Theorem pw2bday 28436

Description: The inverses of powers of two have finite birthdays. (Contributed by Scott Fenton, 7-Aug-2025.)

Assertion

Ref	Expression
pw2bday	⊢ (𝑁 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s𝑁))) ∈ ω)

Proof of Theorem pw2bday

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7456	. . . . . . . 8 ⊢ (𝑚 = 0s → (2s↑s𝑚) = (2s↑s 0s ))
2		2sno 28421	. . . . . . . . 9 ⊢ 2s ∈ No
3		exps0 28428	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
4	2, 3	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
5	1, 4	eqtrdi 2796	. . . . . . 7 ⊢ (𝑚 = 0s → (2s↑s𝑚) = 1s )
6	5	oveq2d 7464	. . . . . 6 ⊢ (𝑚 = 0s → ( 1s /su (2s↑s𝑚)) = ( 1s /su 1s ))
7		1sno 27890	. . . . . . 7 ⊢ 1s ∈ No
8		divs1 28247	. . . . . . 7 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
9	7, 8	ax-mp 5	. . . . . 6 ⊢ ( 1s /su 1s ) = 1s
10	6, 9	eqtrdi 2796	. . . . 5 ⊢ (𝑚 = 0s → ( 1s /su (2s↑s𝑚)) = 1s )
11	10	fveq2d 6924	. . . 4 ⊢ (𝑚 = 0s → ( bday ‘( 1s /su (2s↑s𝑚))) = ( bday ‘ 1s ))
12		bday1s 27894	. . . 4 ⊢ ( bday ‘ 1s ) = 1o
13	11, 12	eqtrdi 2796	. . 3 ⊢ (𝑚 = 0s → ( bday ‘( 1s /su (2s↑s𝑚))) = 1o)
14	13	eleq1d 2829	. 2 ⊢ (𝑚 = 0s → (( bday ‘( 1s /su (2s↑s𝑚))) ∈ ω ↔ 1o ∈ ω))
15		oveq2 7456	. . . . 5 ⊢ (𝑚 = 𝑛 → (2s↑s𝑚) = (2s↑s𝑛))
16	15	oveq2d 7464	. . . 4 ⊢ (𝑚 = 𝑛 → ( 1s /su (2s↑s𝑚)) = ( 1s /su (2s↑s𝑛)))
17	16	fveq2d 6924	. . 3 ⊢ (𝑚 = 𝑛 → ( bday ‘( 1s /su (2s↑s𝑚))) = ( bday ‘( 1s /su (2s↑s𝑛))))
18	17	eleq1d 2829	. 2 ⊢ (𝑚 = 𝑛 → (( bday ‘( 1s /su (2s↑s𝑚))) ∈ ω ↔ ( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω))
19		oveq2 7456	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
20	19	oveq2d 7464	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → ( 1s /su (2s↑s𝑚)) = ( 1s /su (2s↑s(𝑛 +s 1s ))))
21	20	fveq2d 6924	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ( bday ‘( 1s /su (2s↑s𝑚))) = ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))))
22	21	eleq1d 2829	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → (( bday ‘( 1s /su (2s↑s𝑚))) ∈ ω ↔ ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ ω))
23		oveq2 7456	. . . . 5 ⊢ (𝑚 = 𝑁 → (2s↑s𝑚) = (2s↑s𝑁))
24	23	oveq2d 7464	. . . 4 ⊢ (𝑚 = 𝑁 → ( 1s /su (2s↑s𝑚)) = ( 1s /su (2s↑s𝑁)))
25	24	fveq2d 6924	. . 3 ⊢ (𝑚 = 𝑁 → ( bday ‘( 1s /su (2s↑s𝑚))) = ( bday ‘( 1s /su (2s↑s𝑁))))
26	25	eleq1d 2829	. 2 ⊢ (𝑚 = 𝑁 → (( bday ‘( 1s /su (2s↑s𝑚))) ∈ ω ↔ ( bday ‘( 1s /su (2s↑s𝑁))) ∈ ω))
27		1onn 8696	. 2 ⊢ 1o ∈ ω
28		cutpw2 28435	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))}))
29	28	fveq2d 6924	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) = ( bday ‘({ 0s } |s {( 1s /su (2s↑s𝑛))})))
30		0sno 27889	. . . . . . . . 9 ⊢ 0s ∈ No
31	30	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 0s ∈ No )
32	7	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → 1s ∈ No )
33		expscl 28431	. . . . . . . . . 10 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s𝑛) ∈ No )
34	2, 33	mpan 689	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (2s↑s𝑛) ∈ No )
35		2ne0s 28422	. . . . . . . . . 10 ⊢ 2s ≠ 0s
36		expsne0 28432	. . . . . . . . . 10 ⊢ ((2s ∈ No ∧ 2s ≠ 0s ∧ 𝑛 ∈ ℕ0s) → (2s↑s𝑛) ≠ 0s )
37	2, 35, 36	mp3an12 1451	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (2s↑s𝑛) ≠ 0s )
38	32, 34, 37	divscld 28266	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s𝑛)) ∈ No )
39		muls02 28185	. . . . . . . . . . 11 ⊢ ((2s↑s𝑛) ∈ No → ( 0s ·s (2s↑s𝑛)) = 0s )
40	34, 39	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → ( 0s ·s (2s↑s𝑛)) = 0s )
41		0slt1s 27892	. . . . . . . . . 10 ⊢ 0s <s 1s
42	40, 41	eqbrtrdi 5205	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → ( 0s ·s (2s↑s𝑛)) <s 1s )
