Theorem negleft 28075

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 6843	. . . . . 6 ⊢ (𝑦 = ( -us ‘𝑥) → (( -us ‘𝑦) = 𝑥 ↔ ( -us ‘( -us ‘𝑥)) = 𝑥))
2		leftno 27894	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ( L ‘( -us ‘𝐴)) → 𝑥 ∈ No )
3	2	adantl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → 𝑥 ∈ No )
4		negbday 28074	. . . . . . . . . 10 ⊢ (𝑥 ∈ No → ( bday ‘( -us ‘𝑥)) = ( bday ‘𝑥))
5	3, 4	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝑥)) = ( bday ‘𝑥))
6		leftold 27892	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ( L ‘( -us ‘𝐴)) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
7	6	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
8		bdayon 27769	. . . . . . . . . . . 12 ⊢ ( bday ‘( -us ‘𝐴)) ∈ On
9		oldbday 27918	. . . . . . . . . . . 12 ⊢ ((( bday ‘( -us ‘𝐴)) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
10	8, 3, 9	sylancr 593	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
11	7, 10	mpbid 233	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴)))
12		negbday 28074	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
13	12	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
14	11, 13	eleqtrd 2842	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( bday ‘𝑥) ∈ ( bday ‘𝐴))
15	5, 14	eqeltrd 2840	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴))
16		bdayon 27769	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
17	3	negscld 28054	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ No )
18		oldbday 27918	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( -us ‘𝑥) ∈ No ) → (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴)))
19	16, 17, 18	sylancr 593	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴)))
20	15, 19	mpbird 258	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)))
21		negnegs 28061	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
22	21	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝐴)) = 𝐴)
23		leftlt 27870	. . . . . . . . . 10 ⊢ (𝑥 ∈ ( L ‘( -us ‘𝐴)) → 𝑥 <s ( -us ‘𝐴))
24	23	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → 𝑥 <s ( -us ‘𝐴))
25		negscl 28053	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
26	25	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘𝐴) ∈ No )
27	3, 26	ltnegsd 28064	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → (𝑥 <s ( -us ‘𝐴) ↔ ( -us ‘( -us ‘𝐴)) <s ( -us ‘𝑥)))
28	24, 27	mpbid 233	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝐴)) <s ( -us ‘𝑥))
29	22, 28	eqbrtrrd 5103	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → 𝐴 <s ( -us ‘𝑥))
30		elright 27869	. . . . . . 7 ⊢ (( -us ‘𝑥) ∈ ( R ‘𝐴) ↔ (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s ( -us ‘𝑥)))
31	20, 29, 30	sylanbrc 589	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( R ‘𝐴))
32		negnegs 28061	. . . . . . 7 ⊢ (𝑥 ∈ No → ( -us ‘( -us ‘𝑥)) = 𝑥)
33	3, 32	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝑥)) = 𝑥)
34	1, 31, 33	rspcedvdw 3570	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥)
35	34	ex 413	. . . 4 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘( -us ‘𝐴)) → ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥))
36		rightold 27893	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ( R ‘𝐴) → 𝑦 ∈ ( O ‘( bday ‘𝐴)))
37		rightno 27895	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ( R ‘𝐴) → 𝑦 ∈ No )
38		oldbday 27918	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝑦 ∈ No ) → (𝑦 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑦) ∈ ( bday ‘𝐴)))
39	16, 37, 38	sylancr 593	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ( R ‘𝐴) → (𝑦 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑦) ∈ ( bday ‘𝐴)))
40	36, 39	mpbid 233	. . . . . . . . . 10 ⊢ (𝑦 ∈ ( R ‘𝐴) → ( bday ‘𝑦) ∈ ( bday ‘𝐴))
41	40	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( bday ‘𝑦) ∈ ( bday ‘𝐴))
42	37	adantl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → 𝑦 ∈ No )
43		negbday 28074	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → ( bday ‘( -us ‘𝑦)) = ( bday ‘𝑦))
44	42, 43	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( bday ‘( -us ‘𝑦)) = ( bday ‘𝑦))
45	12	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
46	41, 44, 45	3eltr4d 2855	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴)))
47	42	negscld 28054	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( -us ‘𝑦) ∈ No )
48		oldbday 27918	. . . . . . . . 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 593	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴))))
50	46, 49	mpbird 258	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))))
51		rightgt 27871	. . . . . . . . 9 ⊢ (𝑦 ∈ ( R ‘𝐴) → 𝐴 <s 𝑦)
52	51	adantl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → 𝐴 <s 𝑦)
53		simpl 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → 𝐴 ∈ No )
54	53, 42	ltnegsd 28064	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → (𝐴 <s 𝑦 ↔ ( -us ‘𝑦) <s ( -us ‘𝐴)))
55	52, 54	mpbid 233	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( -us ‘𝑦) <s ( -us ‘𝐴))
56		elleft 27868	. . . . . . 7 ⊢ (( -us ‘𝑦) ∈ ( L ‘( -us ‘𝐴)) ↔ (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ∧ ( -us ‘𝑦) <s ( -us ‘𝐴)))
57	50, 55, 56	sylanbrc 589	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( -us ‘𝑦) ∈ ( L ‘( -us ‘𝐴)))
58		eleq1 2828	. . . . . 6 ⊢ (( -us ‘𝑦) = 𝑥 → (( -us ‘𝑦) ∈ ( L ‘( -us ‘𝐴)) ↔ 𝑥 ∈ ( L ‘( -us ‘𝐴))))
59	57, 58	syl5ibcom 246	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → (( -us ‘𝑦) = 𝑥 → 𝑥 ∈ ( L ‘( -us ‘𝐴))))
60	59	rexlimdva 3141	. . . 4 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥 → 𝑥 ∈ ( L ‘( -us ‘𝐴))))
61	35, 60	impbid 213	. . 3 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘( -us ‘𝐴)) ↔ ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥))
62		negsfn 28040	. . . 4 ⊢ -us Fn No
63		rightssno 27891	. . . 4 ⊢ ( R ‘𝐴) ⊆ No
64		fvelimab 6906	. . . 4 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ) → (𝑥 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥))
65	62, 63, 64	mp2an 698	. . 3 ⊢ (𝑥 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥)
66	61, 65	bitr4di 290	. 2 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘( -us ‘𝐴)) ↔ 𝑥 ∈ ( -us “ ( R ‘𝐴))))
67	66	eqrdv 2738	1 ⊢ (𝐴 ∈ No → ( L ‘( -us ‘𝐴)) = ( -us “ ( R ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3064   ⊆ wss 3890   class class class wbr 5079   “ cima 5628  Oncon0 6317   Fn wfn 6487  ‘cfv 6492   No csur 27628   <s clts 27629   bday cbday 27630   O cold 27840   L cleft 27842   R cright 27843   -u_s cnegs 28036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-1o 8402  df-2o 8403  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038

This theorem is referenced by:  zcuts0  28425