Theorem pw2bday 28433

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 7439	. . . . . . . 8 ⊢ (𝑚 = 0s → (2s↑s𝑚) = (2s↑s 0s ))
2		2sno 28418	. . . . . . . . 9 ⊢ 2s ∈ No
3		exps0 28425	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
4	2, 3	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
5	1, 4	eqtrdi 2791	. . . . . . 7 ⊢ (𝑚 = 0s → (2s↑s𝑚) = 1s )
6	5	oveq2d 7447	. . . . . 6 ⊢ (𝑚 = 0s → ( 1s /su (2s↑s𝑚)) = ( 1s /su 1s ))
7		1sno 27887	. . . . . . 7 ⊢ 1s ∈ No
8		divs1 28244	. . . . . . 7 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
9	7, 8	ax-mp 5	. . . . . 6 ⊢ ( 1s /su 1s ) = 1s
10	6, 9	eqtrdi 2791	. . . . 5 ⊢ (𝑚 = 0s → ( 1s /su (2s↑s𝑚)) = 1s )
11	10	fveq2d 6911	. . . 4 ⊢ (𝑚 = 0s → ( bday ‘( 1s /su (2s↑s𝑚))) = ( bday ‘ 1s ))
12		bday1s 27891	. . . 4 ⊢ ( bday ‘ 1s ) = 1o
13	11, 12	eqtrdi 2791	. . 3 ⊢ (𝑚 = 0s → ( bday ‘( 1s /su (2s↑s𝑚))) = 1o)
14	13	eleq1d 2824	. 2 ⊢ (𝑚 = 0s → (( bday ‘( 1s /su (2s↑s𝑚))) ∈ ω ↔ 1o ∈ ω))
15		oveq2 7439	. . . . 5 ⊢ (𝑚 = 𝑛 → (2s↑s𝑚) = (2s↑s𝑛))
16	15	oveq2d 7447	. . . 4 ⊢ (𝑚 = 𝑛 → ( 1s /su (2s↑s𝑚)) = ( 1s /su (2s↑s𝑛)))
17	16	fveq2d 6911	. . 3 ⊢ (𝑚 = 𝑛 → ( bday ‘( 1s /su (2s↑s𝑚))) = ( bday ‘( 1s /su (2s↑s𝑛))))
18	17	eleq1d 2824	. 2 ⊢ (𝑚 = 𝑛 → (( bday ‘( 1s /su (2s↑s𝑚))) ∈ ω ↔ ( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω))
19		oveq2 7439	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
20	19	oveq2d 7447	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → ( 1s /su (2s↑s𝑚)) = ( 1s /su (2s↑s(𝑛 +s 1s ))))
21	20	fveq2d 6911	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ( bday ‘( 1s /su (2s↑s𝑚))) = ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))))
22	21	eleq1d 2824	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → (( bday ‘( 1s /su (2s↑s𝑚))) ∈ ω ↔ ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ ω))
23		oveq2 7439	. . . . 5 ⊢ (𝑚 = 𝑁 → (2s↑s𝑚) = (2s↑s𝑁))
24	23	oveq2d 7447	. . . 4 ⊢ (𝑚 = 𝑁 → ( 1s /su (2s↑s𝑚)) = ( 1s /su (2s↑s𝑁)))
25	24	fveq2d 6911	. . 3 ⊢ (𝑚 = 𝑁 → ( bday ‘( 1s /su (2s↑s𝑚))) = ( bday ‘( 1s /su (2s↑s𝑁))))
26	25	eleq1d 2824	. 2 ⊢ (𝑚 = 𝑁 → (( bday ‘( 1s /su (2s↑s𝑚))) ∈ ω ↔ ( bday ‘( 1s /su (2s↑s𝑁))) ∈ ω))
27		1onn 8677	. 2 ⊢ 1o ∈ ω
28		cutpw2 28432	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))}))
29	28	fveq2d 6911	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) = ( bday ‘({ 0s } |s {( 1s /su (2s↑s𝑛))})))
30		0sno 27886	. . . . . . . . 9 ⊢ 0s ∈ No
31	30	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 0s ∈ No )
32	7	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → 1s ∈ No )
33		expscl 28428	. . . . . . . . . 10 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s𝑛) ∈ No )
34	2, 33	mpan 690	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (2s↑s𝑛) ∈ No )
35		2ne0s 28419	. . . . . . . . . 10 ⊢ 2s ≠ 0s
36		expsne0 28429	. . . . . . . . . 10 ⊢ ((2s ∈ No ∧ 2s ≠ 0s ∧ 𝑛 ∈ ℕ0s) → (2s↑s𝑛) ≠ 0s )
37	2, 35, 36	mp3an12 1450	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (2s↑s𝑛) ≠ 0s )
38	32, 34, 37	divscld 28263	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s𝑛)) ∈ No )
39		muls02 28182	. . . . . . . . . . 11 ⊢ ((2s↑s𝑛) ∈ No → ( 0s ·s (2s↑s𝑛)) = 0s )
40	34, 39	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → ( 0s ·s (2s↑s𝑛)) = 0s )
41		0slt1s 27889	. . . . . . . . . 10 ⊢ 0s <s 1s
42	40, 41	eqbrtrdi 5187	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → ( 0s ·s (2s↑s𝑛)) <s 1s )
43		2nns 28417	. . . . . . . . . . . 12 ⊢ 2s ∈ ℕs
44		nnsgt0 28357	. . . . . . . . . . . 12 ⊢ (2s ∈ ℕs → 0s <s 2s)
45	43, 44	ax-mp 5	. . . . . . . . . . 11 ⊢ 0s <s 2s
46		expsgt0 28430	. . . . . . . . . . 11 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2s↑s𝑛))
47	2, 45, 46	mp3an13 1451	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → 0s <s (2s↑s𝑛))
48	31, 32, 34, 47	sltmuldivd 28268	. . . . . . . . 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 27852	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → { 0s } <<s {( 1s /su (2s↑s𝑛))})
51		suc0 6461	. . . . . . . . . . . 12 ⊢ suc ∅ = {∅}
52		bday0s 27888	. . . . . . . . . . . . 13 ⊢ ( bday ‘ 0s ) = ∅