43		2nns 28420	. . . . . . . . . . . 12 ⊢ 2s ∈ ℕs
44		nnsgt0 28360	. . . . . . . . . . . 12 ⊢ (2s ∈ ℕs → 0s <s 2s)
45	43, 44	ax-mp 5	. . . . . . . . . . 11 ⊢ 0s <s 2s
46		expsgt0 28433	. . . . . . . . . . 11 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2s↑s𝑛))
47	2, 45, 46	mp3an13 1452	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → 0s <s (2s↑s𝑛))
48	31, 32, 34, 47	sltmuldivd 28271	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (( 0s ·s (2s↑s𝑛)) <s 1s ↔ 0s <s ( 1s /su (2s↑s𝑛))))
49	42, 48	mpbid 232	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 0s <s ( 1s /su (2s↑s𝑛)))
50	31, 38, 49	ssltsn 27855	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → { 0s } <<s {( 1s /su (2s↑s𝑛))})
51		suc0 6470	. . . . . . . . . . . 12 ⊢ suc ∅ = {∅}
52		bday0s 27891	. . . . . . . . . . . . 13 ⊢ ( bday ‘ 0s ) = ∅
53	52	sneqi 4659	. . . . . . . . . . . 12 ⊢ {( bday ‘ 0s )} = {∅}
54		bdayfn 27836	. . . . . . . . . . . . 13 ⊢ bday Fn No
55		fnsnfv 7001	. . . . . . . . . . . . 13 ⊢ (( bday Fn No ∧ 0s ∈ No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
56	54, 30, 55	mp2an 691	. . . . . . . . . . . 12 ⊢ {( bday ‘ 0s )} = ( bday “ { 0s })
57	51, 53, 56	3eqtr2i 2774	. . . . . . . . . . 11 ⊢ suc ∅ = ( bday “ { 0s })
58	57	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → suc ∅ = ( bday “ { 0s }))
59		fnsnfv 7001	. . . . . . . . . . . 12 ⊢ (( bday Fn No ∧ ( 1s /su (2s↑s𝑛)) ∈ No ) → {( bday ‘( 1s /su (2s↑s𝑛)))} = ( bday “ {( 1s /su (2s↑s𝑛))}))
60	54, 59	mpan 689	. . . . . . . . . . 11 ⊢ (( 1s /su (2s↑s𝑛)) ∈ No → {( bday ‘( 1s /su (2s↑s𝑛)))} = ( bday “ {( 1s /su (2s↑s𝑛))}))
61	38, 60	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → {( bday ‘( 1s /su (2s↑s𝑛)))} = ( bday “ {( 1s /su (2s↑s𝑛))}))
62	58, 61	uneq12d 4192	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (suc ∅ ∪ {( bday ‘( 1s /su (2s↑s𝑛)))}) = (( bday “ { 0s }) ∪ ( bday “ {( 1s /su (2s↑s𝑛))})))
63		imaundi 6181	. . . . . . . . 9 ⊢ ( bday “ ({ 0s } ∪ {( 1s /su (2s↑s𝑛))})) = (( bday “ { 0s }) ∪ ( bday “ {( 1s /su (2s↑s𝑛))}))
64	62, 63	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → (suc ∅ ∪ {( bday ‘( 1s /su (2s↑s𝑛)))}) = ( bday “ ({ 0s } ∪ {( 1s /su (2s↑s𝑛))})))
65		0ss 4423	. . . . . . . . . 10 ⊢ ∅ ⊆ ( bday ‘( 1s /su (2s↑s𝑛)))
66		ord0 6448	. . . . . . . . . . 11 ⊢ Ord ∅
67		bdayelon 27839	. . . . . . . . . . . 12 ⊢ ( bday ‘( 1s /su (2s↑s𝑛))) ∈ On
68	67	onordi 6506	. . . . . . . . . . 11 ⊢ Ord ( bday ‘( 1s /su (2s↑s𝑛)))
69		ordsucsssuc 7859	. . . . . . . . . . 11 ⊢ ((Ord ∅ ∧ Ord ( bday ‘( 1s /su (2s↑s𝑛)))) → (∅ ⊆ ( bday ‘( 1s /su (2s↑s𝑛))) ↔ suc ∅ ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛)))))
70	66, 68, 69	mp2an 691	. . . . . . . . . 10 ⊢ (∅ ⊆ ( bday ‘( 1s /su (2s↑s𝑛))) ↔ suc ∅ ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))))
71	65, 70	mpbi 230	. . . . . . . . 9 ⊢ suc ∅ ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛)))
72		fvex 6933	. . . . . . . . . . 11 ⊢ ( bday ‘( 1s /su (2s↑s𝑛))) ∈ V
73	72	sucid 6477	. . . . . . . . . 10 ⊢ ( bday ‘( 1s /su (2s↑s𝑛))) ∈ suc ( bday ‘( 1s /su (2s↑s𝑛)))
74		snssi 4833	. . . . . . . . . 10 ⊢ (( bday ‘( 1s /su (2s↑s𝑛))) ∈ suc ( bday ‘( 1s /su (2s↑s𝑛))) → {( bday ‘( 1s /su (2s↑s𝑛)))} ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))))
75	73, 74	ax-mp 5	. . . . . . . . 9 ⊢ {( bday ‘( 1s /su (2s↑s𝑛)))} ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛)))
76	71, 75	unssi 4214	. . . . . . . 8 ⊢ (suc ∅ ∪ {( bday ‘( 1s /su (2s↑s𝑛)))}) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛)))
