Theorem pw2bday 28418

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 28403	. . . . . . . . 9 ⊢ 2s ∈ No
3		exps0 28410	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
4	2, 3	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
5	1, 4	eqtrdi 2793	. . . . . . 7 ⊢ (𝑚 = 0s → (2s↑s𝑚) = 1s )
6	5	oveq2d 7447	. . . . . 6 ⊢ (𝑚 = 0s → ( 1s /su (2s↑s𝑚)) = ( 1s /su 1s ))
7		1sno 27872	. . . . . . 7 ⊢ 1s ∈ No
8		divs1 28229	. . . . . . 7 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
9	7, 8	ax-mp 5	. . . . . 6 ⊢ ( 1s /su 1s ) = 1s
10	6, 9	eqtrdi 2793	. . . . 5 ⊢ (𝑚 = 0s → ( 1s /su (2s↑s𝑚)) = 1s )
11	10	fveq2d 6910	. . . 4 ⊢ (𝑚 = 0s → ( bday ‘( 1s /su (2s↑s𝑚))) = ( bday ‘ 1s ))
12		bday1s 27876	. . . 4 ⊢ ( bday ‘ 1s ) = 1o
13	11, 12	eqtrdi 2793	. . 3 ⊢ (𝑚 = 0s → ( bday ‘( 1s /su (2s↑s𝑚))) = 1o)
14	13	eleq1d 2826	. 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 6910	. . 3 ⊢ (𝑚 = 𝑛 → ( bday ‘( 1s /su (2s↑s𝑚))) = ( bday ‘( 1s /su (2s↑s𝑛))))
18	17	eleq1d 2826	. 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 6910	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ( bday ‘( 1s /su (2s↑s𝑚))) = ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))))
22	21	eleq1d 2826	. 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 6910	. . 3 ⊢ (𝑚 = 𝑁 → ( bday ‘( 1s /su (2s↑s𝑚))) = ( bday ‘( 1s /su (2s↑s𝑁))))
26	25	eleq1d 2826	. 2 ⊢ (𝑚 = 𝑁 → (( bday ‘( 1s /su (2s↑s𝑚))) ∈ ω ↔ ( bday ‘( 1s /su (2s↑s𝑁))) ∈ ω))
27		1onn 8678	. 2 ⊢ 1o ∈ ω
28		cutpw2 28417	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))}))
29	28	fveq2d 6910	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) = ( bday ‘({ 0s } |s {( 1s /su (2s↑s𝑛))})))
30		0sno 27871	. . . . . . . . 9 ⊢ 0s ∈ No
31	30	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 0s ∈ No )
32	7	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → 1s ∈ No )
33		expscl 28413	. . . . . . . . . 10 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s𝑛) ∈ No )
34	2, 33	mpan 690	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (2s↑s𝑛) ∈ No )
35		2ne0s 28404	. . . . . . . . . 10 ⊢ 2s ≠ 0s
36		expsne0 28414	. . . . . . . . . 10 ⊢ ((2s ∈ No ∧ 2s ≠ 0s ∧ 𝑛 ∈ ℕ0s) → (2s↑s𝑛) ≠ 0s )
37	2, 35, 36	mp3an12 1453	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (2s↑s𝑛) ≠ 0s )
38	32, 34, 37	divscld 28248	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s𝑛)) ∈ No )
39		muls02 28167	. . . . . . . . . . 11 ⊢ ((2s↑s𝑛) ∈ No → ( 0s ·s (2s↑s𝑛)) = 0s )
40	34, 39	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → ( 0s ·s (2s↑s𝑛)) = 0s )
41		0slt1s 27874	. . . . . . . . . 10 ⊢ 0s <s 1s
42	40, 41	eqbrtrdi 5182	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → ( 0s ·s (2s↑s𝑛)) <s 1s )
43		2nns 28402	. . . . . . . . . . . 12 ⊢ 2s ∈ ℕs
44		nnsgt0 28342	. . . . . . . . . . . 12 ⊢ (2s ∈ ℕs → 0s <s 2s)
45	43, 44	ax-mp 5	. . . . . . . . . . 11 ⊢ 0s <s 2s
46		expsgt0 28415	. . . . . . . . . . 11 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2s↑s𝑛))
47	2, 45, 46	mp3an13 1454	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → 0s <s (2s↑s𝑛))
