Theorem negleft 28477

Description: The left set of the negative of a surreal is the set of negatives of its right set. (Contributed by Scott Fenton, 21-Feb-2026.)

Assertion

Ref	Expression
negleft	⊢ (𝐴 ∈ No → (L‘( -s‘𝐴)) = ( -s “ (R‘𝐴)))

Proof of Theorem negleft

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveqeq2 6920	. . . . . 6 ⊢ (𝑦 = ( -s‘𝑥) → (( -s‘𝑦) = 𝑥 ↔ ( -s‘( -s‘𝑥)) = 𝑥))
2		leftno 28296	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (L‘( -s‘𝐴)) → 𝑥 ∈ No)
3	2	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → 𝑥 ∈ No)
4		negbday 28476	. . . . . . . . . 10 ⊢ (𝑥 ∈ No → (bday‘( -s‘𝑥)) = (bday‘𝑥))
5	3, 4	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (bday‘( -s‘𝑥)) = (bday‘𝑥))
6		leftold 28294	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (L‘( -s‘𝐴)) → 𝑥 ∈ (O‘(bday‘( -s‘𝐴))))
7	6	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → 𝑥 ∈ (O‘(bday‘( -s‘𝐴))))
8		bdayon 28171	. . . . . . . . . . . 12 ⊢ (bday‘( -s‘𝐴)) ∈ On
9		oldbday 28320	. . . . . . . . . . . 12 ⊢ (((bday‘( -s‘𝐴)) ∈ On ∧ 𝑥 ∈ No) → (𝑥 ∈ (O‘(bday‘( -s‘𝐴))) ↔ (bday‘𝑥) ∈ (bday‘( -s‘𝐴))))
10	8, 3, 9	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (𝑥 ∈ (O‘(bday‘( -s‘𝐴))) ↔ (bday‘𝑥) ∈ (bday‘( -s‘𝐴))))
11	7, 10	mpbid 235	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (bday‘𝑥) ∈ (bday‘( -s‘𝐴)))
12		negbday 28476	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → (bday‘( -s‘𝐴)) = (bday‘𝐴))
13	12	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (bday‘( -s‘𝐴)) = (bday‘𝐴))
14	11, 13	eleqtrd 2866	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (bday‘𝑥) ∈ (bday‘𝐴))
15	5, 14	eqeltrd 2864	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (bday‘( -s‘𝑥)) ∈ (bday‘𝐴))
16		bdayon 28171	. . . . . . . . 9 ⊢ (bday‘𝐴) ∈ On
17	3	negscld 28456	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘𝑥) ∈ No)
18		oldbday 28320	. . . . . . . . 9 ⊢ (((bday‘𝐴) ∈ On ∧ ( -s‘𝑥) ∈ No) → (( -s‘𝑥) ∈ (O‘(bday‘𝐴)) ↔ (bday‘( -s‘𝑥)) ∈ (bday‘𝐴)))
19	16, 17, 18	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (( -s‘𝑥) ∈ (O‘(bday‘𝐴)) ↔ (bday‘( -s‘𝑥)) ∈ (bday‘𝐴)))
20	15, 19	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘𝑥) ∈ (O‘(bday‘𝐴)))
21		negnegs 28463	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -s‘( -s‘𝐴)) = 𝐴)
22	21	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘( -s‘𝐴)) = 𝐴)
23		leftlt 28272	. . . . . . . . . 10 ⊢ (𝑥 ∈ (L‘( -s‘𝐴)) → 𝑥 <s ( -s‘𝐴))
24	23	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → 𝑥 <s ( -s‘𝐴))
25		negscl 28455	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( -s‘𝐴) ∈ No)
26	25	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘𝐴) ∈ No)
27	3, 26	ltnegsd 28466	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (𝑥 <s ( -s‘𝐴) ↔ ( -s‘( -s‘𝐴)) <s ( -s‘𝑥)))
28	24, 27	mpbid 235	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘( -s‘𝐴)) <s ( -s‘𝑥))
29	22, 28	eqbrtrrd 5133	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → 𝐴 <s ( -s‘𝑥))
30		elright 28271	. . . . . . 7 ⊢ (( -s‘𝑥) ∈ (R‘𝐴) ↔ (( -s‘𝑥) ∈ (O‘(bday‘𝐴)) ∧ 𝐴 <s ( -s‘𝑥)))
31	20, 29, 30	sylanbrc 595	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘𝑥) ∈ (R‘𝐴))
32		negnegs 28463	. . . . . . 7 ⊢ (𝑥 ∈ No → ( -s‘( -s‘𝑥)) = 𝑥)
33	3, 32	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘( -s‘𝑥)) = 𝑥)
34	1, 31, 33	rspcedvdw 3583	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ∃𝑦 ∈ (R‘𝐴)( -s‘𝑦) = 𝑥)
35	34	ex 418	. . . 4 ⊢ (𝐴 ∈ No → (𝑥 ∈ (L‘( -s‘𝐴)) → ∃𝑦 ∈ (R‘𝐴)( -s‘𝑦) = 𝑥))