77	64, 76	eqsstrrdi 4064	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ( bday “ ({ 0s } ∪ {( 1s /su (2s↑s𝑛))})) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))))
78	67	onsuci 7875	. . . . . . . 8 ⊢ suc ( bday ‘( 1s /su (2s↑s𝑛))) ∈ On
79		scutbdaybnd 27878	. . . . . . . 8 ⊢ (({ 0s } <<s {( 1s /su (2s↑s𝑛))} ∧ suc ( bday ‘( 1s /su (2s↑s𝑛))) ∈ On ∧ ( bday “ ({ 0s } ∪ {( 1s /su (2s↑s𝑛))})) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛)))) → ( bday ‘({ 0s } |s {( 1s /su (2s↑s𝑛))})) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))))
80	78, 79	mp3an2 1449	. . . . . . 7 ⊢ (({ 0s } <<s {( 1s /su (2s↑s𝑛))} ∧ ( bday “ ({ 0s } ∪ {( 1s /su (2s↑s𝑛))})) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛)))) → ( bday ‘({ 0s } |s {( 1s /su (2s↑s𝑛))})) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))))
81	50, 77, 80	syl2anc 583	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘({ 0s } |s {( 1s /su (2s↑s𝑛))})) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))))
82	29, 81	eqsstrd 4047	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))))
83		bdayelon 27839	. . . . . 6 ⊢ ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ On
84		onsssuc 6485	. . . . . 6 ⊢ ((( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ On ∧ suc ( bday ‘( 1s /su (2s↑s𝑛))) ∈ On) → (( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))) ↔ ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ suc suc ( bday ‘( 1s /su (2s↑s𝑛)))))
85	83, 78, 84	mp2an 691	. . . . 5 ⊢ (( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))) ↔ ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ suc suc ( bday ‘( 1s /su (2s↑s𝑛))))
86	82, 85	sylib 218	. . . 4 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ suc suc ( bday ‘( 1s /su (2s↑s𝑛))))
87		peano2 7929	. . . . 5 ⊢ (( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω → suc ( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω)
88		peano2 7929	. . . . 5 ⊢ (suc ( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω → suc suc ( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω)
89	87, 88	syl 17	. . . 4 ⊢ (( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω → suc suc ( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω)
90		elnn 7914	. . . 4 ⊢ ((( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ suc suc ( bday ‘( 1s /su (2s↑s𝑛))) ∧ suc suc ( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω) → ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ ω)
91	86, 89, 90	syl2an 595	. . 3 ⊢ ((𝑛 ∈ ℕ0s ∧ ( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω) → ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ ω)
92	91	ex 412	. 2 ⊢ (𝑛 ∈ ℕ0s → (( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω → ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ ω))
93	14, 18, 22, 26, 27, 92	n0sind 28355	1 ⊢ (𝑁 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s𝑁))) ∈ ω)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∪ cun 3974   ⊆ wss 3976  ∅c0 4352  {csn 4648   class class class wbr 5166   “ cima 5703  Ord word 6394  Oncon0 6395  suc csuc 6397   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448  ωcom 7903  1_oc1o 8515   No csur 27702   <s cslt 27703   bday cbday 27704   <<s csslt 27843   |s cscut 27845   0_s c0s 27885   1_s c1s 27886   +_s cadds 28010   ·_s cmuls 28150   /_su cdivs 28231  ℕ_0scnn0s 28336  ℕ_scnns 28337  2_sc2s 28412  ↑_scexps 28414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-dc 10515

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-muls 28151  df-divs 28232  df-seqs 28308  df-n0s 28338  df-nns 28339  df-zs 28383  df-2s 28413  df-exps 28415

This theorem is referenced by: (None)