Theorem neg1s 28097

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 27880	. . 3 ⊢ 1s ∈ No
2		negsval 28095	. . 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 27966	. . . . 5 ⊢ ( R ‘ 1s ) = ∅
5	4	imaeq2i 6044	. . . 4 ⊢ ( -us “ ( R ‘ 1s )) = ( -us “ ∅)
6		ima0 6063	. . . 4 ⊢ ( -us “ ∅) = ∅
7	5, 6	eqtri 2784	. . 3 ⊢ ( -us “ ( R ‘ 1s )) = ∅
8		left1s 27965	. . . . 5 ⊢ ( L ‘ 1s ) = { 0s }
9	8	imaeq2i 6044	. . . 4 ⊢ ( -us “ ( L ‘ 1s )) = ( -us “ { 0s })
10		negsfn 28093	. . . . . . 7 ⊢ -us Fn No
11		0no 27879	. . . . . . 7 ⊢ 0s ∈ No
12		fnimapr 6946	. . . . . . 7 ⊢ (( -us Fn No ∧ 0s ∈ No ∧ 0s ∈ No ) → ( -us “ { 0s , 0s }) = {( -us ‘ 0s ), ( -us ‘ 0s )})
13	10, 11, 11, 12	mp3an 1481	. . . . . 6 ⊢ ( -us “ { 0s , 0s }) = {( -us ‘ 0s ), ( -us ‘ 0s )}
14		neg0s 28096	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
15	14, 14	preq12i 4696	. . . . . 6 ⊢ {( -us ‘ 0s ), ( -us ‘ 0s )} = { 0s , 0s }
16	13, 15	eqtri 2784	. . . . 5 ⊢ ( -us “ { 0s , 0s }) = { 0s , 0s }
17		dfsn2 4594	. . . . . 6 ⊢ { 0s } = { 0s , 0s }
18	17	imaeq2i 6044	. . . . 5 ⊢ ( -us “ { 0s }) = ( -us “ { 0s , 0s })
19	16, 18, 17	3eqtr4i 2794	. . . 4 ⊢ ( -us “ { 0s }) = { 0s }
20	9, 19	eqtri 2784	. . 3 ⊢ ( -us “ ( L ‘ 1s )) = { 0s }
21	7, 20	oveq12i 7404	. 2 ⊢ (( -us “ ( R ‘ 1s )) |s ( -us “ ( L ‘ 1s ))) = (∅ |s { 0s })
22	3, 21	eqtri 2784	1 ⊢ ( -us ‘ 1s ) = (∅ |s { 0s })

Syntax hints:   = wceq 1559   ∈ wcel 2141  ∅c0 4285  {csn 4581  {cpr 4583   “ cima 5648   Fn wfn 6512  ‘cfv 6517  (class class class)co 7392   No csur 27681   |s ccuts 27829   0_s c0s 27875   1_s c1s 27876   L cleft 27895   R cright 27896   -u_s cnegs 28089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-1o 8432  df-2o 8433  df-no 27684  df-lts 27685  df-bday 27686  df-les 27786  df-slts 27828  df-cuts 27830  df-0s 27877  df-1s 27878  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec 28008  df-negs 28091

This theorem is referenced by: (None)