Theorem bdaypw2n0bnd 28481

Description: Upper bound for the birthday of a proper fraction of a power of two. This is actually a strict equality when 𝐴 is odd, but we do not need this for the rest of our development. (Contributed by Scott Fenton, 22-Feb-2026.)

Assertion

Ref	Expression
bdaypw2n0bnd	⊢ ((𝐴 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ∧ 𝐴 <s (2s↑s𝑁)) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))

Proof of Theorem bdaypw2n0bnd

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0s0suc 28359	. . 3 ⊢ (𝑁 ∈ ℕ0s → (𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )))
2		n0lts1e0 28385	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → (𝐴 <s 1s ↔ 𝐴 = 0s ))
3		oveq1 7370	. . . . . . . . . 10 ⊢ (𝐴 = 0s → (𝐴 /su 1s ) = ( 0s /su 1s ))
4		0no 27826	. . . . . . . . . . 11 ⊢ 0s ∈ No
5		divs1 28221	. . . . . . . . . . 11 ⊢ ( 0s ∈ No → ( 0s /su 1s ) = 0s )
6	4, 5	ax-mp 5	. . . . . . . . . 10 ⊢ ( 0s /su 1s ) = 0s
7	3, 6	eqtrdi 2791	. . . . . . . . 9 ⊢ (𝐴 = 0s → (𝐴 /su 1s ) = 0s )
8	7	fveq2d 6838	. . . . . . . 8 ⊢ (𝐴 = 0s → ( bday ‘(𝐴 /su 1s )) = ( bday ‘ 0s ))
9		bday0 27828	. . . . . . . 8 ⊢ ( bday ‘ 0s ) = ∅
10	8, 9	eqtrdi 2791	. . . . . . 7 ⊢ (𝐴 = 0s → ( bday ‘(𝐴 /su 1s )) = ∅)
11		0ss 4335	. . . . . . 7 ⊢ ∅ ⊆ suc ∅
12	10, 11	eqsstrdi 3966	. . . . . 6 ⊢ (𝐴 = 0s → ( bday ‘(𝐴 /su 1s )) ⊆ suc ∅)
13	2, 12	biimtrdi 254	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → (𝐴 <s 1s → ( bday ‘(𝐴 /su 1s )) ⊆ suc ∅))
14		oveq2 7371	. . . . . . . 8 ⊢ (𝑁 = 0s → (2s↑s𝑁) = (2s↑s 0s ))
15		2no 28436	. . . . . . . . 9 ⊢ 2s ∈ No
16		exps0 28444	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
18	14, 17	eqtrdi 2791	. . . . . . 7 ⊢ (𝑁 = 0s → (2s↑s𝑁) = 1s )
19	18	breq2d 5091	. . . . . 6 ⊢ (𝑁 = 0s → (𝐴 <s (2s↑s𝑁) ↔ 𝐴 <s 1s ))
20	18	oveq2d 7379	. . . . . . . 8 ⊢ (𝑁 = 0s → (𝐴 /su (2s↑s𝑁)) = (𝐴 /su 1s ))
21	20	fveq2d 6838	. . . . . . 7 ⊢ (𝑁 = 0s → ( bday ‘(𝐴 /su (2s↑s𝑁))) = ( bday ‘(𝐴 /su 1s )))
22		fveq2 6834	. . . . . . . . 9 ⊢ (𝑁 = 0s → ( bday ‘𝑁) = ( bday ‘ 0s ))
23	22, 9	eqtrdi 2791	. . . . . . . 8 ⊢ (𝑁 = 0s → ( bday ‘𝑁) = ∅)
24	23	suceqd 6384	. . . . . . 7 ⊢ (𝑁 = 0s → suc ( bday ‘𝑁) = suc ∅)
25	21, 24	sseq12d 3955	. . . . . 6 ⊢ (𝑁 = 0s → (( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁) ↔ ( bday ‘(𝐴 /su 1s )) ⊆ suc ∅))
26	19, 25	imbi12d 345	. . . . 5 ⊢ (𝑁 = 0s → ((𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁)) ↔ (𝐴 <s 1s → ( bday ‘(𝐴 /su 1s )) ⊆ suc ∅)))
27	13, 26	imbitrrid 247	. . . 4 ⊢ (𝑁 = 0s → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))))
28		bdaypw2n0bndlem 28480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s ∧ 𝐴 <s (2s↑s(𝑥 +s 1s ))) → ( bday ‘(𝐴 /su (2s↑s(𝑥 +s 1s )))) ⊆ suc ( bday ‘(𝑥 +s 1s )))
29	28	3exp 1125	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → (𝑥 ∈ ℕ0s → (𝐴 <s (2s↑s(𝑥 +s 1s )) → ( bday ‘(𝐴 /su (2s↑s(𝑥 +s 1s )))) ⊆ suc ( bday ‘(𝑥 +s 1s )))))
