Theorem bdaypw2n0bnd 28456

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 28334	. . 3 ⊢ (𝑁 ∈ ℕ0s → (𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )))
2		n0lts1e0 28360	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → (𝐴 <s 1s ↔ 𝐴 = 0s ))
3		oveq1 7374	. . . . . . . . . 10 ⊢ (𝐴 = 0s → (𝐴 /su 1s ) = ( 0s /su 1s ))
4		0no 27801	. . . . . . . . . . 11 ⊢ 0s ∈ No
5		divs1 28196	. . . . . . . . . . 11 ⊢ ( 0s ∈ No → ( 0s /su 1s ) = 0s )
6	4, 5	ax-mp 5	. . . . . . . . . 10 ⊢ ( 0s /su 1s ) = 0s
7	3, 6	eqtrdi 2787	. . . . . . . . 9 ⊢ (𝐴 = 0s → (𝐴 /su 1s ) = 0s )
8	7	fveq2d 6844	. . . . . . . 8 ⊢ (𝐴 = 0s → ( bday ‘(𝐴 /su 1s )) = ( bday ‘ 0s ))
9		bday0 27803	. . . . . . . 8 ⊢ ( bday ‘ 0s ) = ∅
10	8, 9	eqtrdi 2787	. . . . . . 7 ⊢ (𝐴 = 0s → ( bday ‘(𝐴 /su 1s )) = ∅)
11		0ss 4340	. . . . . . 7 ⊢ ∅ ⊆ suc ∅
12	10, 11	eqsstrdi 3966	. . . . . 6 ⊢ (𝐴 = 0s → ( bday ‘(𝐴 /su 1s )) ⊆ suc ∅)
13	2, 12	biimtrdi 253	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → (𝐴 <s 1s → ( bday ‘(𝐴 /su 1s )) ⊆ suc ∅))
14		oveq2 7375	. . . . . . . 8 ⊢ (𝑁 = 0s → (2s↑s𝑁) = (2s↑s 0s ))
15		2no 28411	. . . . . . . . 9 ⊢ 2s ∈ No
16		exps0 28419	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
18	14, 17	eqtrdi 2787	. . . . . . 7 ⊢ (𝑁 = 0s → (2s↑s𝑁) = 1s )
19	18	breq2d 5097	. . . . . 6 ⊢ (𝑁 = 0s → (𝐴 <s (2s↑s𝑁) ↔ 𝐴 <s 1s ))
20	18	oveq2d 7383	. . . . . . . 8 ⊢ (𝑁 = 0s → (𝐴 /su (2s↑s𝑁)) = (𝐴 /su 1s ))
21	20	fveq2d 6844	. . . . . . 7 ⊢ (𝑁 = 0s → ( bday ‘(𝐴 /su (2s↑s𝑁))) = ( bday ‘(𝐴 /su 1s )))
22		fveq2 6840	. . . . . . . . 9 ⊢ (𝑁 = 0s → ( bday ‘𝑁) = ( bday ‘ 0s ))
23	22, 9	eqtrdi 2787	. . . . . . . 8 ⊢ (𝑁 = 0s → ( bday ‘𝑁) = ∅)
24	23	suceqd 6390	. . . . . . 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 344	. . . . 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 246	. . . 4 ⊢ (𝑁 = 0s → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))))
28		bdaypw2n0bndlem 28455	. . . . . . . 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 1120	. . . . . . 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 7375	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → (2s↑s𝑁) = (2s↑s(𝑥 +s 1s )))
32	31	breq2d 5097	. . . . . . . 8 ⊢ (𝑁 = (𝑥 +s 1s ) → (𝐴 <s (2s↑s𝑁) ↔ 𝐴 <s (2s↑s(𝑥 +s 1s ))))
33	31	oveq2d 7383	. . . . . . . . . 10 ⊢ (𝑁 = (𝑥 +s 1s ) → (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s(𝑥 +s 1s ))))
34	33	fveq2d 6844	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → ( bday ‘(𝐴 /su (2s↑s𝑁))) = ( bday ‘(𝐴 /su (2s↑s(𝑥 +s 1s )))))
35		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑁 = (𝑥 +s 1s ) → ( bday ‘𝑁) = ( bday ‘(𝑥 +s 1s )))
36	35	suceqd 6390	. . . . . . . . 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 344	. . . . . . 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 340	. . . . . 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 247	. . . . 5 ⊢ (𝑥 ∈ ℕ0s → (𝑁 = (𝑥 +s 1s ) → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁)))))
41	40	rexlimiv 3131	. . . 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 858	. . 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 1114	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ∧ 𝐴 <s (2s↑s𝑁)) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))

Syntax hints:   → wi 4   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ⊆ wss 3889  ∅c0 4273   class class class wbr 5085  suc csuc 6325  ‘cfv 6498  (class class class)co 7367   No csur 27603   <s clts 27604   bday cbday 27605   0_s c0s 27797   1_s c1s 27798   +_s cadds 27951   /_su cdivs 28179  ℕ_0scn0s 28304  2_sc2s 28402  ↑_scexps 28404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-dc 10368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-nadd 8602  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-1s 27800  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-subs 28014  df-muls 28099  df-divs 28180  df-ons 28244  df-seqs 28276  df-n0s 28306  df-nns 28307  df-zs 28371  df-2s 28403  df-exps 28405

This theorem is referenced by:  bdaypw2bnd  28457  z12bdaylem2  28463  z12bdaylem  28476