Theorem negright 28210

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 6880	. . . . . 6 ⊢ (𝑦 = ( -us ‘𝑥) → (( -us ‘𝑦) = 𝑥 ↔ ( -us ‘( -us ‘𝑥)) = 𝑥))
2		rightold 28027	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ( R ‘( -us ‘𝐴)) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
3	2	adantl 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
4		bdayon 27903	. . . . . . . . . . 11 ⊢ ( bday ‘( -us ‘𝐴)) ∈ On
5		rightno 28029	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ( R ‘( -us ‘𝐴)) → 𝑥 ∈ No )
6	5	adantl 486	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → 𝑥 ∈ No )
7		oldbday 28052	. . . . . . . . . . 11 ⊢ ((( bday ‘( -us ‘𝐴)) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
8	4, 6, 7	sylancr 598	. . . . . . . . . 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 28208	. . . . . . . . . 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 28208	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
13	12	adantr 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
14	13	eqcomd 2771	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘𝐴) = ( bday ‘( -us ‘𝐴)))
15	9, 11, 14	3eltr4d 2880	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴))
16		bdayon 27903	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
17	6	negscld 28188	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ No )
18		oldbday 28052	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( -us ‘𝑥) ∈ No ) → (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴)))
19	16, 17, 18	sylancr 598	. . . . . . . 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 28005	. . . . . . . . . 10 ⊢ (𝑥 ∈ ( R ‘( -us ‘𝐴)) → ( -us ‘𝐴) <s 𝑥)
22	21	adantl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝐴) <s 𝑥)
23		simpl 487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → 𝐴 ∈ No )
24	23	negscld 28188	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝐴) ∈ No )
25	24, 6	ltnegsd 28198	. . . . . . . . 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 28195	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
28	27	adantr 485	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝐴)) = 𝐴)
29	26, 28	breqtrd 5131	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) <s 𝐴)
30		elleft 28002	. . . . . . 7 ⊢ (( -us ‘𝑥) ∈ ( L ‘𝐴) ↔ (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ∧ ( -us ‘𝑥) <s 𝐴))
31	20, 29, 30	sylanbrc 594	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( L ‘𝐴))
32		negnegs 28195	. . . . . . 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 3587	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥)
35	34	ex 417	. . . 4 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( R ‘( -us ‘𝐴)) → ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥))
36		leftold 28026	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ ( O ‘( bday ‘𝐴)))
37	36	adantl 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 ∈ ( O ‘( bday ‘𝐴)))
38		leftno 28028	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ No )
39	38	adantl 486	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 ∈ No )
40		oldbday 28052	. . . . . . . . . . 11 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝑦 ∈ No ) → (𝑦 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑦) ∈ ( bday ‘𝐴)))
41	16, 39, 40	sylancr 598	. . . . . . . . . 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 28208	. . . . . . . . . 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 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
46	42, 44, 45	3eltr4d 2880	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴)))
47	39	negscld 28188	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝑦) ∈ No )
48		oldbday 28052	. . . . . . . . 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 598	. . . . . . . 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 28004	. . . . . . . . 9 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 <s 𝐴)
52	51	adantl 486	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 <s 𝐴)
53		simpl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝐴 ∈ No )
54	39, 53	ltnegsd 28198	. . . . . . . 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 28003	. . . . . . 7 ⊢ (( -us ‘𝑦) ∈ ( R ‘( -us ‘𝐴)) ↔ (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ∧ ( -us ‘𝐴) <s ( -us ‘𝑦)))
57	50, 55, 56	sylanbrc 594	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝑦) ∈ ( R ‘( -us ‘𝐴)))
58		eleq1 2853	. . . . . 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 3166	. . . 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 28174	. . . 4 ⊢ -us Fn No
63		leftssno 28024	. . . 4 ⊢ ( L ‘𝐴) ⊆ No
64		fvelimab 6943	. . . 4 ⊢ (( -us Fn No ∧ ( L ‘𝐴) ⊆ No ) → (𝑥 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥))
65	62, 63, 64	mp2an 704	. . 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 2763	1 ⊢ (𝐴 ∈ No → ( R ‘( -us ‘𝐴)) = ( -us “ ( L ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∃wrex 3089   ⊆ wss 3907   class class class wbr 5105   “ cima 5655  Oncon0 6350   Fn wfn 6520  ‘cfv 6525   No csur 27762   <s clts 27763   bday cbday 27764   O cold 27974   L cleft 27976   R cright 27977   -u_s cnegs 28170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-1o 8441  df-2o 8442  df-nadd 8640  df-no 27765  df-lts 27766  df-bday 27767  df-les 27867  df-slts 27909  df-cuts 27911  df-0s 27958  df-made 27978  df-old 27979  df-left 27981  df-right 27982  df-norec 28089  df-norec2 28100  df-adds 28111  df-negs 28172

This theorem is referenced by: (None)