Theorem z12bday 28901

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 28900	. 2 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → (bday‘𝐴) ∈ ω)
2		0no 28225	. . . . . 6 ⊢ 0s ∈ No
3		z12no 28892	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No)
4		lestric 28155	. . . . . 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 28462	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No) → (𝐴 ≤s 0s ↔ ( -s‘ 0s ) ≤s ( -s‘𝐴)))
8	3, 2, 7	sylancl 598	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ≤s 0s ↔ ( -s‘ 0s ) ≤s ( -s‘𝐴)))
9		neg0s 28442	. . . . . . 7 ⊢ ( -s‘ 0s ) = 0s
10	9	breq1i 5113	. . . . . 6 ⊢ (( -s‘ 0s ) ≤s ( -s‘𝐴) ↔ 0s ≤s ( -s‘𝐴))
11	8, 10	bitrdi 290	. . . . 5 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ≤s 0s ↔ 0s ≤s ( -s‘𝐴)))
12		z12negscl 28894	. . . . . . 7 ⊢ (𝐴 ∈ ℤs[1/2] → ( -s‘𝐴) ∈ ℤs[1/2])
13		z12bdaylem 28900	. . . . . . . 8 ⊢ ((( -s‘𝐴) ∈ ℤs[1/2] ∧ 0s ≤s ( -s‘𝐴)) → (bday‘( -s‘𝐴)) ∈ ω)
14	13	ex 418	. . . . . . 7 ⊢ (( -s‘𝐴) ∈ ℤs[1/2] → ( 0s ≤s ( -s‘𝐴) → (bday‘( -s‘𝐴)) ∈ ω))
15	12, 14	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → ( 0s ≤s ( -s‘𝐴) → (bday‘( -s‘𝐴)) ∈ ω))
16		negbday 28473	. . . . . . . 8 ⊢ (𝐴 ∈ No → (bday‘( -s‘𝐴)) = (bday‘𝐴))
17	3, 16	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ ℤs[1/2] → (bday‘( -s‘𝐴)) = (bday‘𝐴))
18	17	eleq1d 2849	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → ((bday‘( -s‘𝐴)) ∈ ω ↔ (bday‘𝐴) ∈ ω))
19	15, 18	sylibd 242	. . . . 5 ⊢ (𝐴 ∈ ℤs[1/2] → ( 0s ≤s ( -s‘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 2147   class class class wbr 5106  ‘cfv 6558  ωcom 7901  Nocsur 28027  bdaycbday 28029   ≤s cles 28131   0_s c0s 28221   -_scnegs 28435  ℤ_s[1/2]cz12s 28830

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-dc 10551

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-subs 28438  df-muls 28523  df-divs 28604  df-ons 28668  df-seqs 28700  df-n0s 28730  df-nns 28731  df-zs 28795  df-2s 28827  df-exps 28829  df-z12s 28831

This theorem is used by:  bdayfin  28903