Theorem neg1s 28418

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 28201	. . 3 ⊢ 1s ∈ No
2		negsval 28416	. . 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 28287	. . . . 5 ⊢ (R‘ 1s ) = ∅
5	4	imaeq2i 6049	. . . 4 ⊢ ( -s “ (R‘ 1s )) = ( -s “ ∅)
6		ima0 6074	. . . 4 ⊢ ( -s “ ∅) = ∅
7	5, 6	eqtri 2783	. . 3 ⊢ ( -s “ (R‘ 1s )) = ∅
8		left1s 28286	. . . . 5 ⊢ (L‘ 1s ) = { 0s }
9	8	imaeq2i 6049	. . . 4 ⊢ ( -s “ (L‘ 1s )) = ( -s “ { 0s })
10		negsfn 28414	. . . . . . 7 ⊢ -s Fn No
11		0no 28200	. . . . . . 7 ⊢ 0s ∈ No
12		fnimapr 6968	. . . . . . 7 ⊢ (( -s Fn No ∧ 0s ∈ No ∧ 0s ∈ No) → ( -s “ { 0s , 0s }) = {( -s‘ 0s ), ( -s‘ 0s )})
13	10, 11, 11, 12	mp3an 1490	. . . . . 6 ⊢ ( -s “ { 0s , 0s }) = {( -s‘ 0s ), ( -s‘ 0s )}
14		neg0s 28417	. . . . . . 7 ⊢ ( -s‘ 0s ) = 0s
15	14, 14	preq12i 4695	. . . . . 6 ⊢ {( -s‘ 0s ), ( -s‘ 0s )} = { 0s , 0s }
16	13, 15	eqtri 2783	. . . . 5 ⊢ ( -s “ { 0s , 0s }) = { 0s , 0s }
17		dfsn2 4593	. . . . . 6 ⊢ { 0s } = { 0s , 0s }
18	17	imaeq2i 6049	. . . . 5 ⊢ ( -s “ { 0s }) = ( -s “ { 0s , 0s })
19	16, 18, 17	3eqtr4i 2793	. . . 4 ⊢ ( -s “ { 0s }) = { 0s }
20	9, 19	eqtri 2783	. . 3 ⊢ ( -s “ (L‘ 1s )) = { 0s }
21	7, 20	oveq12i 7432	. 2 ⊢ (( -s “ (R‘ 1s )) |s ( -s “ (L‘ 1s ))) = (∅ |s { 0s })
22	3, 21	eqtri 2783	1 ⊢ ( -s‘ 1s ) = (∅ |s { 0s })

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  ∅c0 4278  {csn 4584  {cpr 4586   “ cima 5653   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   |s ccuts 28150   0_s c0s 28196   1_s c1s 28197  Lcleft 28216  Rcright 28217   -_scnegs 28410

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-1s 28199  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-negs 28412

This theorem is used by: (None)