Theorem bdaypw2bnd 28851

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 28711	. . 3 ⊢ (𝜑 → 𝑋 ∈ No)
3		bdaypw2bnd.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℕ0s)
4	3	n0nod 28711	. . . 4 ⊢ (𝜑 → 𝑌 ∈ No)
5		bdaypw2bnd.4	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
6	4, 5	pw2divscld 28825	. . 3 ⊢ (𝜑 → (𝑌 /su (2s↑s𝑃)) ∈ No)
7		addbday 28404	. . 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 28850	. . . . 5 ⊢ ((𝑌 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s ∧ 𝑌 <s (2s↑s𝑃)) → (bday‘(𝑌 /su (2s↑s𝑃))) ⊆ suc (bday‘𝑃))
11	3, 5, 9, 10	syl3anc 1398	. . . 4 ⊢ (𝜑 → (bday‘(𝑌 /su (2s↑s𝑃))) ⊆ suc (bday‘𝑃))
12		bdayon 28138	. . . . 5 ⊢ (bday‘(𝑌 /su (2s↑s𝑃))) ∈ On
13		bdayon 28138	. . . . . 6 ⊢ (bday‘𝑃) ∈ On
14	13	onsuci 7850	. . . . 5 ⊢ suc (bday‘𝑃) ∈ On
15		bdayon 28138	. . . . 5 ⊢ (bday‘𝑋) ∈ On
16		naddss2 8700	. . . . 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 1490	. . . 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 28755	. . . . . 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 7436	. . . 4 ⊢ (𝜑 → ((bday‘𝑋) +no (bday‘(𝑃 +s 1s ))) = ((bday‘𝑋) +no suc (bday‘𝑃)))
22		n0on 28722	. . . . . . 7 ⊢ (𝑋 ∈ ℕ0s → 𝑋 ∈ Ons)
23	1, 22	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Ons)
24		peano2n0s 28716	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0s → (𝑃 +s 1s ) ∈ ℕ0s)
25	5, 24	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑃 +s 1s ) ∈ ℕ0s)
26		n0on 28722	. . . . . . 7 ⊢ ((𝑃 +s 1s ) ∈ ℕ0s → (𝑃 +s 1s ) ∈ Ons)
27	25, 26	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑃 +s 1s ) ∈ Ons)
28		addonbday 28665	. . . . . 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 28722	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
33	31, 32	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ Ons)
34		n0addscl 28730	. . . . . . . . . . 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 28722	. . . . . . . . . 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 28653	. . . . . . . . 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 28730	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s) → (𝑋 +s 𝑃) ∈ ℕ0s)
42	1, 5, 41	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s 𝑃) ∈ ℕ0s)
43		n0ltsp1le 28751	. . . . . . . . 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 28711	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ No)
46		1no 28196	. . . . . . . . . . 11 ⊢ 1s ∈ No
47	46	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1s ∈ No)
48	2, 45, 47	addsassd 28392	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 +s 𝑃) +s 1s ) = (𝑋 +s (𝑃 +s 1s )))
49	48	breq1d 5112	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁 ↔ (𝑋 +s (𝑃 +s 1s )) ≤s 𝑁))
50	35	n0nod 28711	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s (𝑃 +s 1s )) ∈ No)
51	31	n0nod 28711	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ No)
52		lenlts 28109	. . . . . . . . 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 28138	. . . . . . . . 9 ⊢ (bday‘(𝑋 +s (𝑃 +s 1s ))) ∈ On
56		bdayon 28138	. . . . . . . . 9 ⊢ (bday‘𝑁) ∈ On
57		ontri1 6397	. . . . . . . . 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 3943	. . . 4 ⊢ (𝜑 → ((bday‘𝑋) +no (bday‘(𝑃 +s 1s ))) ⊆ (bday‘𝑁))
63	21, 62	eqsstrrd 3943	. . 3 ⊢ (𝜑 → ((bday‘𝑋) +no suc (bday‘𝑃)) ⊆ (bday‘𝑁))
64	18, 63	sstrd 3918	. 2 ⊢ (𝜑 → ((bday‘𝑋) +no (bday‘(𝑌 /su (2s↑s𝑃)))) ⊆ (bday‘𝑁))
65	8, 64	sstrd 3918	1 ⊢ (𝜑 → (bday‘(𝑋 +s (𝑌 /su (2s↑s𝑃)))) ⊆ (bday‘𝑁))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   class class class wbr 5102  Oncon0 6362  suc csuc 6364  ‘cfv 6538  (class class class)co 7420   +no cnadd 8674  Nocsur 27997   <s clts 27998  bdaycbday 27999   ≤s cles 28101   1_s c1s 28192   +_s cadds 28345   /_su cdivs 28573  On_scons 28637  ℕ_0scn0s 28698  2_sc2s 28796  ↑_scexps 28798

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-dc 10524

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-nadd 8675  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-1s 28194  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-norec2 28335  df-adds 28346  df-negs 28407  df-subs 28408  df-muls 28493  df-divs 28574  df-ons 28638  df-seqs 28670  df-n0s 28700  df-nns 28701  df-zs 28765  df-2s 28797  df-exps 28799

This theorem is used by:  bdayfinbndlem1  28853