439	. . . . . . 7 ⊢ -s Fn No
11		0no 28225	. . . . . . 7 ⊢ 0s ∈ No
12		fnimapr 6991	. . . . . . 7 ⊢ (( -s Fn No ∧ 0s ∈ No ∧ 0s ∈ No) → ( -s “ { 0s , 0s }) = {( -s‘ 0s ), ( -s‘ 0s )})
13	10, 11, 11, 12	mp3an 1491	. . . . . 6 ⊢ ( -s “ { 0s , 0s }) = {( -s‘ 0s ), ( -s‘ 0s )}
14		neg0s 28442	. . . . . . 7 ⊢ ( -s‘ 0s ) = 0s
15	14, 14	preq12i 4699	. . . . . 6 ⊢ {( -s‘ 0s ), ( -s‘ 0s )} = { 0s , 0s }
16	13, 15	eqtri 2786	. . . . 5 ⊢ ( -s “ { 0s , 0s }) = { 0s , 0s }
17		dfsn2 4597	. . . . . 6 ⊢ { 0s } = { 0s , 0s }
18	17	imaeq2i 6062	. . . . 5 ⊢ ( -s “ { 0s }) = ( -s “ { 0s , 0s })
19	16, 18, 17	3eqtr4i 2796	. . . 4 ⊢ ( -s “ { 0s }) = { 0s }
20	9, 19	eqtri 2786	. . 3 ⊢ ( -s “ (L‘ 1s )) = { 0s }
21	7, 20	oveq12i 7455	. 2 ⊢ (( -s “ (R‘ 1s )) |s ( -s “ (L‘ 1s ))) = (∅ |s { 0s })
22	3, 21	eqtri 2786	1 ⊢ ( -s‘ 1s ) = (∅ |s { 0s })

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  ∅c0 4281  {csn 4588  {cpr 4590   “ cima 5658   Fn wfn 6553  ‘cfv 6558  (class class class)co 7443  Nocsur 28027   |s ccuts 28175   0_s c0s 28221   1_s c1s 28222  Lcleft 28241  Rcright 28242   -_scnegs 28435

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-negs 28437

This theorem is used by: (None)