Theorem z12bdaylem 28554

Description: Lemma for z12bday 28555. 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 486	. . 3 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → 𝐴 ∈ ℤs[1/2])
2		z12no 28546	. . . 4 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No )
3		z12sge0 28553	. . . 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 589	. . 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 234	. 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 1204	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑥 ∈ ℕ0s)
7	6	n0nod 28395	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑥 ∈ No )
8		simpl2 1205	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑦 ∈ ℕ0s)
9	8	n0nod 28395	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑦 ∈ No )
10		simpl3 1206	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑝 ∈ ℕ0s)
11	9, 10	pw2divscld 28509	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (𝑦 /su (2s↑s𝑝)) ∈ No )
12		addbday 28088	. . . . . . . . . 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 593	. . . . . . . . 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 28422	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ0s → ( bday ‘𝑥) ∈ ω)
15	6, 14	syl 17	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝑥) ∈ ω)
16		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑦 <s (2s↑s𝑝))
17		bdaypw2n0bnd 28534	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝))
18	8, 10, 16, 17	syl3anc 1389	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝))
19		n0bday 28422	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℕ0s → ( bday ‘𝑝) ∈ ω)
20		peano2 7866	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝑝) ∈ ω → suc ( bday ‘𝑝) ∈ ω)
21	19, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℕ0s → suc ( bday ‘𝑝) ∈ ω)
22	21	3ad2ant3 1147	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → suc ( bday ‘𝑝) ∈ ω)
23	22	adantr 484	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → suc ( bday ‘𝑝) ∈ ω)
24		bdayon 27822	. . . . . . . . . . . . 13 ⊢ ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ On
25	24	onordi 6455	. . . . . . . . . . . 12 ⊢ Ord ( bday ‘(𝑦 /su (2s↑s𝑝)))
26		ordom 7852	. . . . . . . . . . . 12 ⊢ Ord ω
27		ordtr2 6387	. . . . . . . . . . . 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 702	. . . . . . . . . . 11 ⊢ ((( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝) ∧ suc ( bday ‘𝑝) ∈ ω) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ ω)
29	18, 23, 28	syl2anc 593	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ ω)
30		omnaddcl 8669	. . . . . . . . . 10 ⊢ ((( bday ‘𝑥) ∈ ω ∧ ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ ω) → (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))) ∈ ω)
31	15, 29, 30	syl2anc 593	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))) ∈ ω)
32		bdayon 27822	. . . . . . . . . . 11 ⊢ ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ On
33	32	onordi 6455	. . . . . . . . . 10 ⊢ Ord ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝))))
34		ordtr2 6387	. . . . . . . . . 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 702	. . . . . . . . 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 593	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ ω)
37		fveq2 6863	. . . . . . . . 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 249	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → ( bday ‘𝐴) ∈ ω))
40	39	ex 416	. . . . . 6 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (𝑦 <s (2s↑s𝑝) → (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → ( bday ‘𝐴) ∈ ω)))
41	40	impcomd 415	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → ((𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω))
42	41	3expa 1130	. . . 4 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑝 ∈ ℕ0s) → ((𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω))
43	42	rexlimdva 3162	. . 3 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω))
44	43	rexlimivv 3203	. 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 17	1 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∃wrex 3085   ⊆ wss 3904   class class class wbr 5099  Ord word 6341  suc csuc 6344  ‘cfv 6517  (class class class)co 7392  ωcom 7842   +no cnadd 8630   No csur 27681   <s clts 27682   bday cbday 27683   ≤s cles 27785   0_s c0s 27875   +_s cadds 28029   /_su cdivs 28257  ℕ_0scn0s 28382  2_sc2s 28480  ↑_scexps 28482  ℤ_s[1/2]cz12s 28484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-dc 10400

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-nadd 8631  df-no 27684  df-lts 27685  df-bday 27686  df-les 27786  df-slts 27828  df-cuts 27830  df-0s 27877  df-1s 27878  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec 28008  df-norec2 28019  df-adds 28030  df-negs 28091  df-subs 28092  df-muls 28177  df-divs 28258  df-ons 28322  df-seqs 28354  df-n0s 28384  df-nns 28385  df-zs 28449  df-2s 28481  df-exps 28483  df-z12s 28485

This theorem is referenced by:  z12bday  28555