Theorem zs12bdaylem 28461

Description: Lemma for zs12bday 28462. Handle the non-negative case. (Contributed by Scott Fenton, 22-Feb-2026.)

Assertion

Ref	Expression
zs12bdaylem	⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)

Proof of Theorem zs12bdaylem

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		zs12no 28453	. . . 4 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No )
3		zs12ge0 28460	. . . 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 581	. . 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 1193	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑥 ∈ ℕ0s)
7	6	n0snod 28304	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑥 ∈ No )
8		simpl2 1194	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑦 ∈ ℕ0s)
9	8	n0snod 28304	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑦 ∈ No )
10		simpl3 1195	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → 𝑝 ∈ ℕ0s)
11	9, 10	pw2divscld 28416	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (𝑦 /su (2s↑s𝑝)) ∈ No )
12		addsbday 27998	. . . . . . . . . 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 585	. . . . . . . . 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		n0sbday 28330	. . . . . . . . . . 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		bdaypw2n0sbnd 28441	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝))
18	8, 10, 16, 17	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝))
19		n0sbday 28330	. . . . . . . . . . . . . 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 1136	. . . . . . . . . . . 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		bdayelon 27750	. . . . . . . . . . . . 13 ⊢ ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ On
25	24	onordi 6429	. . . . . . . . . . . 12 ⊢ Ord ( bday ‘(𝑦 /su (2s↑s𝑝)))
26		ordom 7818	. . . . . . . . . . . 12 ⊢ Ord ω
27		ordtr2 6361	. . . . . . . . . . . 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 693	. . . . . . . . . . 11 ⊢ ((( bday ‘(𝑦 /su (2s↑s𝑝))) ⊆ suc ( bday ‘𝑝) ∧ suc ( bday ‘𝑝) ∈ ω) → ( bday ‘(𝑦 /su (2s↑s𝑝))) ∈ ω)
29	18, 23, 28	syl2anc 585	. . . . . . . . . 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 585	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → (( bday ‘𝑥) +no ( bday ‘(𝑦 /su (2s↑s𝑝)))) ∈ ω)
32		bdayelon 27750	. . . . . . . . . . 11 ⊢ ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ On
33	32	onordi 6429	. . . . . . . . . 10 ⊢ Ord ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝))))
34		ordtr2 6361	. . . . . . . . . 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 693	. . . . . . . . 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 585	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))) ∈ ω)
37		fveq2 6833	. . . . . . . . 9 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → ( bday ‘𝐴) = ( bday ‘(𝑥 +s (𝑦 /su (2s↑s𝑝)))))
38	37	eleq1d 2820	. . . . . . . 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 1119	. . . 4 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑝 ∈ ℕ0s) → ((𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω))
43	42	rexlimdva 3136	. . 3 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ( bday ‘𝐴) ∈ ω))
44	43	rexlimivv 3177	. 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 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3059   ⊆ wss 3900   class class class wbr 5097  Ord word 6315  suc csuc 6318  ‘cfv 6491  (class class class)co 7358  ωcom 7808   +no cnadd 8593   No csur 27609   <s cslt 27610   bday cbday 27611   ≤s csle 27714   0_s c0s 27801   +_s cadds 27939   /_su cdivs 28167  ℕ_0scnn0s 28291  2_sc2s 28387  ↑_scexps 28389  ℤ_s[1/2]czs12 28391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-dc 10358

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  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 27612  df-slt 27613  df-bday 27614  df-sle 27715  df-sslt 27756  df-scut 27758  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27918  df-norec2 27929  df-adds 27940  df-negs 28001  df-subs 28002  df-muls 28087  df-divs 28168  df-ons 28231  df-seqs 28263  df-n0s 28293  df-nns 28294  df-zs 28356  df-2s 28388  df-exps 28390  df-zs12 28392

This theorem is referenced by:  zs12bday  28462