Theorem bdaypw2bnd 28624

Description: Birthday bounding rule for non-negative dyadic rationals. (Contributed by Scott Fenton, 25-Feb-2026.)

Hypotheses

Ref	Expression
bdaypw2bnd.1	⊢ (𝜑 → 𝑁 ∈ ℕ0s)
bdaypw2bnd.2	⊢ (𝜑 → 𝑋 ∈ ℕ0s)
bdaypw2bnd.3	⊢ (𝜑 → 𝑌 ∈ ℕ0s)
bdaypw2bnd.4	⊢ (𝜑 → 𝑃 ∈ ℕ0s)
bdaypw2bnd.5	⊢ (𝜑 → 𝑌 <s (2s↑s𝑃))
bdaypw2bnd.6	⊢ (𝜑 → (𝑋 +s 𝑃) <s 𝑁)

Assertion

Ref	Expression
bdaypw2bnd	⊢ (𝜑 → ( bday ‘(𝑋 +s (𝑌 /su (2s↑s𝑃)))) ⊆ ( bday ‘𝑁))

Proof of Theorem bdaypw2bnd

Step	Hyp	Ref	Expression
1		bdaypw2bnd.2	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℕ0s)
2	1	n0nod 28484	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
3		bdaypw2bnd.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℕ0s)
4	3	n0nod 28484	. . . 4 ⊢ (𝜑 → 𝑌 ∈ No )
5		bdaypw2bnd.4	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
6	4, 5	pw2divscld 28598	. . 3 ⊢ (𝜑 → (𝑌 /su (2s↑s𝑃)) ∈ No )
7		addbday 28177	. . 3 ⊢ ((𝑋 ∈ No ∧ (𝑌 /su (2s↑s𝑃)) ∈ No ) → ( bday ‘(𝑋 +s (𝑌 /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑋) +no ( bday ‘(𝑌 /su (2s↑s𝑃)))))
8	2, 6, 7	syl2anc 595	. 2 ⊢ (𝜑 → ( bday ‘(𝑋 +s (𝑌 /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑋) +no ( bday ‘(𝑌 /su (2s↑s𝑃)))))
9		bdaypw2bnd.5	. . . . 5 ⊢ (𝜑 → 𝑌 <s (2s↑s𝑃))
10		bdaypw2n0bnd 28623	. . . . 5 ⊢ ((𝑌 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s ∧ 𝑌 <s (2s↑s𝑃)) → ( bday ‘(𝑌 /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃))
11	3, 5, 9, 10	syl3anc 1396	. . . 4 ⊢ (𝜑 → ( bday ‘(𝑌 /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃))
12		bdayon 27911	. . . . 5 ⊢ ( bday ‘(𝑌 /su (2s↑s𝑃))) ∈ On
13		bdayon 27911	. . . . . 6 ⊢ ( bday ‘𝑃) ∈ On
14	13	onsuci 7835	. . . . 5 ⊢ suc ( bday ‘𝑃) ∈ On
15		bdayon 27911	. . . . 5 ⊢ ( bday ‘𝑋) ∈ On
16		naddss2 8677	. . . . 5 ⊢ ((( bday ‘(𝑌 /su (2s↑s𝑃))) ∈ On ∧ suc ( bday ‘𝑃) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘(𝑌 /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃) ↔ (( bday ‘𝑋) +no ( bday ‘(𝑌 /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑋) +no suc ( bday ‘𝑃))))
17	12, 14, 15, 16	mp3an 1487	. . . 4 ⊢ (( bday ‘(𝑌 /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃) ↔ (( bday ‘𝑋) +no ( bday ‘(𝑌 /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑋) +no suc ( bday ‘𝑃)))
18	11, 17	sylib 221	. . 3 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑌 /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑋) +no suc ( bday ‘𝑃)))
19		bdayn0p1 28528	. . . . . 6 ⊢ (𝑃 ∈ ℕ0s → ( bday ‘(𝑃 +s 1s )) = suc ( bday ‘𝑃))
20	5, 19	syl 18	. . . . 5 ⊢ (𝜑 → ( bday ‘(𝑃 +s 1s )) = suc ( bday ‘𝑃))
21	20	oveq2d 7427	. . . 4 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑃 +s 1s ))) = (( bday ‘𝑋) +no suc ( bday ‘𝑃)))