30	29	com12 32	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s(𝑥 +s 1s )) → ( bday ‘(𝐴 /su (2s↑s(𝑥 +s 1s )))) ⊆ suc ( bday ‘(𝑥 +s 1s )))))
31		oveq2 7371	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → (2s↑s𝑁) = (2s↑s(𝑥 +s 1s )))
32	31	breq2d 5091	. . . . . . . 8 ⊢ (𝑁 = (𝑥 +s 1s ) → (𝐴 <s (2s↑s𝑁) ↔ 𝐴 <s (2s↑s(𝑥 +s 1s ))))
33	31	oveq2d 7379	. . . . . . . . . 10 ⊢ (𝑁 = (𝑥 +s 1s ) → (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s(𝑥 +s 1s ))))
34	33	fveq2d 6838	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → ( bday ‘(𝐴 /su (2s↑s𝑁))) = ( bday ‘(𝐴 /su (2s↑s(𝑥 +s 1s )))))
35		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑁 = (𝑥 +s 1s ) → ( bday ‘𝑁) = ( bday ‘(𝑥 +s 1s )))
36	35	suceqd 6384	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → suc ( bday ‘𝑁) = suc ( bday ‘(𝑥 +s 1s )))
37	34, 36	sseq12d 3955	. . . . . . . 8 ⊢ (𝑁 = (𝑥 +s 1s ) → (( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁) ↔ ( bday ‘(𝐴 /su (2s↑s(𝑥 +s 1s )))) ⊆ suc ( bday ‘(𝑥 +s 1s ))))
38	32, 37	imbi12d 345	. . . . . . 7 ⊢ (𝑁 = (𝑥 +s 1s ) → ((𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁)) ↔ (𝐴 <s (2s↑s(𝑥 +s 1s )) → ( bday ‘(𝐴 /su (2s↑s(𝑥 +s 1s )))) ⊆ suc ( bday ‘(𝑥 +s 1s )))))
39	38	imbi2d 341	. . . . . 6 ⊢ (𝑁 = (𝑥 +s 1s ) → ((𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))) ↔ (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s(𝑥 +s 1s )) → ( bday ‘(𝐴 /su (2s↑s(𝑥 +s 1s )))) ⊆ suc ( bday ‘(𝑥 +s 1s ))))))
40	30, 39	syl5ibrcom 248	. . . . 5 ⊢ (𝑥 ∈ ℕ0s → (𝑁 = (𝑥 +s 1s ) → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁)))))
41	40	rexlimiv 3134	. . . 4 ⊢ (∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s ) → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))))
42	27, 41	jaoi 863	. . 3 ⊢ ((𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))))
43	1, 42	syl 17	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))))
44	43	3imp21 1119	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ∧ 𝐴 <s (2s↑s𝑁)) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))

Syntax hints:   → wi 4   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3064   ⊆ wss 3890  ∅c0 4268   class class class wbr 5079  suc csuc 6319  ‘cfv 6492  (class class class)co 7363   No csur 27628   <s clts 27629   bday cbday 27630   0_s c0s 27822   1_s c1s 27823   +_s cadds 27976   /_su cdivs 28204  ℕ_0scn0s 28329  2_sc2s 28427  ↑_scexps 28429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-dc 10366

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-1s 27825  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038  df-subs 28039  df-muls 28124  df-divs 28205  df-ons 28269  df-seqs 28301  df-n0s 28331  df-nns 28332  df-zs 28396  df-2s 28428  df-exps 28430

This theorem is referenced by:  bdaypw2bnd  28482  z12bdaylem2  28488  z12bdaylem  28501