Theorem z12bday 28481

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 28480	. 2 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)
2		0no 27805	. . . . . 6 ⊢ 0s ∈ No
3		z12no 28472	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No )
4		lestric 27736	. . . . . 6 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
5	2, 3, 4	sylancr 587	. . . . 5 ⊢ (𝐴 ∈ ℤs[1/2] → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
6	5	ord 864	. . . 4 ⊢ (𝐴 ∈ ℤs[1/2] → (¬ 0s ≤s 𝐴 → 𝐴 ≤s 0s ))
7		lenegs 28042	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
8	3, 2, 7	sylancl 586	. . . . . 6 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
9		neg0s 28022	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
10	9	breq1i 5105	. . . . . 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		z12negscl 28474	. . . . . . 7 ⊢ (𝐴 ∈ ℤs[1/2] → ( -us ‘𝐴) ∈ ℤs[1/2])
13		z12bdaylem 28480	. . . . . . . 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		negbday 28053	. . . . . . . 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 2821	. . . . . 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 812	1 ⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1541   ∈ wcel 2113   class class class wbr 5098  ‘cfv 6492  ωcom 7808   No csur 27607   bday cbday 27609   ≤s cles 27712   0_s c0s 27801   -u_s cnegs 28015  ℤ_s[1/2]cz12s 28410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-dc 10356

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-nadd 8594  df-no 27610  df-lts 27611  df-bday 27612  df-les 27713  df-slts 27754  df-cuts 27756  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27934  df-norec2 27945  df-adds 27956  df-negs 28017  df-subs 28018  df-muls 28103  df-divs 28184  df-ons 28248  df-seqs 28280  df-n0s 28310  df-nns 28311  df-zs 28375  df-2s 28407  df-exps 28409  df-z12s 28411

This theorem is referenced by:  bdayfin  28483