Theorem negright 28067

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 6851	. . . . . 6 ⊢ (𝑦 = ( -us ‘𝑥) → (( -us ‘𝑦) = 𝑥 ↔ ( -us ‘( -us ‘𝑥)) = 𝑥))
2		rightold 27884	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ( R ‘( -us ‘𝐴)) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
3	2	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
4		bdayon 27760	. . . . . . . . . . 11 ⊢ ( bday ‘( -us ‘𝐴)) ∈ On
5		rightno 27886	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ( R ‘( -us ‘𝐴)) → 𝑥 ∈ No )
6	5	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → 𝑥 ∈ No )
7		oldbday 27909	. . . . . . . . . . 11 ⊢ ((( bday ‘( -us ‘𝐴)) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
8	4, 6, 7	sylancr 588	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
9	3, 8	mpbid 232	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴)))
10		negbday 28065	. . . . . . . . . 10 ⊢ (𝑥 ∈ No → ( bday ‘( -us ‘𝑥)) = ( bday ‘𝑥))
11	6, 10	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝑥)) = ( bday ‘𝑥))
12		negbday 28065	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
13	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
14	13	eqcomd 2743	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘𝐴) = ( bday ‘( -us ‘𝐴)))
15	9, 11, 14	3eltr4d 2852	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴))
16		bdayon 27760	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
17	6	negscld 28045	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ No )
18		oldbday 27909	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( -us ‘𝑥) ∈ No ) → (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴)))
19	16, 17, 18	sylancr 588	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴)))
20	15, 19	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)))
21		rightgt 27862	. . . . . . . . . 10 ⊢ (𝑥 ∈ ( R ‘( -us ‘𝐴)) → ( -us ‘𝐴) <s 𝑥)
22	21	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝐴) <s 𝑥)
23		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → 𝐴 ∈ No )
24	23	negscld 28045	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝐴) ∈ No )
25	24, 6	ltnegsd 28055	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → (( -us ‘𝐴) <s 𝑥 ↔ ( -us ‘𝑥) <s ( -us ‘( -us ‘𝐴))))
26	22, 25	mpbid 232	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) <s ( -us ‘( -us ‘𝐴)))
27		negnegs 28052	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
28	27	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝐴)) = 𝐴)
29	26, 28	breqtrd 5126	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) <s 𝐴)
30		elleft 27859	. . . . . . 7 ⊢ (( -us ‘𝑥) ∈ ( L ‘𝐴) ↔ (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ∧ ( -us ‘𝑥) <s 𝐴))
31	20, 29, 30	sylanbrc 584	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( L ‘𝐴))
32		negnegs 28052	. . . . . . 7 ⊢ (𝑥 ∈ No → ( -us ‘( -us ‘𝑥)) = 𝑥)
33	6, 32	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝑥)) = 𝑥)
34	1, 31, 33	rspcedvdw 3581	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( R ‘( -us ‘𝐴))) → ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥)
35	34	ex 412	. . . 4 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( R ‘( -us ‘𝐴)) → ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥))
36		leftold 27883	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ ( O ‘( bday ‘𝐴)))
37	36	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 ∈ ( O ‘( bday ‘𝐴)))
38		leftno 27885	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 ∈ No )
39	38	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 ∈ No )
40		oldbday 27909	. . . . . . . . . . 11 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝑦 ∈ No ) → (𝑦 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑦) ∈ ( bday ‘𝐴)))
41	16, 39, 40	sylancr 588	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → (𝑦 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑦) ∈ ( bday ‘𝐴)))
42	37, 41	mpbid 232	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘𝑦) ∈ ( bday ‘𝐴))
43		negbday 28065	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → ( bday ‘( -us ‘𝑦)) = ( bday ‘𝑦))
44	39, 43	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘( -us ‘𝑦)) = ( bday ‘𝑦))
45	12	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
46	42, 44, 45	3eltr4d 2852	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴)))
47	39	negscld 28045	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝑦) ∈ No )
48		oldbday 27909	. . . . . . . . 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 588	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴))))
50	46, 49	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))))
51		leftlt 27861	. . . . . . . . 9 ⊢ (𝑦 ∈ ( L ‘𝐴) → 𝑦 <s 𝐴)
52	51	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝑦 <s 𝐴)
53		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → 𝐴 ∈ No )
54	39, 53	ltnegsd 28055	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → (𝑦 <s 𝐴 ↔ ( -us ‘𝐴) <s ( -us ‘𝑦)))
55	52, 54	mpbid 232	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝐴) <s ( -us ‘𝑦))
56		elright 27860	. . . . . . 7 ⊢ (( -us ‘𝑦) ∈ ( R ‘( -us ‘𝐴)) ↔ (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ∧ ( -us ‘𝐴) <s ( -us ‘𝑦)))
57	50, 55, 56	sylanbrc 584	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → ( -us ‘𝑦) ∈ ( R ‘( -us ‘𝐴)))
58		eleq1 2825	. . . . . 6 ⊢ (( -us ‘𝑦) = 𝑥 → (( -us ‘𝑦) ∈ ( R ‘( -us ‘𝐴)) ↔ 𝑥 ∈ ( R ‘( -us ‘𝐴))))
59	57, 58	syl5ibcom 245	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( L ‘𝐴)) → (( -us ‘𝑦) = 𝑥 → 𝑥 ∈ ( R ‘( -us ‘𝐴))))
60	59	rexlimdva 3139	. . . 4 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥 → 𝑥 ∈ ( R ‘( -us ‘𝐴))))
61	35, 60	impbid 212	. . 3 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( R ‘( -us ‘𝐴)) ↔ ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥))
62		negsfn 28031	. . . 4 ⊢ -us Fn No
63		leftssno 27881	. . . 4 ⊢ ( L ‘𝐴) ⊆ No
64		fvelimab 6914	. . . 4 ⊢ (( -us Fn No ∧ ( L ‘𝐴) ⊆ No ) → (𝑥 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥))
65	62, 63, 64	mp2an 693	. . 3 ⊢ (𝑥 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑦 ∈ ( L ‘𝐴)( -us ‘𝑦) = 𝑥)
66	61, 65	bitr4di 289	. 2 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( R ‘( -us ‘𝐴)) ↔ 𝑥 ∈ ( -us “ ( L ‘𝐴))))
67	66	eqrdv 2735	1 ⊢ (𝐴 ∈ No → ( R ‘( -us ‘𝐴)) = ( -us “ ( L ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3903   class class class wbr 5100   “ cima 5635  Oncon0 6325   Fn wfn 6495  ‘cfv 6500   No csur 27619   <s clts 27620   bday cbday 27621   O cold 27831   L cleft 27833   R cright 27834   -u_s cnegs 28027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-1o 8407  df-2o 8408  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029

This theorem is referenced by: (None)