Theorem negleft 28449

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‘( -s‘𝐴)) = ( -s “ (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 6894	. . . . . 6 ⊢ (𝑦 = ( -s‘𝑥) → (( -s‘𝑦) = 𝑥 ↔ ( -s‘( -s‘𝑥)) = 𝑥))
2		leftno 28268	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (L‘( -s‘𝐴)) → 𝑥 ∈ No)
3	2	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → 𝑥 ∈ No)
4		negbday 28448	. . . . . . . . . 10 ⊢ (𝑥 ∈ No → (bday‘( -s‘𝑥)) = (bday‘𝑥))
5	3, 4	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (bday‘( -s‘𝑥)) = (bday‘𝑥))
6		leftold 28266	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (L‘( -s‘𝐴)) → 𝑥 ∈ (O‘(bday‘( -s‘𝐴))))
7	6	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → 𝑥 ∈ (O‘(bday‘( -s‘𝐴))))
8		bdayon 28143	. . . . . . . . . . . 12 ⊢ (bday‘( -s‘𝐴)) ∈ On
9		oldbday 28292	. . . . . . . . . . . 12 ⊢ (((bday‘( -s‘𝐴)) ∈ On ∧ 𝑥 ∈ No) → (𝑥 ∈ (O‘(bday‘( -s‘𝐴))) ↔ (bday‘𝑥) ∈ (bday‘( -s‘𝐴))))
10	8, 3, 9	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (𝑥 ∈ (O‘(bday‘( -s‘𝐴))) ↔ (bday‘𝑥) ∈ (bday‘( -s‘𝐴))))
11	7, 10	mpbid 235	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (bday‘𝑥) ∈ (bday‘( -s‘𝐴)))
12		negbday 28448	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → (bday‘( -s‘𝐴)) = (bday‘𝐴))
13	12	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (bday‘( -s‘𝐴)) = (bday‘𝐴))
14	11, 13	eleqtrd 2862	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (bday‘𝑥) ∈ (bday‘𝐴))
15	5, 14	eqeltrd 2860	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (bday‘( -s‘𝑥)) ∈ (bday‘𝐴))
16		bdayon 28143	. . . . . . . . 9 ⊢ (bday‘𝐴) ∈ On
17	3	negscld 28428	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘𝑥) ∈ No)
18		oldbday 28292	. . . . . . . . 9 ⊢ (((bday‘𝐴) ∈ On ∧ ( -s‘𝑥) ∈ No) → (( -s‘𝑥) ∈ (O‘(bday‘𝐴)) ↔ (bday‘( -s‘𝑥)) ∈ (bday‘𝐴)))
19	16, 17, 18	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (( -s‘𝑥) ∈ (O‘(bday‘𝐴)) ↔ (bday‘( -s‘𝑥)) ∈ (bday‘𝐴)))
20	15, 19	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘𝑥) ∈ (O‘(bday‘𝐴)))
21		negnegs 28435	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -s‘( -s‘𝐴)) = 𝐴)
22	21	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘( -s‘𝐴)) = 𝐴)
23		leftlt 28244	. . . . . . . . . 10 ⊢ (𝑥 ∈ (L‘( -s‘𝐴)) → 𝑥 <s ( -s‘𝐴))
24	23	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → 𝑥 <s ( -s‘𝐴))
25		negscl 28427	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( -s‘𝐴) ∈ No)
26	25	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘𝐴) ∈ No)
27	3, 26	ltnegsd 28438	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → (𝑥 <s ( -s‘𝐴) ↔ ( -s‘( -s‘𝐴)) <s ( -s‘𝑥)))
28	24, 27	mpbid 235	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘( -s‘𝐴)) <s ( -s‘𝑥))
29	22, 28	eqbrtrrd 5128	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → 𝐴 <s ( -s‘𝑥))
30		elright 28243	. . . . . . 7 ⊢ (( -s‘𝑥) ∈ (R‘𝐴) ↔ (( -s‘𝑥) ∈ (O‘(bday‘𝐴)) ∧ 𝐴 <s ( -s‘𝑥)))
31	20, 29, 30	sylanbrc 595	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘𝑥) ∈ (R‘𝐴))
32		negnegs 28435	. . . . . . 7 ⊢ (𝑥 ∈ No → ( -s‘( -s‘𝑥)) = 𝑥)