22		n0on 28495	. . . . . . 7 ⊢ (𝑋 ∈ ℕ0s → 𝑋 ∈ Ons)
23	1, 22	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Ons)
24		peano2n0s 28489	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0s → (𝑃 +s 1s ) ∈ ℕ0s)
25	5, 24	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑃 +s 1s ) ∈ ℕ0s)
26		n0on 28495	. . . . . . 7 ⊢ ((𝑃 +s 1s ) ∈ ℕ0s → (𝑃 +s 1s ) ∈ Ons)
27	25, 26	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑃 +s 1s ) ∈ Ons)
28		addonbday 28438	. . . . . 6 ⊢ ((𝑋 ∈ Ons ∧ (𝑃 +s 1s ) ∈ Ons) → ( bday ‘(𝑋 +s (𝑃 +s 1s ))) = (( bday ‘𝑋) +no ( bday ‘(𝑃 +s 1s ))))
29	23, 27, 28	syl2anc 595	. . . . 5 ⊢ (𝜑 → ( bday ‘(𝑋 +s (𝑃 +s 1s ))) = (( bday ‘𝑋) +no ( bday ‘(𝑃 +s 1s ))))
30		bdaypw2bnd.6	. . . . . 6 ⊢ (𝜑 → (𝑋 +s 𝑃) <s 𝑁)
31		bdaypw2bnd.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
32		n0on 28495	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
33	31, 32	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ Ons)
34		n0addscl 28503	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℕ0s ∧ (𝑃 +s 1s ) ∈ ℕ0s) → (𝑋 +s (𝑃 +s 1s )) ∈ ℕ0s)
35	1, 25, 34	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 +s (𝑃 +s 1s )) ∈ ℕ0s)
36		n0on 28495	. . . . . . . . . 10 ⊢ ((𝑋 +s (𝑃 +s 1s )) ∈ ℕ0s → (𝑋 +s (𝑃 +s 1s )) ∈ Ons)
37	35, 36	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s (𝑃 +s 1s )) ∈ Ons)
38		onlts 28426	. . . . . . . . 9 ⊢ ((𝑁 ∈ Ons ∧ (𝑋 +s (𝑃 +s 1s )) ∈ Ons) → (𝑁 <s (𝑋 +s (𝑃 +s 1s )) ↔ ( bday ‘𝑁) ∈ ( bday ‘(𝑋 +s (𝑃 +s 1s )))))
39	33, 37, 38	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (𝑁 <s (𝑋 +s (𝑃 +s 1s )) ↔ ( bday ‘𝑁) ∈ ( bday ‘(𝑋 +s (𝑃 +s 1s )))))
40	39	notbid 321	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑁 <s (𝑋 +s (𝑃 +s 1s )) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑋 +s (𝑃 +s 1s )))))
41		n0addscl 28503	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s) → (𝑋 +s 𝑃) ∈ ℕ0s)
42	1, 5, 41	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s 𝑃) ∈ ℕ0s)
43		n0ltsp1le 28524	. . . . . . . . 9 ⊢ (((𝑋 +s 𝑃) ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑋 +s 𝑃) <s 𝑁 ↔ ((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁))
44	42, 31, 43	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 +s 𝑃) <s 𝑁 ↔ ((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁))
45	5	n0nod 28484	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ No )
46		1no 27969	. . . . . . . . . . 11 ⊢ 1s ∈ No
47	46	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1s ∈ No )
48	2, 45, 47	addsassd 28165	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 +s 𝑃) +s 1s ) = (𝑋 +s (𝑃 +s 1s )))
49	48	breq1d 5121	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁 ↔ (𝑋 +s (𝑃 +s 1s )) ≤s 𝑁))
