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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 (2ss𝑁))) ∈ ω)

Proof of Theorem pw2bday
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7439 . . . . . . . 8 (𝑚 = 0s → (2ss𝑚) = (2ss 0s ))
2 2sno 28403 . . . . . . . . 9 2s No
3 exps0 28410 . . . . . . . . 9 (2s No → (2ss 0s ) = 1s )
42, 3ax-mp 5 . . . . . . . 8 (2ss 0s ) = 1s
51, 4eqtrdi 2793 . . . . . . 7 (𝑚 = 0s → (2ss𝑚) = 1s )
65oveq2d 7447 . . . . . 6 (𝑚 = 0s → ( 1s /su (2ss𝑚)) = ( 1s /su 1s ))
7 1sno 27872 . . . . . . 7 1s No
8 divs1 28229 . . . . . . 7 ( 1s No → ( 1s /su 1s ) = 1s )
97, 8ax-mp 5 . . . . . 6 ( 1s /su 1s ) = 1s
106, 9eqtrdi 2793 . . . . 5 (𝑚 = 0s → ( 1s /su (2ss𝑚)) = 1s )
1110fveq2d 6910 . . . 4 (𝑚 = 0s → ( bday ‘( 1s /su (2ss𝑚))) = ( bday ‘ 1s ))
12 bday1s 27876 . . . 4 ( bday ‘ 1s ) = 1o
1311, 12eqtrdi 2793 . . 3 (𝑚 = 0s → ( bday ‘( 1s /su (2ss𝑚))) = 1o)
1413eleq1d 2826 . 2 (𝑚 = 0s → (( bday ‘( 1s /su (2ss𝑚))) ∈ ω ↔ 1o ∈ ω))
15 oveq2 7439 . . . . 5 (𝑚 = 𝑛 → (2ss𝑚) = (2ss𝑛))
1615oveq2d 7447 . . . 4 (𝑚 = 𝑛 → ( 1s /su (2ss𝑚)) = ( 1s /su (2ss𝑛)))
1716fveq2d 6910 . . 3 (𝑚 = 𝑛 → ( bday ‘( 1s /su (2ss𝑚))) = ( bday ‘( 1s /su (2ss𝑛))))
1817eleq1d 2826 . 2 (𝑚 = 𝑛 → (( bday ‘( 1s /su (2ss𝑚))) ∈ ω ↔ ( bday ‘( 1s /su (2ss𝑛))) ∈ ω))
19 oveq2 7439 . . . . 5 (𝑚 = (𝑛 +s 1s ) → (2ss𝑚) = (2ss(𝑛 +s 1s )))
2019oveq2d 7447 . . . 4 (𝑚 = (𝑛 +s 1s ) → ( 1s /su (2ss𝑚)) = ( 1s /su (2ss(𝑛 +s 1s ))))
2120fveq2d 6910 . . 3 (𝑚 = (𝑛 +s 1s ) → ( bday ‘( 1s /su (2ss𝑚))) = ( bday ‘( 1s /su (2ss(𝑛 +s 1s )))))
2221eleq1d 2826 . 2 (𝑚 = (𝑛 +s 1s ) → (( bday ‘( 1s /su (2ss𝑚))) ∈ ω ↔ ( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ∈ ω))
23 oveq2 7439 . . . . 5 (𝑚 = 𝑁 → (2ss𝑚) = (2ss𝑁))
2423oveq2d 7447 . . . 4 (𝑚 = 𝑁 → ( 1s /su (2ss𝑚)) = ( 1s /su (2ss𝑁)))
2524fveq2d 6910 . . 3 (𝑚 = 𝑁 → ( bday ‘( 1s /su (2ss𝑚))) = ( bday ‘( 1s /su (2ss𝑁))))
2625eleq1d 2826 . 2 (𝑚 = 𝑁 → (( bday ‘( 1s /su (2ss𝑚))) ∈ ω ↔ ( bday ‘( 1s /su (2ss𝑁))) ∈ ω))
27 1onn 8678 . 2 1o ∈ ω
28 cutpw2 28417 . . . . . . 7 (𝑛 ∈ ℕ0s → ( 1s /su (2ss(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2ss𝑛))}))
2928fveq2d 6910 . . . . . 6 (𝑛 ∈ ℕ0s → ( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) = ( bday ‘({ 0s } |s {( 1s /su (2ss𝑛))})))
