Theorem z12bdaylem 28714

Description: Lemma for z12bday 28715. Handle the non-negative case. (Contributed by Scott Fenton, 22-Feb-2026.)

Assertion

Ref	Expression
z12bdaylem	⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)

Proof of Theorem z12bdaylem

Dummy variables 𝑥 𝑦 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 488	. . 3 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → 𝐴 ∈ ℤs[1/2])
2		z12no 28706	. . . 4 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No )
3		z12sge0 28713	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
4	2, 3	sylan 592	. . 3 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
5	1, 4	mpbid 235	. 2 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)))
6		simpl1 1210	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑥 ∈ ℕ0s)
7	6	n0nod 28555	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑥 ∈ No )
8		simpl2 1211	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑦 ∈ ℕ0s)
9	8	n0nod 28555	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑦 ∈ No )
10		simpl3 1212	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑝 ∈ ℕ0s)
11	9, 10	pw2divscld 28669	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (𝑦 /su (2s↑s𝑝)) ∈ No )
12		addbday 28248	. . . . . . . . . 10 ⊢ ((𝑥 ∈ No ∧ (𝑦 /su (2s↑s𝑝)) ∈ No ) → ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ⊆ (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))))
13	7, 11, 12	syl2anc 596	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ⊆ (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))))
14		n0bday 28582	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ0s → ( bday ‘𝑥) ∈ ω)
15	6, 14	syl 18	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝑥) ∈ ω)
16		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑦 <s (2s↑s𝑝))
17		bdaypw2n0bnd 28694	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝))
18	8, 10, 16, 17	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝))
19		n0bday 28582	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℕ0s → ( bday ‘𝑝) ∈ ω)
20		peano2 7895	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝑝) ∈ ω → suc ( bday ‘𝑝) ∈ ω)
21	19, 20	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℕ0s → suc ( bday ‘𝑝) ∈ ω)
22	21	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → suc ( bday ‘𝑝) ∈ ω)
23	22	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → suc ( bday ‘𝑝) ∈ ω)
24		bdayon 27982	. . . . . . . . . . . . 13 ⊢ ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ On
25	24	onordi 6481	. . . . . . . . . . . 12 ⊢ Ord ( bday ‘(𝑦 /su (2s↑s𝑝)))
26		ordom 7881	. . . . . . . . . . . 12 ⊢ Ord ω
27		ordtr2 6413	. . . . . . . . . . . 12 ⊢ ((Ord ( bday ‘(𝑦 /su (2s↑s𝑝))) ∧ Ord ω) → ((( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝) ∧ suc ( bday ‘𝑝) ∈ ω) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ ω))
28	25, 26, 27	mp2an 705	. . . . . . . . . . 11 ⊢ ((( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝) ∧ suc ( bday ‘𝑝) ∈ ω) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ ω)
29	18, 23, 28	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ ω)
30		omnaddcl 8699	. . . . . . . . . 10 ⊢ ((( bday ‘𝑥) ∈ ω ∧ ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ ω) → (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))) ∈ ω)
31	15, 29, 30	syl2anc 596	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))) ∈ ω)
32		bdayon 27982	. . . . . . . . . . 11 ⊢ ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ On
33	32	onordi 6481	. . . . . . . . . 10 ⊢ Ord ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝))))
34		ordtr2 6413	. . . . . . . . . 10 ⊢ ((Ord ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∧ Ord ω) → ((( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ⊆ (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))) ∧ (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))) ∈ ω) → ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ ω))
35	33, 26, 34	mp2an 705	. . . . . . . . 9 ⊢ ((( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ⊆ (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))) ∧ (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))) ∈ ω) → ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ ω)
36	13, 31, 35	syl2anc 596	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ ω)
37		fveq2 6888	. . . . . . . . 9 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → ( bday ‘𝐴) = ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))))
38	37	eleq1d 2851	. . . . . . . 8 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → (( bday ‘𝐴) ∈ ω ↔ ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ ω))
39	36, 38	syl5ibrcom 250	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → ( bday ‘𝐴) ∈ ω))
40	39	ex 418	. . . . . 6 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (𝑦 <s (2s↑s𝑝) → (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → ( bday ‘𝐴) ∈ ω)))
41	40	impcomd 417	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → ((𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω))
42	41	3expa 1136	. . . 4 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑝 ∈ ℕ0s) → ((𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω))
43	42	rexlimdva 3169	. . 3 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω))
44	43	rexlimivv 3210	. 2 ⊢ (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω)
45	5, 44	syl 18	1 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3092   ⊆ wss 3908   class class class wbr 5114  Ord word 6366  suc csuc 6369  ‘cfv 6543  (class class class)co 7423  ωcom 7871   +no cnadd 8660   No csur 27841   <s clts 27842   bday cbday 27843   ≤s cles 27945   0_s c0s 28035   +_s cadds 28189   /_su cdivs 28417  ℕ_0scn0s 28542  2_sc2s 28640  ↑_scexps 28642  ℤ_s[1/2]cz12s 28644

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-dc 10448

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-nadd 8661  df-no 27844  df-lts 27845  df-bday 27846  df-les 27946  df-slts 27988  df-cuts 27990  df-0s 28037  df-1s 28038  df-made 28057  df-old 28058  df-left 28060  df-right 28061  df-norec 28168  df-norec2 28179  df-adds 28190  df-negs 28251  df-subs 28252  df-muls 28337  df-divs 28418  df-ons 28482  df-seqs 28514  df-n0s 28544  df-nns 28545  df-zs 28609  df-2s 28641  df-exps 28643  df-z12s 28645

This theorem is used by:  z12bday  28715