Theorem bdaypw2bnd 28457

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 28317	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
3		bdaypw2bnd.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℕ0s)
4	3	n0nod 28317	. . . 4 ⊢ (𝜑 → 𝑌 ∈ No )
5		bdaypw2bnd.4	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
6	4, 5	pw2divscld 28431	. . 3 ⊢ (𝜑 → (𝑌 /su (2s↑s𝑃)) ∈ No )
7		addbday 28010	. . 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 585	. 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 28456	. . . . 5 ⊢ ((𝑌 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s ∧ 𝑌 <s (2s↑s𝑃)) → ( bday ‘(𝑌 /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃))
11	3, 5, 9, 10	syl3anc 1374	. . . 4 ⊢ (𝜑 → ( bday ‘(𝑌 /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃))
12		bdayon 27744	. . . . 5 ⊢ ( bday ‘(𝑌 /su (2s↑s𝑃))) ∈ On
13		bdayon 27744	. . . . . 6 ⊢ ( bday ‘𝑃) ∈ On
14	13	onsuci 7790	. . . . 5 ⊢ suc ( bday ‘𝑃) ∈ On
15		bdayon 27744	. . . . 5 ⊢ ( bday ‘𝑋) ∈ On
16		naddss2 8626	. . . . 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 1464	. . . 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 218	. . 3 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑌 /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑋) +no suc ( bday ‘𝑃)))
19		bdayn0p1 28361	. . . . . 6 ⊢ (𝑃 ∈ ℕ0s → ( bday ‘(𝑃 +s 1s )) = suc ( bday ‘𝑃))
20	5, 19	syl 17	. . . . 5 ⊢ (𝜑 → ( bday ‘(𝑃 +s 1s )) = suc ( bday ‘𝑃))
21	20	oveq2d 7383	. . . 4 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑃 +s 1s ))) = (( bday ‘𝑋) +no suc ( bday ‘𝑃)))
22		n0on 28328	. . . . . . 7 ⊢ (𝑋 ∈ ℕ0s → 𝑋 ∈ Ons)
23	1, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Ons)
24		peano2n0s 28322	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0s → (𝑃 +s 1s ) ∈ ℕ0s)
25	5, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑃 +s 1s ) ∈ ℕ0s)
26		n0on 28328	. . . . . . 7 ⊢ ((𝑃 +s 1s ) ∈ ℕ0s → (𝑃 +s 1s ) ∈ Ons)
27	25, 26	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑃 +s 1s ) ∈ Ons)
28		addonbday 28271	. . . . . 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 585	. . . . 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 28328	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
33	31, 32	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ Ons)
34		n0addscl 28336	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℕ0s ∧ (𝑃 +s 1s ) ∈ ℕ0s) → (𝑋 +s (𝑃 +s 1s )) ∈ ℕ0s)
35	1, 25, 34	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 +s (𝑃 +s 1s )) ∈ ℕ0s)
36		n0on 28328	. . . . . . . . . 10 ⊢ ((𝑋 +s (𝑃 +s 1s )) ∈ ℕ0s → (𝑋 +s (𝑃 +s 1s )) ∈ Ons)
37	35, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s (𝑃 +s 1s )) ∈ Ons)
38		onlts 28259	. . . . . . . . 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 585	. . . . . . . 8 ⊢ (𝜑 → (𝑁 <s (𝑋 +s (𝑃 +s 1s )) ↔ ( bday ‘𝑁) ∈ ( bday ‘(𝑋 +s (𝑃 +s 1s )))))
40	39	notbid 318	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑁 <s (𝑋 +s (𝑃 +s 1s )) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑋 +s (𝑃 +s 1s )))))
41		n0addscl 28336	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s) → (𝑋 +s 𝑃) ∈ ℕ0s)
42	1, 5, 41	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s 𝑃) ∈ ℕ0s)
43		n0ltsp1le 28357	. . . . . . . . 9 ⊢ (((𝑋 +s 𝑃) ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑋 +s 𝑃) <s 𝑁 ↔ ((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁))
44	42, 31, 43	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 +s 𝑃) <s 𝑁 ↔ ((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁))
45	5	n0nod 28317	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ No )
46		1no 27802	. . . . . . . . . . 11 ⊢ 1s ∈ No
47	46	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1s ∈ No )
48	2, 45, 47	addsassd 27998	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 +s 𝑃) +s 1s ) = (𝑋 +s (𝑃 +s 1s )))
49	48	breq1d 5095	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁 ↔ (𝑋 +s (𝑃 +s 1s )) ≤s 𝑁))
50	35	n0nod 28317	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s (𝑃 +s 1s )) ∈ No )
51	31	n0nod 28317	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ No )
52		lenlts 27716	. . . . . . . . 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 585	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 +s (𝑃 +s 1s )) ≤s 𝑁 ↔ ¬ 𝑁 <s (𝑋 +s (𝑃 +s 1s ))))
54	44, 49, 53	3bitrd 305	. . . . . . 7 ⊢ (𝜑 → ((𝑋 +s 𝑃) <s 𝑁 ↔ ¬ 𝑁 <s (𝑋 +s (𝑃 +s 1s ))))
55		bdayon 27744	. . . . . . . . 9 ⊢ ( bday ‘(𝑋 +s (𝑃 +s 1s ))) ∈ On
56		bdayon 27744	. . . . . . . . 9 ⊢ ( bday ‘𝑁) ∈ On
57		ontri1 6357	. . . . . . . . 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 693	. . . . . . . 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 311	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑃) <s 𝑁 ↔ ( bday ‘(𝑋 +s (𝑃 +s 1s ))) ⊆ ( bday ‘𝑁)))
61	30, 60	mpbid 232	. . . . 5 ⊢ (𝜑 → ( bday ‘(𝑋 +s (𝑃 +s 1s ))) ⊆ ( bday ‘𝑁))
62	29, 61	eqsstrrd 3957	. . . 4 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑃 +s 1s ))) ⊆ ( bday ‘𝑁))
63	21, 62	eqsstrrd 3957	. . 3 ⊢ (𝜑 → (( bday ‘𝑋) +no suc ( bday ‘𝑃)) ⊆ ( bday ‘𝑁))
64	18, 63	sstrd 3932	. 2 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑌 /su (2s↑s𝑃)))) ⊆ ( bday ‘𝑁))
65	8, 64	sstrd 3932	1 ⊢ (𝜑 → ( bday ‘(𝑋 +s (𝑌 /su (2s↑s𝑃)))) ⊆ ( bday ‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889   class class class wbr 5085  Oncon0 6323  suc csuc 6325  ‘cfv 6498  (class class class)co 7367   +no cnadd 8601   No csur 27603   <s clts 27604   bday cbday 27605   ≤s cles 27708   1_s c1s 27798   +_s cadds 27951   /_su cdivs 28179  On_scons 28243  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-isom 6507  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:  bdayfinbndlem1  28459