36		rightold 28295	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (R‘𝐴) → 𝑦 ∈ (O‘(bday‘𝐴)))
37		rightno 28297	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (R‘𝐴) → 𝑦 ∈ No)
38		oldbday 28320	. . . . . . . . . . . 12 ⊢ (((bday‘𝐴) ∈ On ∧ 𝑦 ∈ No) → (𝑦 ∈ (O‘(bday‘𝐴)) ↔ (bday‘𝑦) ∈ (bday‘𝐴)))
39	16, 37, 38	sylancr 599	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (R‘𝐴) → (𝑦 ∈ (O‘(bday‘𝐴)) ↔ (bday‘𝑦) ∈ (bday‘𝐴)))
40	36, 39	mpbid 235	. . . . . . . . . 10 ⊢ (𝑦 ∈ (R‘𝐴) → (bday‘𝑦) ∈ (bday‘𝐴))
41	40	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (bday‘𝑦) ∈ (bday‘𝐴))
42	37	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → 𝑦 ∈ No)
43		negbday 28476	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → (bday‘( -s‘𝑦)) = (bday‘𝑦))
44	42, 43	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (bday‘( -s‘𝑦)) = (bday‘𝑦))
45	12	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (bday‘( -s‘𝐴)) = (bday‘𝐴))
46	41, 44, 45	3eltr4d 2879	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (bday‘( -s‘𝑦)) ∈ (bday‘( -s‘𝐴)))
47	42	negscld 28456	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → ( -s‘𝑦) ∈ No)
48		oldbday 28320	. . . . . . . . 9 ⊢ (((bday‘( -s‘𝐴)) ∈ On ∧ ( -s‘𝑦) ∈ No) → (( -s‘𝑦) ∈ (O‘(bday‘( -s‘𝐴))) ↔ (bday‘( -s‘𝑦)) ∈ (bday‘( -s‘𝐴))))
49	8, 47, 48	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (( -s‘𝑦) ∈ (O‘(bday‘( -s‘𝐴))) ↔ (bday‘( -s‘𝑦)) ∈ (bday‘( -s‘𝐴))))
50	46, 49	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → ( -s‘𝑦) ∈ (O‘(bday‘( -s‘𝐴))))
51		rightgt 28273	. . . . . . . . 9 ⊢ (𝑦 ∈ (R‘𝐴) → 𝐴 <s 𝑦)
52	51	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → 𝐴 <s 𝑦)
53		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → 𝐴 ∈ No)
54	53, 42	ltnegsd 28466	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (𝐴 <s 𝑦 ↔ ( -s‘𝑦) <s ( -s‘𝐴)))
55	52, 54	mpbid 235	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → ( -s‘𝑦) <s ( -s‘𝐴))
56		elleft 28270	. . . . . . 7 ⊢ (( -s‘𝑦) ∈ (L‘( -s‘𝐴)) ↔ (( -s‘𝑦) ∈ (O‘(bday‘( -s‘𝐴))) ∧ ( -s‘𝑦) <s ( -s‘𝐴)))
57	50, 55, 56	sylanbrc 595	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → ( -s‘𝑦) ∈ (L‘( -s‘𝐴)))
58		eleq1 2852	. . . . . 6 ⊢ (( -s‘𝑦) = 𝑥 → (( -s‘𝑦) ∈ (L‘( -s‘𝐴)) ↔ 𝑥 ∈ (L‘( -s‘𝐴))))
59	57, 58	syl5ibcom 248	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (( -s‘𝑦) = 𝑥 → 𝑥 ∈ (L‘( -s‘𝐴))))
60	59	rexlimdva 3167	. . . 4 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ (R‘𝐴)( -s‘𝑦) = 𝑥 → 𝑥 ∈ (L‘( -s‘𝐴))))
61	35, 60	impbid 215	. . 3 ⊢ (𝐴 ∈ No → (𝑥 ∈ (L‘( -s‘𝐴)) ↔ ∃𝑦 ∈ (R‘𝐴)( -s‘𝑦) = 𝑥))
62		negsfn 28442	. . . 4 ⊢ -s Fn No
63		rightssno 28293	. . . 4 ⊢ (R‘𝐴) ⊆ No
64		fvelimab 6983	. . . 4 ⊢ (( -s Fn No ∧ (R‘𝐴) ⊆ No) → (𝑥 ∈ ( -s “ (R‘𝐴)) ↔ ∃𝑦 ∈ (R‘𝐴)( -s‘𝑦) = 𝑥))
65	62, 63, 64	mp2an 705	. . 3 ⊢ (𝑥 ∈ ( -s “ (R‘𝐴)) ↔ ∃𝑦 ∈ (R‘𝐴)( -s‘𝑦) = 𝑥)
66	61, 65	bitr4di 292	. 2 ⊢ (𝐴 ∈ No → (𝑥 ∈ (L‘( -s‘𝐴)) ↔ 𝑥 ∈ ( -s “ (R‘𝐴))))
67	66	eqrdv 2761	1 ⊢ (𝐴 ∈ No → (L‘( -s‘𝐴)) = ( -s “ (R‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5106   “ cima 5659  Oncon0 6383   Fn wfn 6556  ‘cfv 6561  Nocsur 28030   <s clts 28031  bdaycbday 28032  Ocold 28242  Lcleft 28244  Rcright 28245   -_scnegs 28438

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 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-nadd 8706  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-0s 28226  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec 28357  df-norec2 28368  df-adds 28379  df-negs 28440

This theorem is used by:  zcuts0  28827