53	52	sneqi 4642	. . . . . . . . . . . 12 ⊢ {( bday ‘ 0s )} = {∅}
54		bdayfn 27833	. . . . . . . . . . . . 13 ⊢ bday Fn No
55		fnsnfv 6988	. . . . . . . . . . . . 13 ⊢ (( bday Fn No ∧ 0s ∈ No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
56	54, 30, 55	mp2an 692	. . . . . . . . . . . 12 ⊢ {( bday ‘ 0s )} = ( bday “ { 0s })
57	51, 53, 56	3eqtr2i 2769	. . . . . . . . . . 11 ⊢ suc ∅ = ( bday “ { 0s })
58	57	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → suc ∅ = ( bday “ { 0s }))
59		fnsnfv 6988	. . . . . . . . . . . 12 ⊢ (( bday Fn No ∧ ( 1s /su (2s↑s𝑛)) ∈ No ) → {( bday ‘( 1s /su (2s↑s𝑛)))} = ( bday “ {( 1s /su (2s↑s𝑛))}))
60	54, 59	mpan 690	. . . . . . . . . . 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 4179	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (suc ∅ ∪ {( bday ‘( 1s /su (2s↑s𝑛)))}) = (( bday “ { 0s }) ∪ ( bday “ {( 1s /su (2s↑s𝑛))})))
63		imaundi 6172	. . . . . . . . 9 ⊢ ( bday “ ({ 0s } ∪ {( 1s /su (2s↑s𝑛))})) = (( bday “ { 0s }) ∪ ( bday “ {( 1s /su (2s↑s𝑛))}))
64	62, 63	eqtr4di 2793	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → (suc ∅ ∪ {( bday ‘( 1s /su (2s↑s𝑛)))}) = ( bday “ ({ 0s } ∪ {( 1s /su (2s↑s𝑛))})))
65		0ss 4406	. . . . . . . . . 10 ⊢ ∅ ⊆ ( bday ‘( 1s /su (2s↑s𝑛)))
66		ord0 6439	. . . . . . . . . . 11 ⊢ Ord ∅
67		bdayelon 27836	. . . . . . . . . . . 12 ⊢ ( bday ‘( 1s /su (2s↑s𝑛))) ∈ On
68	67	onordi 6497	. . . . . . . . . . 11 ⊢ Ord ( bday ‘( 1s /su (2s↑s𝑛)))
69		ordsucsssuc 7843	. . . . . . . . . . 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 692	. . . . . . . . . 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 6920	. . . . . . . . . . 11 ⊢ ( bday ‘( 1s /su (2s↑s𝑛))) ∈ V
73	72	sucid 6468	. . . . . . . . . 10 ⊢ ( bday ‘( 1s /su (2s↑s𝑛))) ∈ suc ( bday ‘( 1s /su (2s↑s𝑛)))
74		snssi 4813	. . . . . . . . . 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 4201	. . . . . . . 8 ⊢ (suc ∅ ∪ {( bday ‘( 1s /su (2s↑s𝑛)))}) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛)))
77	64, 76	eqsstrrdi 4051	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ( bday “ ({ 0s } ∪ {( 1s /su (2s↑s𝑛))})) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))))
78	67	onsuci 7859	. . . . . . . 8 ⊢ suc ( bday ‘( 1s /su (2s↑s𝑛))) ∈ On
79		scutbdaybnd 27875	. . . . . . . 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 1448	. . . . . . 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 584	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘({ 0s } |s {( 1s /su (2s↑s𝑛))})) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))))
82	29, 81	eqsstrd 4034	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))))
83		bdayelon 27836	. . . . . 6 ⊢ ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ On
84		onsssuc 6476	. . . . . 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 692	. . . . 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 7913	. . . . 5 ⊢ (( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω → suc ( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω)
88		peano2 7913	. . . . 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 7898	. . . 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 596	. . 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 28352	1 ⊢ (𝑁 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s𝑁))) ∈ ω)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106   ≠ wne 2938   ∪ cun 3961   ⊆ wss 3963  ∅c0 4339  {csn 4631   class class class wbr 5148   “ cima 5692  Ord word 6385  Oncon0 6386  suc csuc 6388   Fn wfn 6558  ‘cfv 6563  (class class class)co 7431  ωcom 7887  1_oc1o 8498   No csur 27699   <s cslt 27700   bday cbday 27701   <<s csslt 27840   |s cscut 27842   0_s c0s 27882   1_s c1s 27883   +_s cadds 28007   ·_s cmuls 28147   /_su cdivs 28228  ℕ_0scnn0s 28333  ℕ_scnns 28334  2_sc2s 28409  ↑_scexps 28411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-dc 10484

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148  df-divs 28229  df-seqs 28305  df-n0s 28335  df-nns 28336  df-zs 28380  df-2s 28410  df-exps 28412

This theorem is referenced by: (None)