Theorem zs12bday 28464

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
zs12bday	⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)

Proof of Theorem zs12bday

Step	Hyp	Ref	Expression
1		zs12bdaylem 28463	. 2 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)
2		0sno 27807	. . . . . 6 ⊢ 0s ∈ No
3		zs12no 28455	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No )
4		sletric 27740	. . . . . 6 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
5	2, 3, 4	sylancr 588	. . . . 5 ⊢ (𝐴 ∈ ℤs[1/2] → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
6	5	ord 865	. . . 4 ⊢ (𝐴 ∈ ℤs[1/2] → (¬ 0s ≤s 𝐴 → 𝐴 ≤s 0s ))
7		sleneg 28028	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
8	3, 2, 7	sylancl 587	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
9		negs0s 28008	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
10	9	breq1i 5106	. . . . . 6 ⊢ (( -us ‘ 0s ) ≤s ( -us ‘𝐴) ↔ 0s ≤s ( -us ‘𝐴))
11	8, 10	bitrdi 287	. . . . 5 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ≤s 0s ↔ 0s ≤s ( -us ‘𝐴)))
12		zs12negscl 28457	. . . . . . 7 ⊢ (𝐴 ∈ ℤs[1/2] → ( -us ‘𝐴) ∈ ℤs[1/2])
13		zs12bdaylem 28463	. . . . . . . 8 ⊢ ((( -us ‘𝐴) ∈ ℤs[1/2] ∧ 0s ≤s ( -us ‘𝐴)) → ( bday ‘( -us ‘𝐴)) ∈ ω)
14	13	ex 412	. . . . . . 7 ⊢ (( -us ‘𝐴) ∈ ℤs[1/2] → ( 0s ≤s ( -us ‘𝐴) → ( bday ‘( -us ‘𝐴)) ∈ ω))
15	12, 14	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → ( 0s ≤s ( -us ‘𝐴) → ( bday ‘( -us ‘𝐴)) ∈ ω))
16		negsbday 28039	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
17	3, 16	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
18	17	eleq1d 2822	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → (( bday ‘( -us ‘𝐴)) ∈ ω ↔ ( bday ‘𝐴) ∈ ω))
19	15, 18	sylibd 239	. . . . 5 ⊢ (𝐴 ∈ ℤs[1/2] → ( 0s ≤s ( -us ‘𝐴) → ( bday ‘𝐴) ∈ ω))
20	11, 19	sylbid 240	. . . 4 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ≤s 0s → ( bday ‘𝐴) ∈ ω))
21	6, 20	syld 47	. . 3 ⊢ (𝐴 ∈ ℤs[1/2] → (¬ 0s ≤s 𝐴 → ( bday ‘𝐴) ∈ ω))
22	21	imp 406	. 2 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ ¬ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)
23	1, 22	pm2.61dan 813	1 ⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 848   = wceq 1542   ∈ wcel 2114   class class class wbr 5099  ‘cfv 6493  ωcom 7810   No csur 27611   bday cbday 27613   ≤s csle 27716   0_s c0s 27803   -u_s cnegs 28001  ℤ_s[1/2]czs12 28393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-dc 10360

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-nadd 8596  df-no 27614  df-slt 27615  df-bday 27616  df-sle 27717  df-sslt 27758  df-scut 27760  df-0s 27805  df-1s 27806  df-made 27825  df-old 27826  df-left 27828  df-right 27829  df-norec 27920  df-norec2 27931  df-adds 27942  df-negs 28003  df-subs 28004  df-muls 28089  df-divs 28170  df-ons 28233  df-seqs 28265  df-n0s 28295  df-nns 28296  df-zs 28358  df-2s 28390  df-exps 28392  df-zs12 28394

This theorem is referenced by:  bdayfin  28466