50	35	n0nod 28484	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s (𝑃 +s 1s )) ∈ No )
51	31	n0nod 28484	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ No )
52		lenlts 27882	. . . . . . . . 9 ⊢ (((𝑋 +s (𝑃 +s 1s )) ∈ No ∧ 𝑁 ∈ No ) → ((𝑋 +s (𝑃 +s 1s )) ≤s 𝑁 ↔ ¬ 𝑁 <s (𝑋 +s (𝑃 +s 1s ))))
53	50, 51, 52	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 +s (𝑃 +s 1s )) ≤s 𝑁 ↔ ¬ 𝑁 <s (𝑋 +s (𝑃 +s 1s ))))
54	44, 49, 53	3bitrd 308	. . . . . . 7 ⊢ (𝜑 → ((𝑋 +s 𝑃) <s 𝑁 ↔ ¬ 𝑁 <s (𝑋 +s (𝑃 +s 1s ))))
55		bdayon 27911	. . . . . . . . 9 ⊢ ( bday ‘(𝑋 +s (𝑃 +s 1s ))) ∈ On
56		bdayon 27911	. . . . . . . . 9 ⊢ ( bday ‘𝑁) ∈ On
57		ontri1 6396	. . . . . . . . 9 ⊢ ((( bday ‘(𝑋 +s (𝑃 +s 1s ))) ∈ On ∧ ( bday ‘𝑁) ∈ On) → (( bday ‘(𝑋 +s (𝑃 +s 1s ))) ⊆ ( bday ‘𝑁) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑋 +s (𝑃 +s 1s )))))
58	55, 56, 57	mp2an 704	. . . . . . . 8 ⊢ (( bday ‘(𝑋 +s (𝑃 +s 1s ))) ⊆ ( bday ‘𝑁) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑋 +s (𝑃 +s 1s ))))
59	58	a1i 11	. . . . . . 7 ⊢ (𝜑 → (( bday ‘(𝑋 +s (𝑃 +s 1s ))) ⊆ ( bday ‘𝑁) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑋 +s (𝑃 +s 1s )))))
60	40, 54, 59	3bitr4d 314	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑃) <s 𝑁 ↔ ( bday ‘(𝑋 +s (𝑃 +s 1s ))) ⊆ ( bday ‘𝑁)))
61	30, 60	mpbid 235	. . . . 5 ⊢ (𝜑 → ( bday ‘(𝑋 +s (𝑃 +s 1s ))) ⊆ ( bday ‘𝑁))
62	29, 61	eqsstrrd 3978	. . . 4 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑃 +s 1s ))) ⊆ ( bday ‘𝑁))
63	21, 62	eqsstrrd 3978	. . 3 ⊢ (𝜑 → (( bday ‘𝑋) +no suc ( bday ‘𝑃)) ⊆ ( bday ‘𝑁))
64	18, 63	sstrd 3953	. 2 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑌 /su (2s↑s𝑃)))) ⊆ ( bday ‘𝑁))
65	8, 64	sstrd 3953	1 ⊢ (𝜑 → ( bday ‘(𝑋 +s (𝑌 /su (2s↑s𝑃)))) ⊆ ( bday ‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149   ⊆ wss 3911   class class class wbr 5111  Oncon0 6361  suc csuc 6363  ‘cfv 6537  (class class class)co 7411   +no cnadd 8651   No csur 27770   <s clts 27771   bday cbday 27772   ≤s cles 27874   1_s c1s 27965   +_s cadds 28118   /_su cdivs 28346  On_scons 28410  ℕ_0scn0s 28471  2_sc2s 28569  ↑_scexps 28571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-dc 10430

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-oadd 8457  df-nadd 8652  df-no 27773  df-lts 27774  df-bday 27775  df-les 27875  df-slts 27917  df-cuts 27919  df-0s 27966  df-1s 27967  df-made 27986  df-old 27987  df-left 27989  df-right 27990  df-norec 28097  df-norec2 28108  df-adds 28119  df-negs 28180  df-subs 28181  df-muls 28266  df-divs 28347  df-ons 28411  df-seqs 28443  df-n0s 28473  df-nns 28474  df-zs 28538  df-2s 28570  df-exps 28572

This theorem is referenced by:  bdayfinbndlem1  28626