Theorem z12bdaylem 28852

Description: Lemma for z12bday 28853. 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 28844	. . . 4 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No )
3		z12sge0 28851	. . . 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 28693	. . . . . . . . . 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 28693	. . . . . . . . . . 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 28807	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (𝑦 /su (2s↑s𝑝)) ∈ No )
12		addbday 28386	. . . . . . . . . 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 28720	. . . . . . . . . . 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 28832	. . . . . . . . . . . 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 28720	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℕ0s → ( bday ‘𝑝) ∈ ω)
20		peano2 7890	. . . . . . . . . . . . . 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 28120	. . . . . . . . . . . . 13 ⊢ ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ On
25	24	onordi 6469	. . . . . . . . . . . 12 ⊢ Ord ( bday ‘(𝑦 /su (2s↑s𝑝)))
26		ordom 7876	. . . . . . . . . . . 12 ⊢ Ord ω
27		ordtr2 6401	. . . . . . . . . . . 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 8697	. . . . . . . . . 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 28120	. . . . . . . . . . 11 ⊢ ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ On
33	32	onordi 6469	. . . . . . . . . 10 ⊢ Ord ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝))))
34		ordtr2 6401	. . . . . . . . . 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 6877	. . . . . . . . 9 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → ( bday ‘𝐴) = ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))))
38	37	eleq1d 2846	. . . . . . . 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 3164	. . 3 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω))
44	43	rexlimivv 3205	. 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 2145  ∃wrex 3087   ⊆ wss 3899   class class class wbr 5103  Ord word 6354  suc csuc 6357  ‘cfv 6531  (class class class)co 7412  ωcom 7866   +no cnadd 8658   No csur 27979   <s clts 27980   bday cbday 27981   ≤s cles 28083   0_s c0s 28173   +_s cadds 28327   /_su cdivs 28555  ℕ_0scn0s 28680  2_sc2s 28778  ↑_scexps 28780  ℤ_s[1/2]cz12s 28782

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-dc 10505

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-oadd 8464  df-nadd 8659  df-no 27982  df-lts 27983  df-bday 27984  df-les 28084  df-slts 28126  df-cuts 28128  df-0s 28175  df-1s 28176  df-made 28195  df-old 28196  df-left 28198  df-right 28199  df-norec 28306  df-norec2 28317  df-adds 28328  df-negs 28389  df-subs 28390  df-muls 28475  df-divs 28556  df-ons 28620  df-seqs 28652  df-n0s 28682  df-nns 28683  df-zs 28747  df-2s 28779  df-exps 28781  df-z12s 28783

This theorem is used by:  z12bday  28853