48	31, 32, 34, 47	sltmuldivd 28253	. . . . . . . . 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 27837	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → { 0s } <<s {( 1s /su (2s↑s𝑛))})
51		suc0 6459	. . . . . . . . . . . 12 ⊢ suc ∅ = {∅}
52		bday0s 27873	. . . . . . . . . . . . 13 ⊢ ( bday ‘ 0s ) = ∅
53	52	sneqi 4637	. . . . . . . . . . . 12 ⊢ {( bday ‘ 0s )} = {∅}
54		bdayfn 27818	. . . . . . . . . . . . 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 2771	. . . . . . . . . . 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 4169	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (suc ∅ ∪ {( bday ‘( 1s /su (2s↑s𝑛)))}) = (( bday “ { 0s }) ∪ ( bday “ {( 1s /su (2s↑s𝑛))})))
63		imaundi 6169	. . . . . . . . 9 ⊢ ( bday “ ({ 0s } ∪ {( 1s /su (2s↑s𝑛))})) = (( bday “ { 0s }) ∪ ( bday “ {( 1s /su (2s↑s𝑛))}))
64	62, 63	eqtr4di 2795	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → (suc ∅ ∪ {( bday ‘( 1s /su (2s↑s𝑛)))}) = ( bday “ ({ 0s } ∪ {( 1s /su (2s↑s𝑛))})))
65		0ss 4400	. . . . . . . . . 10 ⊢ ∅ ⊆ ( bday ‘( 1s /su (2s↑s𝑛)))
66		ord0 6437	. . . . . . . . . . 11 ⊢ Ord ∅
67		bdayelon 27821	. . . . . . . . . . . 12 ⊢ ( bday ‘( 1s /su (2s↑s𝑛))) ∈ On
68	67	onordi 6495	. . . . . . . . . . 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 6919	. . . . . . . . . . 11 ⊢ ( bday ‘( 1s /su (2s↑s𝑛))) ∈ V
73	72	sucid 6466	. . . . . . . . . 10 ⊢ ( bday ‘( 1s /su (2s↑s𝑛))) ∈ suc ( bday ‘( 1s /su (2s↑s𝑛)))
74		snssi 4808	. . . . . . . . . 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 4191	. . . . . . . 8 ⊢ (suc ∅ ∪ {( bday ‘( 1s /su (2s↑s𝑛)))}) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛)))
77	64, 76	eqsstrrdi 4029	. . . . . . 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 27860	. . . . . . . 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 1451	. . . . . . 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 4018	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ⊆ suc ( bday ‘( 1s /su (2s↑s𝑛))))
83		bdayelon 27821	. . . . . 6 ⊢ ( bday ‘( 1s /su (2s↑s(𝑛 +s 1s )))) ∈ On
84		onsssuc 6474	. . . . . 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 7912	. . . . 5 ⊢ (( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω → suc ( bday ‘( 1s /su (2s↑s𝑛))) ∈ ω)
88		peano2 7912	. . . . 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 28337	1 ⊢ (𝑁 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s𝑁))) ∈ ω)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108   ≠ wne 2940   ∪ cun 3949   ⊆ wss 3951  ∅c0 4333  {csn 4626   class class class wbr 5143   “ cima 5688  Ord word 6383  Oncon0 6384  suc csuc 6386   Fn wfn 6556  ‘cfv 6561  (class class class)co 7431  ωcom 7887  1_oc1o 8499   No csur 27684   <s cslt 27685   bday cbday 27686   <<s csslt 27825   |s cscut 27827   0_s c0s 27867   1_s c1s 27868   +_s cadds 27992   ·_s cmuls 28132   /_su cdivs 28213  ℕ_0scnn0s 28318  ℕ_scnns 28319  2_sc2s 28394  ↑_scexps 28396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-dc 10486

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133  df-divs 28214  df-seqs 28290  df-n0s 28320  df-nns 28321  df-zs 28365  df-2s 28395  df-exps 28397

This theorem is referenced by: (None)