30 0sno 27871 . . . . . . . . 9 0s No
3130a1i 11 . . . . . . . 8 (𝑛 ∈ ℕ0s → 0s No )
327a1i 11 . . . . . . . . 9 (𝑛 ∈ ℕ0s → 1s No )
33 expscl 28413 . . . . . . . . . 10 ((2s No 𝑛 ∈ ℕ0s) → (2ss𝑛) ∈ No )
342, 33mpan 690 . . . . . . . . 9 (𝑛 ∈ ℕ0s → (2ss𝑛) ∈ No )
35 2ne0s 28404 . . . . . . . . . 10 2s ≠ 0s
36 expsne0 28414 . . . . . . . . . 10 ((2s No ∧ 2s ≠ 0s𝑛 ∈ ℕ0s) → (2ss𝑛) ≠ 0s )
372, 35, 36mp3an12 1453 . . . . . . . . 9 (𝑛 ∈ ℕ0s → (2ss𝑛) ≠ 0s )
3832, 34, 37divscld 28248 . . . . . . . 8 (𝑛 ∈ ℕ0s → ( 1s /su (2ss𝑛)) ∈ No )
39 muls02 28167 . . . . . . . . . . 11 ((2ss𝑛) ∈ No → ( 0s ·s (2ss𝑛)) = 0s )
4034, 39syl 17 . . . . . . . . . 10 (𝑛 ∈ ℕ0s → ( 0s ·s (2ss𝑛)) = 0s )
41 0slt1s 27874 . . . . . . . . . 10 0s <s 1s
4240, 41eqbrtrdi 5182 . . . . . . . . 9 (𝑛 ∈ ℕ0s → ( 0s ·s (2ss𝑛)) <s 1s )
43 2nns 28402 . . . . . . . . . . . 12 2s ∈ ℕs
44 nnsgt0 28342 . . . . . . . . . . . 12 (2s ∈ ℕs → 0s <s 2s)
4543, 44ax-mp 5 . . . . . . . . . . 11 0s <s 2s
46 expsgt0 28415 . . . . . . . . . . 11 ((2s No 𝑛 ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2ss𝑛))
472, 45, 46mp3an13 1454 . . . . . . . . . 10 (𝑛 ∈ ℕ0s → 0s <s (2ss𝑛))
4831, 32, 34, 47sltmuldivd 28253 . . . . . . . . 9 (𝑛 ∈ ℕ0s → (( 0s ·s (2ss𝑛)) <s 1s ↔ 0s <s ( 1s /su (2ss𝑛))))
4942, 48mpbid 232 . . . . . . . 8 (𝑛 ∈ ℕ0s → 0s <s ( 1s /su (2ss𝑛)))
5031, 38, 49ssltsn 27837 . . . . . . 7 (𝑛 ∈ ℕ0s → { 0s } <<s {( 1s /su (2ss𝑛))})
51 suc0 6459 . . . . . . . . . . . 12 suc ∅ = {∅}
52 bday0s 27873 . . . . . . . . . . . . 13 ( bday ‘ 0s ) = ∅
5352sneqi 4637 . . . . . . . . . . . 12 {( bday ‘ 0s )} = {∅}
54 bdayfn 27818 . . . . . . . . . . . . 13 bday Fn No
55 fnsnfv 6988 . . . . . . . . . . . . 13 (( bday Fn No ∧ 0s No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
5654, 30, 55mp2an 692 . . . . . . . . . . . 12 {( bday ‘ 0s )} = ( bday “ { 0s })
5751, 53, 563eqtr2i 2771 . . . . . . . . . . 11 suc ∅ = ( bday “ { 0s })
5857a1i 11 . . . . . . . . . 10 (𝑛 ∈ ℕ0s → suc ∅ = ( bday “ { 0s }))
59 fnsnfv 6988 . . . . . . . . . . . 12 (( bday Fn No ∧ ( 1s /su (2ss𝑛)) ∈ No ) → {( bday ‘( 1s /su (2ss𝑛)))} = ( bday “ {( 1s /su (2ss𝑛))}))
6054, 59mpan 690 . . . . . . . . . . 11 (( 1s /su (2ss𝑛)) ∈ No → {( bday ‘( 1s /su (2ss𝑛)))} = ( bday “ {( 1s /su (2ss𝑛))}))
