Theorem z12bday 28758

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 28757	. 2 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)
2		0no 28082	. . . . . 6 ⊢ 0s ∈ No
3		z12no 28749	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No )
4		lestric 28012	. . . . . 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 28319	. . . . . . 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 28299	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
10	9	breq1i 5114	. . . . . 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 28751	. . . . . . 7 ⊢ (𝐴 ∈ ℤs[1/2] → ( -us ‘𝐴) ∈ ℤs[1/2])
13		z12bdaylem 28757	. . . . . . . 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 28330	. . . . . . . 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 2847	. . . . . 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 2145   class class class wbr 5107  ‘cfv 6537  ωcom 7866   No csur 27884   bday cbday 27886   ≤s cles 27988   0_s c0s 28078   -u_s cnegs 28292  ℤ_s[1/2]cz12s 28687

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-dc 10452

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-nadd 8658  df-no 27887  df-lts 27888  df-bday 27889  df-les 27989  df-slts 28031  df-cuts 28033  df-0s 28080  df-1s 28081  df-made 28100  df-old 28101  df-left 28103  df-right 28104  df-norec 28211  df-norec2 28222  df-adds 28233  df-negs 28294  df-subs 28295  df-muls 28380  df-divs 28461  df-ons 28525  df-seqs 28557  df-n0s 28587  df-nns 28588  df-zs 28652  df-2s 28684  df-exps 28686  df-z12s 28688

This theorem is used by:  bdayfin  28760