Theorem bdaypw2bnd 28878

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 28738	. . 3 ⊢ (𝜑 → 𝑋 ∈ No)
3		bdaypw2bnd.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℕ0s)
4	3	n0nod 28738	. . . 4 ⊢ (𝜑 → 𝑌 ∈ No)
5		bdaypw2bnd.4	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
6	4, 5	pw2divscld 28852	. . 3 ⊢ (𝜑 → (𝑌 /su (2s↑s𝑃)) ∈ No)
7		addbday 28431	. . 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 596	. 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 28877	. . . . 5 ⊢ ((𝑌 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s ∧ 𝑌 <s (2s↑s𝑃)) → (bday‘(𝑌 /su (2s↑s𝑃))) ⊆ suc (bday‘𝑃))
11	3, 5, 9, 10	syl3anc 1399	. . . 4 ⊢ (𝜑 → (bday‘(𝑌 /su (2s↑s𝑃))) ⊆ suc (bday‘𝑃))
12		bdayon 28165	. . . . 5 ⊢ (bday‘(𝑌 /su (2s↑s𝑃))) ∈ On
13		bdayon 28165	. . . . . 6 ⊢ (bday‘𝑃) ∈ On
14	13	onsuci 7871	. . . . 5 ⊢ suc (bday‘𝑃) ∈ On
15		bdayon 28165	. . . . 5 ⊢ (bday‘𝑋) ∈ On
16		naddss2 8725	. . . . 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 1491	. . . 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 28782	. . . . . 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 7456	. . . 4 ⊢ (𝜑 → ((bday‘𝑋) +no (bday‘(𝑃 +s 1s ))) = ((bday‘𝑋) +no suc (bday‘𝑃)))
22		n0on 28749	. . . . . . 7 ⊢ (𝑋 ∈ ℕ0s → 𝑋 ∈ Ons)
23	1, 22	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Ons)
24		peano2n0s 28743	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0s → (𝑃 +s 1s ) ∈ ℕ0s)
25	5, 24	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑃 +s 1s ) ∈ ℕ0s)
26		n0on 28749	. . . . . . 7 ⊢ ((𝑃 +s 1s ) ∈ ℕ0s → (𝑃 +s 1s ) ∈ Ons)
27	25, 26	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑃 +s 1s ) ∈ Ons)
28		addonbday 28692	. . . . . 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 596	. . . . 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 28749	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
33	31, 32	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ Ons)
34		n0addscl 28757	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℕ0s ∧ (𝑃 +s 1s ) ∈ ℕ0s) → (𝑋 +s (𝑃 +s 1s )) ∈ ℕ0s)
35	1, 25, 34	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 +s (𝑃 +s 1s )) ∈ ℕ0s)
36		n0on 28749	. . . . . . . . . 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 28680	. . . . . . . . 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 596	. . . . . . . 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 28757	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s) → (𝑋 +s 𝑃) ∈ ℕ0s)
42	1, 5, 41	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s 𝑃) ∈ ℕ0s)
43		n0ltsp1le 28778	. . . . . . . . 9 ⊢ (((𝑋 +s 𝑃) ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑋 +s 𝑃) <s 𝑁 ↔ ((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁))
44	42, 31, 43	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 +s 𝑃) <s 𝑁 ↔ ((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁))
45	5	n0nod 28738	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ No)
46		1no 28223	. . . . . . . . . . 11 ⊢ 1s ∈ No
47	46	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1s ∈ No)
48	2, 45, 47	addsassd 28419	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 +s 𝑃) +s 1s ) = (𝑋 +s (𝑃 +s 1s )))
49	48	breq1d 5116	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁 ↔ (𝑋 +s (𝑃 +s 1s )) ≤s 𝑁))
50	35	n0nod 28738	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s (𝑃 +s 1s )) ∈ No)
51	31	n0nod 28738	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ No)
52		lenlts 28136	. . . . . . . . 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 596	. . . . . . . 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 28165	. . . . . . . . 9 ⊢ (bday‘(𝑋 +s (𝑃 +s 1s ))) ∈ On
56		bdayon 28165	. . . . . . . . 9 ⊢ (bday‘𝑁) ∈ On
57		ontri1 6412	. . . . . . . . 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 705	. . . . . . . 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 3948	. . . 4 ⊢ (𝜑 → ((bday‘𝑋) +no (bday‘(𝑃 +s 1s ))) ⊆ (bday‘𝑁))
63	21, 62	eqsstrrd 3948	. . 3 ⊢ (𝜑 → ((bday‘𝑋) +no suc (bday‘𝑃)) ⊆ (bday‘𝑁))
64	18, 63	sstrd 3923	. 2 ⊢ (𝜑 → ((bday‘𝑋) +no (bday‘(𝑌 /su (2s↑s𝑃)))) ⊆ (bday‘𝑁))
65	8, 64	sstrd 3923	1 ⊢ (𝜑 → (bday‘(𝑋 +s (𝑌 /su (2s↑s𝑃)))) ⊆ (bday‘𝑁))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   class class class wbr 5106  Oncon0 6377  suc csuc 6379  ‘cfv 6555  (class class class)co 7440   +no cnadd 8699  Nocsur 28024   <s clts 28025  bdaycbday 28026   ≤s cles 28128   1_s c1s 28219   +_s cadds 28372   /_su cdivs 28600  On_scons 28664  ℕ_0scn0s 28725  2_sc2s 28823  ↑_scexps 28825

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 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-dc 10548

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 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-nadd 8700  df-no 28027  df-lts 28028  df-bday 28029  df-les 28129  df-slts 28171  df-cuts 28173  df-0s 28220  df-1s 28221  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-norec 28351  df-norec2 28362  df-adds 28373  df-negs 28434  df-subs 28435  df-muls 28520  df-divs 28601  df-ons 28665  df-seqs 28697  df-n0s 28727  df-nns 28728  df-zs 28792  df-2s 28824  df-exps 28826

This theorem is used by:  bdayfinbndlem1  28880