Theorem negright 28332

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

Assertion

Ref	Expression
negright	⊢ (𝐴 ∈ No → ( R ‘( -us ‘𝐴)) = ( -us “ ( L ‘𝐴)))

Proof of Theorem negright

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

Step	Hyp	Ref	Expression
1		fveqeq2 6891	. . . . . 6 ⊢ (𝑦 = ( -us ‘𝑥) → (( -us ‘𝑦) = 𝑥 ↔ ( -us ‘( -us ‘𝑥)) = 𝑥))
2		rightold 28149	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ( R ‘( -us ‘𝐴)) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
3	2	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
4		bdayon 28025	. . . . . . . . . . 11 ⊢ ( bday ‘( -us ‘𝐴)) ∈ On
5		rightno 28151	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ( R ‘( -us ‘𝐴)) → 𝑥 ∈ No )
6	5	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → 𝑥 ∈ No )
7		oldbday 28174	. . . . . . . . . . 11 ⊢ ((( bday ‘( -us ‘𝐴)) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
8	4, 6, 7	sylancr 599	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
9	3, 8	mpbid 235	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴)))
10		negbday 28330	. . . . . . . . . 10 ⊢ (𝑥 ∈ No → ( bday ‘( -us ‘𝑥)) = ( bday ‘𝑥))
11	6, 10	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝑥)) = ( bday ‘𝑥))
12		negbday 28330	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
13	12	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
14	13	eqcomd 2768	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘𝐴) = ( bday ‘( -us ‘𝐴)))
15	9, 11, 14	3eltr4d 2877	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴))
16		bdayon 28025	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
17	6	negscld 28310	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ No )
18		oldbday 28174	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( -us ‘𝑥) ∈ No ) → (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴)))
19	16, 17, 18	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴)))
20	15, 19	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)))
21		rightgt 28127	. . . . . . . . . 10 ⊢ (𝑥 ∈ ( R ‘( -us ‘𝐴)) → ( -us ‘𝐴) <s 𝑥)
22	21	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝐴) <s 𝑥)
23		simpl 488	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → 𝐴 ∈ No )
24	23	negscld 28310	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝐴) ∈ No )
25	24, 6	ltnegsd 28320	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → (( -us ‘𝐴) <s 𝑥 ↔ ( -us ‘𝑥) <s ( -us ‘( -us ‘𝐴))))
26	22, 25	mpbid 235	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) <s ( -us ‘( -us ‘𝐴)))
27		negnegs 28317	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
28	27	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝐴)) = 𝐴)
29	26, 28	breqtrd 5135	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) <s 𝐴)
30		elleft 28124	. . . . . . 7 ⊢ (( -us ‘𝑥) ∈ ( L ‘𝐴) ↔ (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ∧ ( -us ‘𝑥) <s 𝐴))
31	20, 29, 30	sylanbrc 595	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( L ‘𝐴))
32		negnegs 28317	. . . . . . 7 ⊢ (𝑥 ∈ No → ( -us ‘( -us ‘𝑥)) = 𝑥)
33	6, 32	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝑥)) = 𝑥)
34	1, 31, 33	rspcedvdw 3582	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥)
35	34	ex 418	. . . 4 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( R ‘( -us ‘𝐴)) → ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥))
36		leftold 28148	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ ( O ‘( bday ‘𝐴)))
37	36	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 ∈ ( O ‘( bday ‘𝐴)))
38		leftno 28150	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ No )
39	38	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 ∈ No )
40		oldbday 28174	. . . . . . . . . . 11 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝑦 ∈ No ) → (𝑦 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑦) ∈ ( bday ‘𝐴)))
41	16, 39, 40	sylancr 599	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → (𝑦 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑦) ∈ ( bday ‘𝐴)))
42	37, 41	mpbid 235	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘𝑦) ∈ ( bday ‘𝐴))
43		negbday 28330	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → ( bday ‘( -us ‘𝑦)) = ( bday ‘𝑦))
44	39, 43	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘( -us ‘𝑦)) = ( bday ‘𝑦))
45	12	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
46	42, 44, 45	3eltr4d 2877	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴)))
47	39	negscld 28310	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝑦) ∈ No )
48		oldbday 28174	. . . . . . . . 9 ⊢ ((( bday ‘( -us ‘𝐴)) ∈ On ∧ ( -us ‘𝑦) ∈ No ) → (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴))))
49	4, 47, 48	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴))))
50	46, 49	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))))
51		leftlt 28126	. . . . . . . . 9 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 <s 𝐴)
52	51	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 <s 𝐴)
53		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝐴 ∈ No )
54	39, 53	ltnegsd 28320	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → (𝑦 <s 𝐴 ↔ ( -us ‘𝐴) <s ( -us ‘𝑦)))
55	52, 54	mpbid 235	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝐴) <s ( -us ‘𝑦))
56		elright 28125	. . . . . . 7 ⊢ (( -us ‘𝑦) ∈ ( R ‘( -us ‘𝐴)) ↔ (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ∧ ( -us ‘𝐴) <s ( -us ‘𝑦)))
57	50, 55, 56	sylanbrc 595	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝑦) ∈ ( R ‘( -us ‘𝐴)))
58		eleq1 2850	. . . . . 6 ⊢ (( -us ‘𝑦) = 𝑥 → (( -us ‘𝑦) ∈ ( R ‘( -us ‘𝐴)) ↔ 𝑥 ∈ ( R ‘( -us ‘𝐴))))
59	57, 58	syl5ibcom 248	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → (( -us ‘𝑦) = 𝑥 → 𝑥 ∈ ( R ‘( -us ‘𝐴))))
60	59	rexlimdva 3165	. . . 4 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥 → 𝑥 ∈ ( R ‘( -us ‘𝐴))))
61	35, 60	impbid 215	. . 3 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( R ‘( -us ‘𝐴)) ↔ ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥))
62		negsfn 28296	. . . 4 ⊢ -us Fn No
63		leftssno 28146	. . . 4 ⊢ ( L ‘𝐴) ⊆ No
64		fvelimab 6954	. . . 4 ⊢ (( -us Fn No ∧ ( L ‘𝐴) ⊆ No ) → (𝑥 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥))
65	62, 63, 64	mp2an 705	. . 3 ⊢ (𝑥 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥)
66	61, 65	bitr4di 292	. 2 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( R ‘( -us ‘𝐴)) ↔ 𝑥 ∈ ( -us “ ( L ‘𝐴))))
67	66	eqrdv 2760	1 ⊢ (𝐴 ∈ No → ( R ‘( -us ‘𝐴)) = ( -us “ ( L ‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   ⊆ wss 3902   class class class wbr 5107   “ cima 5662  Oncon0 6361   Fn wfn 6532  ‘cfv 6537   No csur 27884   <s clts 27885   bday cbday 27886   O cold 28096   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-ot 4596  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-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-1o 8459  df-2o 8460  df-nadd 8658  df-no 27887  df-lts 27888  df-bday 27889  df-les 27989  df-slts 28031  df-cuts 28033  df-0s 28080  df-made 28100  df-old 28101  df-left 28103  df-right 28104  df-norec 28211  df-norec2 28222  df-adds 28233  df-negs 28294

This theorem is used by: (None)