Theorem negbdaylem 28286

Description: Lemma for negbday 28287. Bound the birthday of the negative of a surreal number above. (Contributed by Scott Fenton, 8-Mar-2025.)

Assertion

Ref	Expression
negbdaylem	⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) ⊆ ( bday ‘𝐴))

Proof of Theorem negbdaylem

Dummy variables 𝑥 𝑥_𝑂 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6893	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘( -us ‘𝑥)) = ( bday ‘( -us ‘𝑥𝑂)))
2		fveq2 6888	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘𝑥) = ( bday ‘𝑥𝑂))
3	1, 2	sseq12d 3973	. 2 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥) ↔ ( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)))
4		2fveq3 6893	. . 3 ⊢ (𝑥 = 𝐴 → ( bday ‘( -us ‘𝑥)) = ( bday ‘( -us ‘𝐴)))
5		fveq2 6888	. . 3 ⊢ (𝑥 = 𝐴 → ( bday ‘𝑥) = ( bday ‘𝐴))
6	4, 5	sseq12d 3973	. 2 ⊢ (𝑥 = 𝐴 → (( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥) ↔ ( bday ‘( -us ‘𝐴)) ⊆ ( bday ‘𝐴)))
7		negsval 28255	. . . . . 6 ⊢ (𝑥 ∈ No → ( -us ‘𝑥) = (( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥))))
8	7	fveq2d 6892	. . . . 5 ⊢ (𝑥 ∈ No → ( bday ‘( -us ‘𝑥)) = ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))))
9	8	adantr 486	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday ‘( -us ‘𝑥)) = ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))))
10		negcut2 28270	. . . . 5 ⊢ (𝑥 ∈ No → ( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)))
11		lrold 28127	. . . . . . . . . 10 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday ‘𝑥))
12		uncom 4115	. . . . . . . . . 10 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
13	11, 12	eqtr3i 2791	. . . . . . . . 9 ⊢ ( O ‘( bday ‘𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
14	13	imaeq2i 6065	. . . . . . . 8 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) = ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥)))
15		imaundi 6152	. . . . . . . 8 ⊢ ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
16	14, 15	eqtri 2789	. . . . . . 7 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
17	16	imaeq2i 6065	. . . . . 6 ⊢ ( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) = ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥))))
18	11	raleqi 3324	. . . . . . 7 ⊢ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂))
19		oldbdayim 28119	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 ∈ ( O ‘( bday ‘𝑥)) → ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥))
20	19	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))) → ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥))
21		bdayon 27982	. . . . . . . . . . . . 13 ⊢ ( bday ‘( -us ‘𝑥𝑂)) ∈ On
22		bdayon 27982	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑥) ∈ On
23		ontr2 6416	. . . . . . . . . . . . 13 ⊢ ((( bday ‘( -us ‘𝑥𝑂)) ∈ On ∧ ( bday ‘𝑥) ∈ On) → ((( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) ∧ ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥)) → ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
24	21, 22, 23	mp2an 705	. . . . . . . . . . . 12 ⊢ ((( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) ∧ ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥)) → ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
25	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))) → ((( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) ∧ ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥)) → ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
26	20, 25	mpan2d 707	. . . . . . . . . 10 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))) → (( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) → ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
27	26	ralimdva 3180	. . . . . . . . 9 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) → ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
28	27	imp 412	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
29		bdayfun 27977	. . . . . . . . . 10 ⊢ Fun bday
30		imassrn 6078	. . . . . . . . . . 11 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) ⊆ ran -us
31		bdaydm 27979	. . . . . . . . . . . 12 ⊢ dom bday = No
32		negsfo 28283	. . . . . . . . . . . . 13 ⊢ -us : No –onto→ No
33		forn 6802	. . . . . . . . . . . . 13 ⊢ ( -us : No –onto→ No → ran -us = No )
34	32, 33	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran -us = No
35	31, 34	eqtr4i 2792	. . . . . . . . . . 11 ⊢ dom bday = ran -us
36	30, 35	sseqtrri 3989	. . . . . . . . . 10 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) ⊆ dom bday
37		funimass4 6952	. . . . . . . . . 10 ⊢ ((Fun bday ∧ ( -us “ ( O ‘( bday ‘𝑥))) ⊆ dom bday ) → (( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥) ↔ ∀𝑦 ∈ ( -us “ ( O ‘( bday ‘𝑥)))( bday ‘𝑦) ∈ ( bday ‘𝑥)))
38	29, 36, 37	mp2an 705	. . . . . . . . 9 ⊢ (( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥) ↔ ∀𝑦 ∈ ( -us “ ( O ‘( bday ‘𝑥)))( bday ‘𝑦) ∈ ( bday ‘𝑥))
39		negsfn 28253	. . . . . . . . . 10 ⊢ -us Fn No
40		oldssno 28071	. . . . . . . . . 10 ⊢ ( O ‘( bday ‘𝑥)) ⊆ No
41		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑦 = ( -us ‘𝑥𝑂) → ( bday ‘𝑦) = ( bday ‘( -us ‘𝑥𝑂)))
42	41	eleq1d 2851	. . . . . . . . . . 11 ⊢ (𝑦 = ( -us ‘𝑥𝑂) → (( bday ‘𝑦) ∈ ( bday ‘𝑥) ↔ ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
43	42	ralima 7242	. . . . . . . . . 10 ⊢ (( -us Fn No ∧ ( O ‘( bday ‘𝑥)) ⊆ No ) → (∀𝑦 ∈ ( -us “ ( O ‘( bday ‘𝑥)))( bday ‘𝑦) ∈ ( bday ‘𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
44	39, 40, 43	mp2an 705	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ( -us “ ( O ‘( bday ‘𝑥)))( bday ‘𝑦) ∈ ( bday ‘𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
45	38, 44	bitri 278	. . . . . . . 8 ⊢ (( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
46	28, 45	sylibr 237	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥))
47	18, 46	sylan2b 606	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥))
48	17, 47	eqsstrrid 3979	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥))
49		cutbdaybnd 28025	. . . . . 6 ⊢ ((( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)) ∧ ( bday ‘𝑥) ∈ On ∧ ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥)) → ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥))
50	22, 49	mp3an2 1478	. . . . 5 ⊢ ((( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)) ∧ ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥)) → ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥))
51	10, 48, 50	syl2an2r 698	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥))
52	9, 51	eqsstrd 3974	. . 3 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥))
53	52	ex 418	. 2 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) → ( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥)))
54	3, 6, 53	noinds 28175	1 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) ⊆ ( bday ‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082   ∪ cun 3906   ⊆ wss 3908   class class class wbr 5114  dom cdm 5666  ran crn 5667   “ cima 5669  Oncon0 6367  Fun wfun 6537   Fn wfn 6538  –onto→wfo 6541  ‘cfv 6543  (class class class)co 7423   No csur 27841   bday cbday 27843   <<s cslts 27987   |s ccuts 27989   O cold 28053   L cleft 28055   R cright 28056   -u_s cnegs 28249

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-1o 8462  df-2o 8463  df-nadd 8661  df-no 27844  df-lts 27845  df-bday 27846  df-les 27946  df-slts 27988  df-cuts 27990  df-0s 28037  df-made 28057  df-old 28058  df-left 28060  df-right 28061  df-norec 28168  df-norec2 28179  df-adds 28190  df-negs 28251

This theorem is used by:  negbday  28287