Theorem bdaypw2n0bnd 28737

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 28615	. . 3 ⊢ (𝑁 ∈ ℕ0s → (𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )))
2		n0lts1e0 28641	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → (𝐴 <s 1s ↔ 𝐴 = 0s ))
3		oveq1 7424	. . . . . . . . . 10 ⊢ (𝐴 = 0s → (𝐴 /su 1s ) = ( 0s /su 1s ))
4		0no 28082	. . . . . . . . . . 11 ⊢ 0s ∈ No
5		divs1 28477	. . . . . . . . . . 11 ⊢ ( 0s ∈ No → ( 0s /su 1s ) = 0s )
6	4, 5	ax-mp 5	. . . . . . . . . 10 ⊢ ( 0s /su 1s ) = 0s
7	3, 6	eqtrdi 2813	. . . . . . . . 9 ⊢ (𝐴 = 0s → (𝐴 /su 1s ) = 0s )
8	7	fveq2d 6886	. . . . . . . 8 ⊢ (𝐴 = 0s → ( bday ‘(𝐴 /su 1s )) = ( bday ‘ 0s ))
9		bday0 28084	. . . . . . . 8 ⊢ ( bday ‘ 0s ) = ∅
10	8, 9	eqtrdi 2813	. . . . . . 7 ⊢ (𝐴 = 0s → ( bday ‘(𝐴 /su 1s )) = ∅)
11		0ss 4353	. . . . . . 7 ⊢ ∅ ⊆ suc ∅
12	10, 11	eqsstrdi 3978	. . . . . 6 ⊢ (𝐴 = 0s → ( bday ‘(𝐴 /su 1s )) ⊆ suc ∅)
13	2, 12	biimtrdi 256	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → (𝐴 <s 1s → ( bday ‘(𝐴 /su 1s )) ⊆ suc ∅))
14		oveq2 7425	. . . . . . . 8 ⊢ (𝑁 = 0s → (2s↑s𝑁) = (2s↑s 0s ))
15		2no 28692	. . . . . . . . 9 ⊢ 2s ∈ No
16		exps0 28700	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
18	14, 17	eqtrdi 2813	. . . . . . 7 ⊢ (𝑁 = 0s → (2s↑s𝑁) = 1s )
19	18	breq2d 5119	. . . . . 6 ⊢ (𝑁 = 0s → (𝐴 <s (2s↑s𝑁) ↔ 𝐴 <s 1s ))
20	18	oveq2d 7433	. . . . . . . 8 ⊢ (𝑁 = 0s → (𝐴 /su (2s↑s𝑁)) = (𝐴 /su 1s ))
21	20	fveq2d 6886	. . . . . . 7 ⊢ (𝑁 = 0s → ( bday ‘(𝐴 /su (2s↑s𝑁))) = ( bday ‘(𝐴 /su 1s )))
22		fveq2 6882	. . . . . . . . 9 ⊢ (𝑁 = 0s → ( bday ‘𝑁) = ( bday ‘ 0s ))
23	22, 9	eqtrdi 2813	. . . . . . . 8 ⊢ (𝑁 = 0s → ( bday ‘𝑁) = ∅)
24	23	suceqd 6429	. . . . . . 7 ⊢ (𝑁 = 0s → suc ( bday ‘𝑁) = suc ∅)
25	21, 24	sseq12d 3967	. . . . . 6 ⊢ (𝑁 = 0s → (( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁) ↔ ( bday ‘(𝐴 /su 1s )) ⊆ suc ∅))
26	19, 25	imbi12d 347	. . . . 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 249	. . . 4 ⊢ (𝑁 = 0s → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))))
28		bdaypw2n0bndlem 28736	. . . . . . . 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 1137	. . . . . . 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 33	. . . . . 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 7425	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → (2s↑s𝑁) = (2s↑s(𝑥 +s 1s )))
32	31	breq2d 5119	. . . . . . . 8 ⊢ (𝑁 = (𝑥 +s 1s ) → (𝐴 <s (2s↑s𝑁) ↔ 𝐴 <s (2s↑s(𝑥 +s 1s ))))
33	31	oveq2d 7433	. . . . . . . . . 10 ⊢ (𝑁 = (𝑥 +s 1s ) → (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s(𝑥 +s 1s ))))
34	33	fveq2d 6886	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → ( bday ‘(𝐴 /su (2s↑s𝑁))) = ( bday ‘(𝐴 /su (2s↑s(𝑥 +s 1s )))))
35		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑁 = (𝑥 +s 1s ) → ( bday ‘𝑁) = ( bday ‘(𝑥 +s 1s )))
36	35	suceqd 6429	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → suc ( bday ‘𝑁) = suc ( bday ‘(𝑥 +s 1s )))
37	34, 36	sseq12d 3967	. . . . . . . 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 347	. . . . . . 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 343	. . . . . 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 250	. . . . 5 ⊢ (𝑥 ∈ ℕ0s → (𝑁 = (𝑥 +s 1s ) → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁)))))
41	40	rexlimiv 3158	. . . 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 871	. . 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 18	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))))
44	43	3imp21 1131	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ∧ 𝐴 <s (2s↑s𝑁)) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   ⊆ wss 3902  ∅c0 4282   class class class wbr 5107  suc csuc 6363  ‘cfv 6537  (class class class)co 7417   No csur 27884   <s clts 27885   bday cbday 27886   0_s c0s 28078   1_s c1s 28079   +_s cadds 28232   /_su cdivs 28460  ℕ_0scn0s 28585  2_sc2s 28683  ↑_scexps 28685

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-dc 10452

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-nadd 8658  df-no 27887  df-lts 27888  df-bday 27889  df-les 27989  df-slts 28031  df-cuts 28033  df-0s 28080  df-1s 28081  df-made 28100  df-old 28101  df-left 28103  df-right 28104  df-norec 28211  df-norec2 28222  df-adds 28233  df-negs 28294  df-subs 28295  df-muls 28380  df-divs 28461  df-ons 28525  df-seqs 28557  df-n0s 28587  df-nns 28588  df-zs 28652  df-2s 28684  df-exps 28686

This theorem is used by:  bdaypw2bnd  28738  z12bdaylem2  28744  z12bdaylem  28757