Theorem negleft 28128

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 6872	. . . . . 6 ⊢ (𝑦 = ( -us ‘𝑥) → (( -us ‘𝑦) = 𝑥 ↔ ( -us ‘( -us ‘𝑥)) = 𝑥))
2		leftno 27947	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ( L ‘( -us ‘𝐴)) → 𝑥 ∈ No )
3	2	adantl 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → 𝑥 ∈ No )
4		negbday 28127	. . . . . . . . . 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 27945	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ( L ‘( -us ‘𝐴)) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
7	6	adantl 485	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → 𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))))
8		bdayon 27822	. . . . . . . . . . . 12 ⊢ ( bday ‘( -us ‘𝐴)) ∈ On
9		oldbday 27971	. . . . . . . . . . . 12 ⊢ ((( bday ‘( -us ‘𝐴)) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
10	8, 3, 9	sylancr 596	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → (𝑥 ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴))))
11	7, 10	mpbid 234	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( bday ‘𝑥) ∈ ( bday ‘( -us ‘𝐴)))
12		negbday 28127	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
13	12	adantr 484	. . . . . . . . . 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 27822	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
17	3	negscld 28107	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ No )
18		oldbday 27971	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( -us ‘𝑥) ∈ No ) → (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴)))
19	16, 17, 18	sylancr 596	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘( -us ‘𝑥)) ∈ ( bday ‘𝐴)))
20	15, 19	mpbird 259	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)))
21		negnegs 28114	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
22	21	adantr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝐴)) = 𝐴)
23		leftlt 27923	. . . . . . . . . 10 ⊢ (𝑥 ∈ ( L ‘( -us ‘𝐴)) → 𝑥 <s ( -us ‘𝐴))
24	23	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → 𝑥 <s ( -us ‘𝐴))
25		negscl 28106	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
26	25	adantr 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘𝐴) ∈ No )
27	3, 26	ltnegsd 28117	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → (𝑥 <s ( -us ‘𝐴) ↔ ( -us ‘( -us ‘𝐴)) <s ( -us ‘𝑥)))
28	24, 27	mpbid 234	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘( -us ‘𝐴)) <s ( -us ‘𝑥))
29	22, 28	eqbrtrrd 5123	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → 𝐴 <s ( -us ‘𝑥))
30		elright 27922	. . . . . . 7 ⊢ (( -us ‘𝑥) ∈ ( R ‘𝐴) ↔ (( -us ‘𝑥) ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s ( -us ‘𝑥)))
31	20, 29, 30	sylanbrc 592	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ( -us ‘𝑥) ∈ ( R ‘𝐴))
32		negnegs 28114	. . . . . . 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 3584	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ ( L ‘( -us ‘𝐴))) → ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥)
35	34	ex 416	. . . 4 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘( -us ‘𝐴)) → ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥))
36		rightold 27946	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ( R ‘𝐴) → 𝑦 ∈ ( O ‘( bday ‘𝐴)))
37		rightno 27948	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ( R ‘𝐴) → 𝑦 ∈ No )
38		oldbday 27971	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝑦 ∈ No ) → (𝑦 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑦) ∈ ( bday ‘𝐴)))
39	16, 37, 38	sylancr 596	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ( R ‘𝐴) → (𝑦 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝑦) ∈ ( bday ‘𝐴)))
40	36, 39	mpbid 234	. . . . . . . . . 10 ⊢ (𝑦 ∈ ( R ‘𝐴) → ( bday ‘𝑦) ∈ ( bday ‘𝐴))
41	40	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( bday ‘𝑦) ∈ ( bday ‘𝐴))
42	37	adantl 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → 𝑦 ∈ No )
43		negbday 28127	. . . . . . . . . 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 484	. . . . . . . . 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 28107	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( -us ‘𝑦) ∈ No )
48		oldbday 27971	. . . . . . . . 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 596	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ↔ ( bday ‘( -us ‘𝑦)) ∈ ( bday ‘( -us ‘𝐴))))
50	46, 49	mpbird 259	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))))
51		rightgt 27924	. . . . . . . . 9 ⊢ (𝑦 ∈ ( R ‘𝐴) → 𝐴 <s 𝑦)
52	51	adantl 485	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → 𝐴 <s 𝑦)
53		simpl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → 𝐴 ∈ No )
54	53, 42	ltnegsd 28117	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → (𝐴 <s 𝑦 ↔ ( -us ‘𝑦) <s ( -us ‘𝐴)))
55	52, 54	mpbid 234	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → ( -us ‘𝑦) <s ( -us ‘𝐴))
56		elleft 27921	. . . . . . 7 ⊢ (( -us ‘𝑦) ∈ ( L ‘( -us ‘𝐴)) ↔ (( -us ‘𝑦) ∈ ( O ‘( bday ‘( -us ‘𝐴))) ∧ ( -us ‘𝑦) <s ( -us ‘𝐴)))
57	50, 55, 56	sylanbrc 592	. . . . . 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 247	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ ( R ‘𝐴)) → (( -us ‘𝑦) = 𝑥 → 𝑥 ∈ ( L ‘( -us ‘𝐴))))
60	59	rexlimdva 3162	. . . 4 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥 → 𝑥 ∈ ( L ‘( -us ‘𝐴))))
61	35, 60	impbid 214	. . 3 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘( -us ‘𝐴)) ↔ ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥))
62		negsfn 28093	. . . 4 ⊢ -us Fn No
63		rightssno 27944	. . . 4 ⊢ ( R ‘𝐴) ⊆ No
64		fvelimab 6935	. . . 4 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ) → (𝑥 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥))
65	62, 63, 64	mp2an 702	. . 3 ⊢ (𝑥 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑦 ∈ ( R ‘𝐴)( -us ‘𝑦) = 𝑥)
66	61, 65	bitr4di 291	. 2 ⊢ (𝐴 ∈ No → (𝑥 ∈ ( L ‘( -us ‘𝐴)) ↔ 𝑥 ∈ ( -us “ ( R ‘𝐴))))
67	66	eqrdv 2759	1 ⊢ (𝐴 ∈ No → ( L ‘( -us ‘𝐴)) = ( -us “ ( R ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085   ⊆ wss 3904   class class class wbr 5099   “ cima 5648  Oncon0 6342   Fn wfn 6512  ‘cfv 6517   No csur 27681   <s clts 27682   bday cbday 27683   O cold 27893   L cleft 27895   R cright 27896   -u_s cnegs 28089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-1o 8432  df-2o 8433  df-nadd 8631  df-no 27684  df-lts 27685  df-bday 27686  df-les 27786  df-slts 27828  df-cuts 27830  df-0s 27877  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec 28008  df-norec2 28019  df-adds 28030  df-negs 28091

This theorem is referenced by:  zcuts0  28478