Theorem bdaypw2n0bnd 28883

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 28761	. . 3 ⊢ (𝑁 ∈ ℕ0s → (𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )))
2		n0lts1e0 28787	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → (𝐴 <s 1s ↔ 𝐴 = 0s ))
3		oveq1 7453	. . . . . . . . . 10 ⊢ (𝐴 = 0s → (𝐴 /su 1s ) = ( 0s /su 1s ))
4		0no 28228	. . . . . . . . . . 11 ⊢ 0s ∈ No
5		divs1 28623	. . . . . . . . . . 11 ⊢ ( 0s ∈ No → ( 0s /su 1s ) = 0s )
6	4, 5	ax-mp 5	. . . . . . . . . 10 ⊢ ( 0s /su 1s ) = 0s
7	3, 6	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝐴 = 0s → (𝐴 /su 1s ) = 0s )
8	7	fveq2d 6915	. . . . . . . 8 ⊢ (𝐴 = 0s → (bday‘(𝐴 /su 1s )) = (bday‘ 0s ))
9		bday0 28230	. . . . . . . 8 ⊢ (bday‘ 0s ) = ∅
10	8, 9	eqtrdi 2814	. . . . . . 7 ⊢ (𝐴 = 0s → (bday‘(𝐴 /su 1s )) = ∅)
11		0ss 4353	. . . . . . 7 ⊢ ∅ ⊆ suc ∅
12	10, 11	eqsstrdi 3957	. . . . . 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 7454	. . . . . . . 8 ⊢ (𝑁 = 0s → (2s↑s𝑁) = (2s↑s 0s ))
15		2no 28838	. . . . . . . . 9 ⊢ 2s ∈ No
16		exps0 28846	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
18	14, 17	eqtrdi 2814	. . . . . . 7 ⊢ (𝑁 = 0s → (2s↑s𝑁) = 1s )
19	18	breq2d 5118	. . . . . 6 ⊢ (𝑁 = 0s → (𝐴 <s (2s↑s𝑁) ↔ 𝐴 <s 1s ))
20	18	oveq2d 7462	. . . . . . . 8 ⊢ (𝑁 = 0s → (𝐴 /su (2s↑s𝑁)) = (𝐴 /su 1s ))
21	20	fveq2d 6915	. . . . . . 7 ⊢ (𝑁 = 0s → (bday‘(𝐴 /su (2s↑s𝑁))) = (bday‘(𝐴 /su 1s )))
22		fveq2 6911	. . . . . . . . 9 ⊢ (𝑁 = 0s → (bday‘𝑁) = (bday‘ 0s ))
23	22, 9	eqtrdi 2814	. . . . . . . 8 ⊢ (𝑁 = 0s → (bday‘𝑁) = ∅)
24	23	suceqd 6451	. . . . . . 7 ⊢ (𝑁 = 0s → suc (bday‘𝑁) = suc ∅)
25	21, 24	sseq12d 3946	. . . . . 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 28882	. . . . . . . 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 7454	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → (2s↑s𝑁) = (2s↑s(𝑥 +s 1s )))
32	31	breq2d 5118	. . . . . . . 8 ⊢ (𝑁 = (𝑥 +s 1s ) → (𝐴 <s (2s↑s𝑁) ↔ 𝐴 <s (2s↑s(𝑥 +s 1s ))))
33	31	oveq2d 7462	. . . . . . . . . 10 ⊢ (𝑁 = (𝑥 +s 1s ) → (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s(𝑥 +s 1s ))))
34	33	fveq2d 6915	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → (bday‘(𝐴 /su (2s↑s𝑁))) = (bday‘(𝐴 /su (2s↑s(𝑥 +s 1s )))))
35		fveq2 6911	. . . . . . . . . 10 ⊢ (𝑁 = (𝑥 +s 1s ) → (bday‘𝑁) = (bday‘(𝑥 +s 1s )))
36	35	suceqd 6451	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → suc (bday‘𝑁) = suc (bday‘(𝑥 +s 1s )))
37	34, 36	sseq12d 3946	. . . . . . . 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 3160	. . . 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 2147  ∃wrex 3090   ⊆ wss 3900  ∅c0 4281   class class class wbr 5106  suc csuc 6385  ‘cfv 6561  (class class class)co 7446  Nocsur 28030   <s clts 28031  bdaycbday 28032   0_s c0s 28224   1_s c1s 28225   +_s cadds 28378   /_su cdivs 28606  ℕ_0scn0s 28731  2_sc2s 28829  ↑_scexps 28831

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-dc 10554

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-oadd 8511  df-nadd 8706  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-0s 28226  df-1s 28227  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec 28357  df-norec2 28368  df-adds 28379  df-negs 28440  df-subs 28441  df-muls 28526  df-divs 28607  df-ons 28671  df-seqs 28703  df-n0s 28733  df-nns 28734  df-zs 28798  df-2s 28830  df-exps 28832

This theorem is used by:  bdaypw2bnd  28884  z12bdaylem2  28890  z12bdaylem  28903