Theorem z12bday 28636

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 28635	. 2 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)
2		0no 27960	. . . . . 6 ⊢ 0s ∈ No
3		z12no 28627	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No )
4		lestric 27890	. . . . . 6 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
5	2, 3, 4	sylancr 598	. . . . 5 ⊢ (𝐴 ∈ ℤs[1/2] → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
6	5	ord 877	. . . 4 ⊢ (𝐴 ∈ ℤs[1/2] → (¬ 0s ≤s 𝐴 → 𝐴 ≤s 0s ))
7		lenegs 28197	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
8	3, 2, 7	sylancl 597	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
9		neg0s 28177	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
10	9	breq1i 5112	. . . . . 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 28629	. . . . . . 7 ⊢ (𝐴 ∈ ℤs[1/2] → ( -us ‘𝐴) ∈ ℤs[1/2])
13		z12bdaylem 28635	. . . . . . . 8 ⊢ ((( -us ‘𝐴) ∈ ℤs[1/2] ∧ 0s ≤s ( -us ‘𝐴)) → ( bday ‘( -us ‘𝐴)) ∈ ω)
14	13	ex 417	. . . . . . 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 28208	. . . . . . . 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 2850	. . . . . 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 411	. 2 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ ¬ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)
23	1, 22	pm2.61dan 824	1 ⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1563   ∈ wcel 2145   class class class wbr 5105  ‘cfv 6525  ωcom 7850   No csur 27762   bday cbday 27764   ≤s cles 27866   0_s c0s 27956   -u_s cnegs 28170  ℤ_s[1/2]cz12s 28565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-dc 10418

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-oadd 8445  df-nadd 8640  df-no 27765  df-lts 27766  df-bday 27767  df-les 27867  df-slts 27909  df-cuts 27911  df-0s 27958  df-1s 27959  df-made 27978  df-old 27979  df-left 27981  df-right 27982  df-norec 28089  df-norec2 28100  df-adds 28111  df-negs 28172  df-subs 28173  df-muls 28258  df-divs 28339  df-ons 28403  df-seqs 28435  df-n0s 28465  df-nns 28466  df-zs 28530  df-2s 28562  df-exps 28564  df-z12s 28566

This theorem is referenced by:  bdayfin  28638