6138, 60syl 17 . . . . . . . . . 10 (𝑛 ∈ ℕ0s → {( bday ‘( 1s /su (2ss𝑛)))} = ( bday “ {( 1s /su (2ss𝑛))}))
6258, 61uneq12d 4169 . . . . . . . . 9 (𝑛 ∈ ℕ0s → (suc ∅ ∪ {( bday ‘( 1s /su (2ss𝑛)))}) = (( bday “ { 0s }) ∪ ( bday “ {( 1s /su (2ss𝑛))})))
63 imaundi 6169 . . . . . . . . 9 ( bday “ ({ 0s } ∪ {( 1s /su (2ss𝑛))})) = (( bday “ { 0s }) ∪ ( bday “ {( 1s /su (2ss𝑛))}))
6462, 63eqtr4di 2795 . . . . . . . 8 (𝑛 ∈ ℕ0s → (suc ∅ ∪ {( bday ‘( 1s /su (2ss𝑛)))}) = ( bday “ ({ 0s } ∪ {( 1s /su (2ss𝑛))})))
65 0ss 4400 . . . . . . . . . 10 ∅ ⊆ ( bday ‘( 1s /su (2ss𝑛)))
66 ord0 6437 . . . . . . . . . . 11 Ord ∅
67 bdayelon 27821 . . . . . . . . . . . 12 ( bday ‘( 1s /su (2ss𝑛))) ∈ On
6867onordi 6495 . . . . . . . . . . 11 Ord ( bday ‘( 1s /su (2ss𝑛)))
69 ordsucsssuc 7843 . . . . . . . . . . 11 ((Ord ∅ ∧ Ord ( bday ‘( 1s /su (2ss𝑛)))) → (∅ ⊆ ( bday ‘( 1s /su (2ss𝑛))) ↔ suc ∅ ⊆ suc ( bday ‘( 1s /su (2ss𝑛)))))
7066, 68, 69mp2an 692 . . . . . . . . . 10 (∅ ⊆ ( bday ‘( 1s /su (2ss𝑛))) ↔ suc ∅ ⊆ suc ( bday ‘( 1s /su (2ss𝑛))))
7165, 70mpbi 230 . . . . . . . . 9 suc ∅ ⊆ suc ( bday ‘( 1s /su (2ss𝑛)))
72 fvex 6919 . . . . . . . . . . 11 ( bday ‘( 1s /su (2ss𝑛))) ∈ V
7372sucid 6466 . . . . . . . . . 10 ( bday ‘( 1s /su (2ss𝑛))) ∈ suc ( bday ‘( 1s /su (2ss𝑛)))
74 snssi 4808 . . . . . . . . . 10 (( bday ‘( 1s /su (2ss𝑛))) ∈ suc ( bday ‘( 1s /su (2ss𝑛))) → {( bday ‘( 1s /su (2ss𝑛)))} ⊆ suc ( bday ‘( 1s /su (2ss𝑛))))
7573, 74ax-mp 5 . . . . . . . . 9 {( bday ‘( 1s /su (2ss𝑛)))} ⊆ suc ( bday ‘( 1s /su (2ss𝑛)))
7671, 75unssi 4191 . . . . . . . 8 (suc ∅ ∪ {( bday ‘( 1s /su (2ss𝑛)))}) ⊆ suc ( bday ‘( 1s /su (2ss𝑛)))
7764, 76eqsstrrdi 4029 . . . . . . 7 (𝑛 ∈ ℕ0s → ( bday “ ({ 0s } ∪ {( 1s /su (2ss𝑛))})) ⊆ suc ( bday ‘( 1s /su (2ss𝑛))))
7867onsuci 7859 . . . . . . . 8 suc ( bday ‘( 1s /su (2ss𝑛))) ∈ On
79 scutbdaybnd 27860 . . . . . . . 8 (({ 0s } <<s {( 1s /su (2ss𝑛))} ∧ suc ( bday ‘( 1s /su (2ss𝑛))) ∈ On ∧ ( bday “ ({ 0s } ∪ {( 1s /su (2ss𝑛))})) ⊆ suc ( bday ‘( 1s /su (2ss𝑛)))) → ( bday ‘({ 0s } |s {( 1s /su (2ss𝑛))})) ⊆ suc ( bday ‘( 1s /su (2ss𝑛))))
8078, 79mp3an2 1451 . . . . . . 7 (({ 0s } <<s {( 1s /su (2ss𝑛))} ∧ ( bday “ ({ 0s } ∪ {( 1s /su (2ss𝑛))})) ⊆ suc ( bday ‘( 1s /su (2ss𝑛)))) → ( bday ‘({ 0s } |s {( 1s /su (2ss𝑛))})) ⊆ suc ( bday ‘( 1s /su (2ss𝑛))))
