Theorem z12bday 28876

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 28875	. 2 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → (bday‘𝐴) ∈ ω)
2		0no 28200	. . . . . 6 ⊢ 0s ∈ No
3		z12no 28867	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No)
4		lestric 28130	. . . . . 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 28437	. . . . . . 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 28417	. . . . . . 7 ⊢ ( -s‘ 0s ) = 0s
10	9	breq1i 5109	. . . . . 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 28869	. . . . . . 7 ⊢ (𝐴 ∈ ℤs[1/2] → ( -s‘𝐴) ∈ ℤs[1/2])
13		z12bdaylem 28875	. . . . . . . 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 28448	. . . . . . . 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 2845	. . . . . 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 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6538  ωcom 7877  Nocsur 28002  bdaycbday 28004   ≤s cles 28106   0_s c0s 28196   -_scnegs 28410  ℤ_s[1/2]cz12s 28805

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  ax-dc 10527

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-lim 6367  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-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-oadd 8483  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-1s 28199  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-norec2 28340  df-adds 28351  df-negs 28412  df-subs 28413  df-muls 28498  df-divs 28579  df-ons 28643  df-seqs 28675  df-n0s 28705  df-nns 28706  df-zs 28770  df-2s 28802  df-exps 28804  df-z12s 28806

This theorem is used by:  bdayfin  28878