Theorem bdaypw2n0bnd 28534

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 28412	. . 3 ⊢ (𝑁 ∈ ℕ0s → (𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )))
2		n0lts1e0 28438	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → (𝐴 <s 1s ↔ 𝐴 = 0s ))
3		oveq1 7399	. . . . . . . . . 10 ⊢ (𝐴 = 0s → (𝐴 /su 1s ) = ( 0s /su 1s ))
4		0no 27879	. . . . . . . . . . 11 ⊢ 0s ∈ No
5		divs1 28274	. . . . . . . . . . 11 ⊢ ( 0s ∈ No → ( 0s /su 1s ) = 0s )
6	4, 5	ax-mp 5	. . . . . . . . . 10 ⊢ ( 0s /su 1s ) = 0s
7	3, 6	eqtrdi 2812	. . . . . . . . 9 ⊢ (𝐴 = 0s → (𝐴 /su 1s ) = 0s )
8	7	fveq2d 6867	. . . . . . . 8 ⊢ (𝐴 = 0s → ( bday ‘(𝐴 /su 1s )) = ( bday ‘ 0s ))
9		bday0 27881	. . . . . . . 8 ⊢ ( bday ‘ 0s ) = ∅
10	8, 9	eqtrdi 2812	. . . . . . 7 ⊢ (𝐴 = 0s → ( bday ‘(𝐴 /su 1s )) = ∅)
11		0ss 4353	. . . . . . 7 ⊢ ∅ ⊆ suc ∅
12	10, 11	eqsstrdi 3980	. . . . . 6 ⊢ (𝐴 = 0s → ( bday ‘(𝐴 /su 1s )) ⊆ suc ∅)
13	2, 12	biimtrdi 255	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → (𝐴 <s 1s → ( bday ‘(𝐴 /su 1s )) ⊆ suc ∅))
14		oveq2 7400	. . . . . . . 8 ⊢ (𝑁 = 0s → (2s↑s𝑁) = (2s↑s 0s ))
15		2no 28489	. . . . . . . . 9 ⊢ 2s ∈ No
16		exps0 28497	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
18	14, 17	eqtrdi 2812	. . . . . . 7 ⊢ (𝑁 = 0s → (2s↑s𝑁) = 1s )
19	18	breq2d 5111	. . . . . 6 ⊢ (𝑁 = 0s → (𝐴 <s (2s↑s𝑁) ↔ 𝐴 <s 1s ))
20	18	oveq2d 7408	. . . . . . . 8 ⊢ (𝑁 = 0s → (𝐴 /su (2s↑s𝑁)) = (𝐴 /su 1s ))
21	20	fveq2d 6867	. . . . . . 7 ⊢ (𝑁 = 0s → ( bday ‘(𝐴 /su (2s↑s𝑁))) = ( bday ‘(𝐴 /su 1s )))
22		fveq2 6863	. . . . . . . . 9 ⊢ (𝑁 = 0s → ( bday ‘𝑁) = ( bday ‘ 0s ))
23	22, 9	eqtrdi 2812	. . . . . . . 8 ⊢ (𝑁 = 0s → ( bday ‘𝑁) = ∅)
24	23	suceqd 6409	. . . . . . 7 ⊢ (𝑁 = 0s → suc ( bday ‘𝑁) = suc ∅)
25	21, 24	sseq12d 3969	. . . . . 6 ⊢ (𝑁 = 0s → (( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁) ↔ ( bday ‘(𝐴 /su 1s )) ⊆ suc ∅))
26	19, 25	imbi12d 346	. . . . 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 248	. . . 4 ⊢ (𝑁 = 0s → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))))
28		bdaypw2n0bndlem 28533	. . . . . . . 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 1131	. . . . . . 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 7400	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → (2s↑s𝑁) = (2s↑s(𝑥 +s 1s )))
32	31	breq2d 5111	. . . . . . . 8 ⊢ (𝑁 = (𝑥 +s 1s ) → (𝐴 <s (2s↑s𝑁) ↔ 𝐴 <s (2s↑s(𝑥 +s 1s ))))
33	31	oveq2d 7408	. . . . . . . . . 10 ⊢ (𝑁 = (𝑥 +s 1s ) → (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s(𝑥 +s 1s ))))
34	33	fveq2d 6867	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → ( bday ‘(𝐴 /su (2s↑s𝑁))) = ( bday ‘(𝐴 /su (2s↑s(𝑥 +s 1s )))))
35		fveq2 6863	. . . . . . . . . 10 ⊢ (𝑁 = (𝑥 +s 1s ) → ( bday ‘𝑁) = ( bday ‘(𝑥 +s 1s )))
36	35	suceqd 6409	. . . . . . . . 9 ⊢ (𝑁 = (𝑥 +s 1s ) → suc ( bday ‘𝑁) = suc ( bday ‘(𝑥 +s 1s )))
37	34, 36	sseq12d 3969	. . . . . . . 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 346	. . . . . . 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 342	. . . . . 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 249	. . . . 5 ⊢ (𝑥 ∈ ℕ0s → (𝑁 = (𝑥 +s 1s ) → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s𝑁) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁)))))
41	40	rexlimiv 3155	. . . 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 868	. . 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 1125	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ∧ 𝐴 <s (2s↑s𝑁)) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))

Syntax hints:   → wi 4   ∨ wo 858   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∃wrex 3085   ⊆ wss 3904  ∅c0 4285   class class class wbr 5099  suc csuc 6344  ‘cfv 6517  (class class class)co 7392   No csur 27681   <s clts 27682   bday cbday 27683   0_s c0s 27875   1_s c1s 27876   +_s cadds 28029   /_su cdivs 28257  ℕ_0scn0s 28382  2_sc2s 28480  ↑_scexps 28482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-dc 10400

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-nadd 8631  df-no 27684  df-lts 27685  df-bday 27686  df-les 27786  df-slts 27828  df-cuts 27830  df-0s 27877  df-1s 27878  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec 28008  df-norec2 28019  df-adds 28030  df-negs 28091  df-subs 28092  df-muls 28177  df-divs 28258  df-ons 28322  df-seqs 28354  df-n0s 28384  df-nns 28385  df-zs 28449  df-2s 28481  df-exps 28483

This theorem is referenced by:  bdaypw2bnd  28535  z12bdaylem2  28541  z12bdaylem  28554