33	3, 32	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ( -s‘( -s‘𝑥)) = 𝑥)
34	1, 31, 33	rspcedvdw 3579	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ (L‘( -s‘𝐴))) → ∃𝑦 ∈ (R‘𝐴)( -s‘𝑦) = 𝑥)
35	34	ex 418	. . . 4 ⊢ (𝐴 ∈ No → (𝑥 ∈ (L‘( -s‘𝐴)) → ∃𝑦 ∈ (R‘𝐴)( -s‘𝑦) = 𝑥))
36		rightold 28267	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (R‘𝐴) → 𝑦 ∈ (O‘(bday‘𝐴)))
37		rightno 28269	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (R‘𝐴) → 𝑦 ∈ No)
38		oldbday 28292	. . . . . . . . . . . 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 28448	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → (bday‘( -s‘𝑦)) = (bday‘𝑦))
44	42, 43	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (bday‘( -s‘𝑦)) = (bday‘𝑦))
45	12	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (bday‘( -s‘𝐴)) = (bday‘𝐴))
46	41, 44, 45	3eltr4d 2875	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (bday‘( -s‘𝑦)) ∈ (bday‘( -s‘𝐴)))
47	42	negscld 28428	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → ( -s‘𝑦) ∈ No)
48		oldbday 28292	. . . . . . . . 9 ⊢ (((bday‘( -s‘𝐴)) ∈ On ∧ ( -s‘𝑦) ∈ No) → (( -s‘𝑦) ∈ (O‘(bday‘( -s‘𝐴))) ↔ (bday‘( -s‘𝑦)) ∈ (bday‘( -s‘𝐴))))
49	8, 47, 48	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (( -s‘𝑦) ∈ (O‘(bday‘( -s‘𝐴))) ↔ (bday‘( -s‘𝑦)) ∈ (bday‘( -s‘𝐴))))
50	46, 49	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → ( -s‘𝑦) ∈ (O‘(bday‘( -s‘𝐴))))
51		rightgt 28245	. . . . . . . . 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 28438	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (𝐴 <s 𝑦 ↔ ( -s‘𝑦) <s ( -s‘𝐴)))
55	52, 54	mpbid 235	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → ( -s‘𝑦) <s ( -s‘𝐴))
56		elleft 28242	. . . . . . 7 ⊢ (( -s‘𝑦) ∈ (L‘( -s‘𝐴)) ↔ (( -s‘𝑦) ∈ (O‘(bday‘( -s‘𝐴))) ∧ ( -s‘𝑦) <s ( -s‘𝐴)))
57	50, 55, 56	sylanbrc 595	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → ( -s‘𝑦) ∈ (L‘( -s‘𝐴)))
58		eleq1 2848	. . . . . 6 ⊢ (( -s‘𝑦) = 𝑥 → (( -s‘𝑦) ∈ (L‘( -s‘𝐴)) ↔ 𝑥 ∈ (L‘( -s‘𝐴))))
59	57, 58	syl5ibcom 248	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ (R‘𝐴)) → (( -s‘𝑦) = 𝑥 → 𝑥 ∈ (L‘( -s‘𝐴))))
60	59	rexlimdva 3163	. . . 4 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ (R‘𝐴)( -s‘𝑦) = 𝑥 → 𝑥 ∈ (L‘( -s‘𝐴))))
61	35, 60	impbid 215	. . 3 ⊢ (𝐴 ∈ No → (𝑥 ∈ (L‘( -s‘𝐴)) ↔ ∃𝑦 ∈ (R‘𝐴)( -s‘𝑦) = 𝑥))
62		negsfn 28414	. . . 4 ⊢ -s Fn No
63		rightssno 28265	. . . 4 ⊢ (R‘𝐴) ⊆ No
64		fvelimab 6957	. . . 4 ⊢ (( -s Fn No ∧ (R‘𝐴) ⊆ No) → (𝑥 ∈ ( -s “ (R‘𝐴)) ↔ ∃𝑦 ∈ (R‘𝐴)( -s‘𝑦) = 𝑥))
65	62, 63, 64	mp2an 705	. . 3 ⊢ (𝑥 ∈ ( -s “ (R‘𝐴)) ↔ ∃𝑦 ∈ (R‘𝐴)( -s‘𝑦) = 𝑥)
66	61, 65	bitr4di 292	. 2 ⊢ (𝐴 ∈ No → (𝑥 ∈ (L‘( -s‘𝐴)) ↔ 𝑥 ∈ ( -s “ (R‘𝐴))))
67	66	eqrdv 2758	1 ⊢ (𝐴 ∈ No → (L‘( -s‘𝐴)) = ( -s “ (R‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895   class class class wbr 5102   “ cima 5653  Oncon0 6362   Fn wfn 6533  ‘cfv 6538  Nocsur 28002   <s clts 28003  bdaycbday 28004  Ocold 28214  Lcleft 28216  Rcright 28217   -_scnegs 28410

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-norec2 28340  df-adds 28351  df-negs 28412

This theorem is used by:  zcuts0  28799