Theorem negsright 28028

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
negsright	⊢ (𝐴 ∈ No → ( R ‘( -us ‘𝐴)) = ( -us “ ( L ‘𝐴)))

Proof of Theorem negsright

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

Step	Hyp	Ref	Expression
1		fveqeq2 6841	. . . . . 6 ⊢ (𝑦 = ( -us ‘𝑥) → (( -us ‘𝑦) = 𝑥 ↔ ( -us ‘( -us ‘𝑥)) = 𝑥))
2		rightssold 27852	. . . . . . . . . . . 12 ⊢ ( R ‘( -us ‘𝐴)) ⊆ ( O ‘( bday ‘( -us ‘𝐴)))
3	2	sseli 3927	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ( R ‘( -us ‘𝐴)) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
4	3	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
5		bdayelon 27742	. . . . . . . . . . 11 ⊢ ( bday ‘( -us ‘𝐴)) ∈ On
6		rightssno 27854	. . . . . . . . . . . . 13 ⊢ ( R ‘( -us ‘𝐴)) ⊆ No
7	6	sseli 3927	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ( R ‘( -us ‘𝐴)) → 𝑥 ∈ No )
8	7	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → 𝑥 ∈ No )
9		oldbday 27873	. . . . . . . . . . 11 ⊢ ((( bday ‘( -us ‘𝐴)) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
10	5, 8, 9	sylancr 587	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
11	4, 10	mpbid 232	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴)))
12		negsbday 28026	. . . . . . . . . 10 ⊢ (𝑥 ∈ No → ( bday ‘( -us ‘𝑥)) = ( bday ‘𝑥))
13	8, 12	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝑥)) = ( bday ‘𝑥))
14		negsbday 28026	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
15	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
16	15	eqcomd 2740	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘𝐴) = ( bday ‘( -us ‘𝐴)))
17	11, 13, 16	3eltr4d 2849	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴))
18		bdayelon 27742	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
19	8	negscld 28006	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ No )
20		oldbday 27873	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( -us ‘𝑥) ∈ No ) → (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴)))
21	18, 19, 20	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴)))
22	17, 21	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)))
23		rightgt 27836	. . . . . . . . . 10 ⊢ (𝑥 ∈ ( R ‘( -us ‘𝐴)) → ( -us ‘𝐴) <s 𝑥)
24	23	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝐴) <s 𝑥)
25		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → 𝐴 ∈ No )
26	25	negscld 28006	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝐴) ∈ No )
27	26, 8	sltnegd 28016	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → (( -us ‘𝐴) <s 𝑥 ↔ ( -us ‘𝑥) <s ( -us ‘( -us ‘𝐴))))
28	24, 27	mpbid 232	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) <s ( -us ‘( -us ‘𝐴)))
29		negnegs 28013	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
30	29	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝐴)) = 𝐴)
31	28, 30	breqtrd 5122	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) <s 𝐴)
32		elleft 27833	. . . . . . 7 ⊢ (( -us ‘𝑥) ∈ ( L ‘𝐴) ↔ (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ∧ ( -us ‘𝑥) <s 𝐴))
33	22, 31, 32	sylanbrc 583	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( L ‘𝐴))
34		negnegs 28013	. . . . . . 7 ⊢ (𝑥 ∈ No → ( -us ‘( -us ‘𝑥)) = 𝑥)
35	8, 34	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝑥)) = 𝑥)
36	1, 33, 35	rspcedvdw 3577	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥)
37	36	ex 412	. . . 4 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( R ‘( -us ‘𝐴)) → ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥))
38		leftssold 27851	. . . . . . . . . . . 12 ⊢ ( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
39	38	sseli 3927	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ ( O ‘( bday ‘𝐴)))
40	39	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 ∈ ( O ‘( bday ‘𝐴)))
41		leftssno 27853	. . . . . . . . . . . . 13 ⊢ ( L ‘𝐴) ⊆ No
42	41	sseli 3927	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ No )
43	42	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 ∈ No )
44		oldbday 27873	. . . . . . . . . . 11 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝑦 ∈ No ) → (𝑦 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑦) ∈ ( bday ‘𝐴)))
45	18, 43, 44	sylancr 587	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → (𝑦 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑦) ∈ ( bday ‘𝐴)))
46	40, 45	mpbid 232	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘𝑦) ∈ ( bday ‘𝐴))
47		negsbday 28026	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → ( bday ‘( -us ‘𝑦)) = ( bday ‘𝑦))
48	43, 47	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘( -us ‘𝑦)) = ( bday ‘𝑦))
49	14	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
50	46, 48, 49	3eltr4d 2849	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴)))
51	43	negscld 28006	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝑦) ∈ No )
52		oldbday 27873	. . . . . . . . 9 ⊢ ((( bday ‘( -us ‘𝐴)) ∈ On ∧ ( -us ‘𝑦) ∈ No ) → (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴))))
53	5, 51, 52	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴))))
54	50, 53	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))))
55		leftlt 27835	. . . . . . . . 9 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 <s 𝐴)
56	55	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 <s 𝐴)
57		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝐴 ∈ No )
58	43, 57	sltnegd 28016	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → (𝑦 <s 𝐴 ↔ ( -us ‘𝐴) <s ( -us ‘𝑦)))
59	56, 58	mpbid 232	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝐴) <s ( -us ‘𝑦))
60		elright 27834	. . . . . . 7 ⊢ (( -us ‘𝑦) ∈ ( R ‘( -us ‘𝐴)) ↔ (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ∧ ( -us ‘𝐴) <s ( -us ‘𝑦)))
61	54, 59, 60	sylanbrc 583	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝑦) ∈ ( R ‘( -us ‘𝐴)))
62		eleq1 2822	. . . . . 6 ⊢ (( -us ‘𝑦) = 𝑥 → (( -us ‘𝑦) ∈ ( R ‘( -us ‘𝐴)) ↔ 𝑥 ∈ ( R ‘( -us ‘𝐴))))
63	61, 62	syl5ibcom 245	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → (( -us ‘𝑦) = 𝑥 → 𝑥 ∈ ( R ‘( -us ‘𝐴))))
64	63	rexlimdva 3135	. . . 4 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥 → 𝑥 ∈ ( R ‘( -us ‘𝐴))))
65	37, 64	impbid 212	. . 3 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( R ‘( -us ‘𝐴)) ↔ ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥))
66		negsfn 27992	. . . 4 ⊢ -us Fn No
67		fvelimab 6904	. . . 4 ⊢ (( -us Fn No ∧ ( L ‘𝐴) ⊆ No ) → (𝑥 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥))
68	66, 41, 67	mp2an 692	. . 3 ⊢ (𝑥 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥)
69	65, 68	bitr4di 289	. 2 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( R ‘( -us ‘𝐴)) ↔ 𝑥 ∈ ( -us “ ( L ‘𝐴))))
70	69	eqrdv 2732	1 ⊢ (𝐴 ∈ No → ( R ‘( -us ‘𝐴)) = ( -us “ ( L ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058   ⊆ wss 3899   class class class wbr 5096   “ cima 5625  Oncon0 6315   Fn wfn 6485  ‘cfv 6490   No csur 27605   <s cslt 27606   bday cbday 27607   O cold 27811   L cleft 27813   R cright 27814   -u_s cnegs 27988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-ot 4587  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-nadd 8592  df-no 27608  df-slt 27609  df-bday 27610  df-sle 27711  df-sslt 27748  df-scut 27750  df-0s 27795  df-made 27815  df-old 27816  df-left 27818  df-right 27819  df-norec 27908  df-norec2 27919  df-adds 27930  df-negs 27990

This theorem is referenced by: (None)