Theorem negleft 28426

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 ‘( -us ‘𝐴)) = ( -us “ ( 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 6886	. . . . . 6 ⊢ (𝑦 = ( -us ‘𝑥) → (( -us ‘𝑦) = 𝑥 ↔ ( -us ‘( -us ‘𝑥)) = 𝑥))
2		leftno 28245	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ( L ‘( -us ‘𝐴)) → 𝑥 ∈ No )
3	2	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → 𝑥 ∈ No )
4		negbday 28425	. . . . . . . . . 10 ⊢ (𝑥 ∈ No → ( bday ‘( -us ‘𝑥)) = ( bday ‘𝑥))
5	3, 4	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝑥)) = ( bday ‘𝑥))
6		leftold 28243	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ( L ‘( -us ‘𝐴)) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
7	6	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
8		bdayon 28120	. . . . . . . . . . . 12 ⊢ ( bday ‘( -us ‘𝐴)) ∈ On
9		oldbday 28269	. . . . . . . . . . . 12 ⊢ ((( bday ‘( -us ‘𝐴)) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
10	8, 3, 9	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
11	7, 10	mpbid 235	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴)))
12		negbday 28425	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
13	12	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
14	11, 13	eleqtrd 2863	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( bday ‘𝑥) ∈ ( bday ‘𝐴))
15	5, 14	eqeltrd 2861	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴))
16		bdayon 28120	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
17	3	negscld 28405	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ No )
18		oldbday 28269	. . . . . . . . 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 ∈ ( L ‘( -us ‘𝐴))) → (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴)))
20	15, 19	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)))
21		negnegs 28412	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
22	21	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝐴)) = 𝐴)
23		leftlt 28221	. . . . . . . . . 10 ⊢ (𝑥 ∈ ( L ‘( -us ‘𝐴)) → 𝑥 <s ( -us ‘𝐴))
24	23	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → 𝑥 <s ( -us ‘𝐴))
25		negscl 28404	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
26	25	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘𝐴) ∈ No )
27	3, 26	ltnegsd 28415	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → (𝑥 <s ( -us ‘𝐴) ↔ ( -us ‘( -us ‘𝐴)) <s ( -us ‘𝑥)))
28	24, 27	mpbid 235	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝐴)) <s ( -us ‘𝑥))
29	22, 28	eqbrtrrd 5129	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → 𝐴 <s ( -us ‘𝑥))
30		elright 28220	. . . . . . 7 ⊢ (( -us ‘𝑥) ∈ ( R ‘𝐴) ↔ (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s ( -us ‘𝑥)))
31	20, 29, 30	sylanbrc 595	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( R ‘𝐴))
32		negnegs 28412	. . . . . . 7 ⊢ (𝑥 ∈ No → ( -us ‘( -us ‘𝑥)) = 𝑥)
33	3, 32	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝑥)) = 𝑥)
34	1, 31, 33	rspcedvdw 3580	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥)
35	34	ex 418	. . . 4 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘( -us ‘𝐴)) → ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥))
36		rightold 28244	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ( R ‘𝐴) → 𝑦 ∈ ( O ‘( bday ‘𝐴)))
37		rightno 28246	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ( R ‘𝐴) → 𝑦 ∈ No )
38		oldbday 28269	. . . . . . . . . . . 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 28425	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → ( bday ‘( -us ‘𝑦)) = ( bday ‘𝑦))
44	42, 43	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( bday ‘( -us ‘𝑦)) = ( bday ‘𝑦))
45	12	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
46	41, 44, 45	3eltr4d 2876	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴)))
47	42	negscld 28405	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( -us ‘𝑦) ∈ No )
48		oldbday 28269	. . . . . . . . 9 ⊢ ((( bday ‘( -us ‘𝐴)) ∈ On ∧ ( -us ‘𝑦) ∈ No ) → (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴))))
49	8, 47, 48	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴))))
50	46, 49	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))))
51		rightgt 28222	. . . . . . . . 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 28415	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → (𝐴 <s 𝑦 ↔ ( -us ‘𝑦) <s ( -us ‘𝐴)))
55	52, 54	mpbid 235	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( -us ‘𝑦) <s ( -us ‘𝐴))
56		elleft 28219	. . . . . . 7 ⊢ (( -us ‘𝑦) ∈ ( L ‘( -us ‘𝐴)) ↔ (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ∧ ( -us ‘𝑦) <s ( -us ‘𝐴)))
57	50, 55, 56	sylanbrc 595	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( -us ‘𝑦) ∈ ( L ‘( -us ‘𝐴)))
58		eleq1 2849	. . . . . 6 ⊢ (( -us ‘𝑦) = 𝑥 → (( -us ‘𝑦) ∈ ( L ‘( -us ‘𝐴)) ↔ 𝑥 ∈ ( L ‘( -us ‘𝐴))))
59	57, 58	syl5ibcom 248	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → (( -us ‘𝑦) = 𝑥 → 𝑥 ∈ ( L ‘( -us ‘𝐴))))
60	59	rexlimdva 3164	. . . 4 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥 → 𝑥 ∈ ( L ‘( -us ‘𝐴))))
61	35, 60	impbid 215	. . 3 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘( -us ‘𝐴)) ↔ ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥))
62		negsfn 28391	. . . 4 ⊢ -us Fn No
63		rightssno 28242	. . . 4 ⊢ ( R ‘𝐴) ⊆ No
64		fvelimab 6949	. . . 4 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ) → (𝑥 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥))
65	62, 63, 64	mp2an 705	. . 3 ⊢ (𝑥 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥)
66	61, 65	bitr4di 292	. 2 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘( -us ‘𝐴)) ↔ 𝑥 ∈ ( -us “ ( R ‘𝐴))))
67	66	eqrdv 2759	1 ⊢ (𝐴 ∈ No → ( L ‘( -us ‘𝐴)) = ( -us “ ( R ‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087   ⊆ wss 3899   class class class wbr 5103   “ cima 5654  Oncon0 6355   Fn wfn 6526  ‘cfv 6531   No csur 27979   <s clts 27980   bday cbday 27981   O cold 28191   L cleft 28193   R cright 28194   -u_s cnegs 28387

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8460  df-2o 8461  df-nadd 8659  df-no 27982  df-lts 27983  df-bday 27984  df-les 28084  df-slts 28126  df-cuts 28128  df-0s 28175  df-made 28195  df-old 28196  df-left 28198  df-right 28199  df-norec 28306  df-norec2 28317  df-adds 28328  df-negs 28389

This theorem is used by:  zcuts0  28776