Theorem z12bdaylem 28875

Description: Lemma for z12bday 28876. 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 28867	. . . 4 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No)
3		z12sge0 28874	. . . 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 28716	. . . . . . . . . 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 28716	. . . . . . . . . . 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 28830	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (𝑦 /su (2s↑s𝑝)) ∈ No)
12		addbday 28409	. . . . . . . . . 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 28743	. . . . . . . . . . 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 28855	. . . . . . . . . . . 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 28743	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℕ0s → (bday‘𝑝) ∈ ω)
20		peano2 7901	. . . . . . . . . . . . . 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 28143	. . . . . . . . . . . . 13 ⊢ (bday‘(𝑦 /su (2s↑s𝑝))) ∈ On
25	24	onordi 6476	. . . . . . . . . . . 12 ⊢ Ord (bday‘(𝑦 /su (2s↑s𝑝)))
26		ordom 7887	. . . . . . . . . . . 12 ⊢ Ord ω
27		ordtr2 6408	. . . . . . . . . . . 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 8716	. . . . . . . . . 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 28143	. . . . . . . . . . 11 ⊢ (bday‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ On
33	32	onordi 6476	. . . . . . . . . 10 ⊢ Ord (bday‘(𝑥 +s (𝑦 /su (2s↑s𝑝))))
34		ordtr2 6408	. . . . . . . . . 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 6885	. . . . . . . . 9 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → (bday‘𝐴) = (bday‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))))
38	37	eleq1d 2845	. . . . . . . 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 3163	. . 3 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → (bday‘𝐴) ∈ ω))
44	43	rexlimivv 3204	. 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 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895   class class class wbr 5102  Ord word 6361  suc csuc 6364  ‘cfv 6538  (class class class)co 7420  ωcom 7877   +no cnadd 8677  Nocsur 28002   <s clts 28003  bdaycbday 28004   ≤s cles 28106   0_s c0s 28196   +_s cadds 28350   /_su cdivs 28578  ℕ_0scn0s 28703  2_sc2s 28801  ↑_scexps 28803  ℤ_s[1/2]cz12s 28805

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-dc 10527

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-oadd 8483  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-1s 28199  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-norec2 28340  df-adds 28351  df-negs 28412  df-subs 28413  df-muls 28498  df-divs 28579  df-ons 28643  df-seqs 28675  df-n0s 28705  df-nns 28706  df-zs 28770  df-2s 28802  df-exps 28804  df-z12s 28806

This theorem is used by:  z12bday  28876