Theorem z12bday 28715

Description: A dyadic fraction has a finite birthday. (Contributed by Scott Fenton, 20-Aug-2025.) (Proof shortened by Scott Fenton, 22-Feb-2026.)

Assertion

Ref	Expression
z12bday	⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)

Proof of Theorem z12bday

Step	Hyp	Ref	Expression
1		z12bdaylem 28714	. 2 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)
2		0no 28039	. . . . . 6 ⊢ 0s ∈ No
3		z12no 28706	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No )
4		lestric 27969	. . . . . 6 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
5	2, 3, 4	sylancr 599	. . . . 5 ⊢ (𝐴 ∈ ℤs[1/2] → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
6	5	ord 878	. . . 4 ⊢ (𝐴 ∈ ℤs[1/2] → (¬ 0s ≤s 𝐴 → 𝐴 ≤s 0s ))
7		lenegs 28276	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
8	3, 2, 7	sylancl 598	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
9		neg0s 28256	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
10	9	breq1i 5121	. . . . . 6 ⊢ (( -us ‘ 0s ) ≤s ( -us ‘𝐴) ↔ 0s ≤s ( -us ‘𝐴))
11	8, 10	bitrdi 290	. . . . 5 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ≤s 0s ↔ 0s ≤s ( -us ‘𝐴)))
12		z12negscl 28708	. . . . . . 7 ⊢ (𝐴 ∈ ℤs[1/2] → ( -us ‘𝐴) ∈ ℤs[1/2])
13		z12bdaylem 28714	. . . . . . . 8 ⊢ ((( -us ‘𝐴) ∈ ℤs[1/2] ∧ 0s ≤s ( -us ‘𝐴)) → ( bday ‘( -us ‘𝐴)) ∈ ω)
14	13	ex 418	. . . . . . 7 ⊢ (( -us ‘𝐴) ∈ ℤs[1/2] → ( 0s ≤s ( -us ‘𝐴) → ( bday ‘( -us ‘𝐴)) ∈ ω))
15	12, 14	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → ( 0s ≤s ( -us ‘𝐴) → ( bday ‘( -us ‘𝐴)) ∈ ω))
16		negbday 28287	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
17	3, 16	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
18	17	eleq1d 2851	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → (( bday ‘( -us ‘𝐴)) ∈ ω ↔ ( bday ‘𝐴) ∈ ω))
19	15, 18	sylibd 242	. . . . 5 ⊢ (𝐴 ∈ ℤs[1/2] → ( 0s ≤s ( -us ‘𝐴) → ( bday ‘𝐴) ∈ ω))
20	11, 19	sylbid 243	. . . 4 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ≤s 0s → ( bday ‘𝐴) ∈ ω))
21	6, 20	syld 48	. . 3 ⊢ (𝐴 ∈ ℤs[1/2] → (¬ 0s ≤s 𝐴 → ( bday ‘𝐴) ∈ ω))
22	21	imp 412	. 2 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ ¬ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)
23	1, 22	pm2.61dan 825	1 ⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1570   ∈ wcel 2146   class class class wbr 5114  ‘cfv 6543  ωcom 7871   No csur 27841   bday cbday 27843   ≤s cles 27945   0_s c0s 28035   -u_s cnegs 28249  ℤ_s[1/2]cz12s 28644

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  ax-dc 10448

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-lim 6372  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-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  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-1s 28038  df-made 28057  df-old 28058  df-left 28060  df-right 28061  df-norec 28168  df-norec2 28179  df-adds 28190  df-negs 28251  df-subs 28252  df-muls 28337  df-divs 28418  df-ons 28482  df-seqs 28514  df-n0s 28544  df-nns 28545  df-zs 28609  df-2s 28641  df-exps 28643  df-z12s 28645

This theorem is used by:  bdayfin  28717