Theorem z12bdaylem 28480

Description: Lemma for z12bday 28481. 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 482	. . 3 ⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → 𝐴 ∈ ℤs[1/2])
2		z12no 28472	. . . 4 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No )
3		z12sge0 28479	. . . 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 580	. . 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 232	. 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 1192	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑥 ∈ ℕ0s)
7	6	n0nod 28321	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑥 ∈ No )
8		simpl2 1193	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑦 ∈ ℕ0s)
9	8	n0nod 28321	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑦 ∈ No )
10		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑝 ∈ ℕ0s)
11	9, 10	pw2divscld 28435	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (𝑦 /su (2s↑s𝑝)) ∈ No )
12		addbday 28014	. . . . . . . . . 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 584	. . . . . . . . 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 28348	. . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑦 <s (2s↑s𝑝))
17		bdaypw2n0bnd 28460	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝))
18	8, 10, 16, 17	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝))
19		n0bday 28348	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℕ0s → ( bday ‘𝑝) ∈ ω)
20		peano2 7832	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝑝) ∈ ω → suc ( bday ‘𝑝) ∈ ω)
21	19, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℕ0s → suc ( bday ‘𝑝) ∈ ω)
22	21	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → suc ( bday ‘𝑝) ∈ ω)
23	22	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → suc ( bday ‘𝑝) ∈ ω)
24		bdayon 27748	. . . . . . . . . . . . 13 ⊢ ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ On
25	24	onordi 6430	. . . . . . . . . . . 12 ⊢ Ord ( bday ‘(𝑦 /su (2s↑s𝑝)))
26		ordom 7818	. . . . . . . . . . . 12 ⊢ Ord ω
27		ordtr2 6362	. . . . . . . . . . . 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 692	. . . . . . . . . . 11 ⊢ ((( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝) ∧ suc ( bday ‘𝑝) ∈ ω) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ ω)
29	18, 23, 28	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ ω)
30		omnaddcl 8631	. . . . . . . . . 10 ⊢ ((( bday ‘𝑥) ∈ ω ∧ ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ ω) → (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))) ∈ ω)
31	15, 29, 30	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))) ∈ ω)
32		bdayon 27748	. . . . . . . . . . 11 ⊢ ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ On
33	32	onordi 6430	. . . . . . . . . 10 ⊢ Ord ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝))))
34		ordtr2 6362	. . . . . . . . . 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 692	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ ω)
37		fveq2 6834	. . . . . . . . 9 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → ( bday ‘𝐴) = ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))))
38	37	eleq1d 2821	. . . . . . . 8 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → (( bday ‘𝐴) ∈ ω ↔ ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ ω))
39	36, 38	syl5ibrcom 247	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → ( bday ‘𝐴) ∈ ω))
40	39	ex 412	. . . . . 6 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (𝑦 <s (2s↑s𝑝) → (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → ( bday ‘𝐴) ∈ ω)))
41	40	impcomd 411	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → ((𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω))
42	41	3expa 1118	. . . 4 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑝 ∈ ℕ0s) → ((𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω))
43	42	rexlimdva 3137	. . 3 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω))
44	43	rexlimivv 3178	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3060   ⊆ wss 3901   class class class wbr 5098  Ord word 6316  suc csuc 6319  ‘cfv 6492  (class class class)co 7358  ωcom 7808   +no cnadd 8593   No csur 27607   <s clts 27608   bday cbday 27609   ≤s cles 27712   0_s c0s 27801   +_s cadds 27955   /_su cdivs 28183  ℕ_0scn0s 28308  2_sc2s 28406  ↑_scexps 28408  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:  z12bday  28481