8150, 77, 80syl2anc 584 . . . . . 6 (𝑛 ∈ ℕ0s → ( bday ‘({ 0s } |s {( 1s /su (2ss𝑛))})) ⊆ suc ( bday ‘( 1s /su (2ss𝑛))))
8229, 81eqsstrd 4018 . . . . 5 (𝑛 ∈ ℕ0s → ( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ⊆ suc ( bday ‘( 1s /su (2ss𝑛))))
83 bdayelon 27821 . . . . . 6 ( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ∈ On
84 onsssuc 6474 . . . . . 6 ((( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ∈ On ∧ suc ( bday ‘( 1s /su (2ss𝑛))) ∈ On) → (( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ⊆ suc ( bday ‘( 1s /su (2ss𝑛))) ↔ ( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ∈ suc suc ( bday ‘( 1s /su (2ss𝑛)))))
8583, 78, 84mp2an 692 . . . . 5 (( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ⊆ suc ( bday ‘( 1s /su (2ss𝑛))) ↔ ( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ∈ suc suc ( bday ‘( 1s /su (2ss𝑛))))
8682, 85sylib 218 . . . 4 (𝑛 ∈ ℕ0s → ( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ∈ suc suc ( bday ‘( 1s /su (2ss𝑛))))
87 peano2 7912 . . . . 5 (( bday ‘( 1s /su (2ss𝑛))) ∈ ω → suc ( bday ‘( 1s /su (2ss𝑛))) ∈ ω)
88 peano2 7912 . . . . 5 (suc ( bday ‘( 1s /su (2ss𝑛))) ∈ ω → suc suc ( bday ‘( 1s /su (2ss𝑛))) ∈ ω)
8987, 88syl 17 . . . 4 (( bday ‘( 1s /su (2ss𝑛))) ∈ ω → suc suc ( bday ‘( 1s /su (2ss𝑛))) ∈ ω)
90 elnn 7898 . . . 4 ((( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ∈ suc suc ( bday ‘( 1s /su (2ss𝑛))) ∧ suc suc ( bday ‘( 1s /su (2ss𝑛))) ∈ ω) → ( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ∈ ω)
9186, 89, 90syl2an 596 . . 3 ((𝑛 ∈ ℕ0s ∧ ( bday ‘( 1s /su (2ss𝑛))) ∈ ω) → ( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ∈ ω)
9291ex 412 . 2 (𝑛 ∈ ℕ0s → (( bday ‘( 1s /su (2ss𝑛))) ∈ ω → ( bday ‘( 1s /su (2ss(𝑛 +s 1s )))) ∈ ω))
9314, 18, 22, 26, 27, 92n0sind 28337 1 (𝑁 ∈ ℕ0s → ( bday ‘( 1s /su (2ss𝑁))) ∈ ω)
Colors of variables: wff setvar class
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  1oc1o 8499   No csur 27684   <s cslt 27685   bday cbday 27686   <<s csslt 27825   |s cscut 27827   0s c0s 27867   1s c1s 27868   +s cadds 27992   ·s cmuls 28132   /su cdivs 28213  0scnn0s 28318  scnns 28319  2sc2s 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)
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