Theorem bdaypw2bnd 28475

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 28335	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
3		bdaypw2bnd.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℕ0s)
4	3	n0nod 28335	. . . 4 ⊢ (𝜑 → 𝑌 ∈ No )
5		bdaypw2bnd.4	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
6	4, 5	pw2divscld 28449	. . 3 ⊢ (𝜑 → (𝑌 /su (2s↑s𝑃)) ∈ No )
7		addbday 28028	. . 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 28474	. . . . 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 27762	. . . . 5 ⊢ ( bday ‘(𝑌 /su (2s↑s𝑃))) ∈ On
13		bdayon 27762	. . . . . 6 ⊢ ( bday ‘𝑃) ∈ On
14	13	onsuci 7785	. . . . 5 ⊢ suc ( bday ‘𝑃) ∈ On
15		bdayon 27762	. . . . 5 ⊢ ( bday ‘𝑋) ∈ On
16		naddss2 8621	. . . . 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 28379	. . . . . 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 7378	. . . 4 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑃 +s 1s ))) = (( bday ‘𝑋) +no suc ( bday ‘𝑃)))
22		n0on 28346	. . . . . . 7 ⊢ (𝑋 ∈ ℕ0s → 𝑋 ∈ Ons)
23	1, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Ons)
24		peano2n0s 28340	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0s → (𝑃 +s 1s ) ∈ ℕ0s)
25	5, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑃 +s 1s ) ∈ ℕ0s)
26		n0on 28346	. . . . . . 7 ⊢ ((𝑃 +s 1s ) ∈ ℕ0s → (𝑃 +s 1s ) ∈ Ons)
27	25, 26	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑃 +s 1s ) ∈ Ons)
28		addonbday 28289	. . . . . 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 28346	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
33	31, 32	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ Ons)
34		n0addscl 28354	. . . . . . . . . . 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 28346	. . . . . . . . . 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 28277	. . . . . . . . 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 28354	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s) → (𝑋 +s 𝑃) ∈ ℕ0s)
42	1, 5, 41	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s 𝑃) ∈ ℕ0s)
43		n0ltsp1le 28375	. . . . . . . . 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 28335	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ No )
46		1no 27820	. . . . . . . . . . 11 ⊢ 1s ∈ No
47	46	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1s ∈ No )
48	2, 45, 47	addsassd 28016	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 +s 𝑃) +s 1s ) = (𝑋 +s (𝑃 +s 1s )))
49	48	breq1d 5096	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁 ↔ (𝑋 +s (𝑃 +s 1s )) ≤s 𝑁))
50	35	n0nod 28335	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s (𝑃 +s 1s )) ∈ No )
51	31	n0nod 28335	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ No )
52		lenlts 27734	. . . . . . . . 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 27762	. . . . . . . . 9 ⊢ ( bday ‘(𝑋 +s (𝑃 +s 1s ))) ∈ On
56		bdayon 27762	. . . . . . . . 9 ⊢ ( bday ‘𝑁) ∈ On
57		ontri1 6353	. . . . . . . . 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 3958	. . . 4 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑃 +s 1s ))) ⊆ ( bday ‘𝑁))
63	21, 62	eqsstrrd 3958	. . 3 ⊢ (𝜑 → (( bday ‘𝑋) +no suc ( bday ‘𝑃)) ⊆ ( bday ‘𝑁))
64	18, 63	sstrd 3933	. 2 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑌 /su (2s↑s𝑃)))) ⊆ ( bday ‘𝑁))
65	8, 64	sstrd 3933	1 ⊢ (𝜑 → ( bday ‘(𝑋 +s (𝑌 /su (2s↑s𝑃)))) ⊆ ( bday ‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890   class class class wbr 5086  Oncon0 6319  suc csuc 6321  ‘cfv 6494  (class class class)co 7362   +no cnadd 8596   No csur 27621   <s clts 27622   bday cbday 27623   ≤s cles 27726   1_s c1s 27816   +_s cadds 27969   /_su cdivs 28197  On_scons 28261  ℕ_0scn0s 28322  2_sc2s 28420  ↑_scexps 28422

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-dc 10363

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-oadd 8404  df-nadd 8597  df-no 27624  df-lts 27625  df-bday 27626  df-les 27727  df-slts 27768  df-cuts 27770  df-0s 27817  df-1s 27818  df-made 27837  df-old 27838  df-left 27840  df-right 27841  df-norec 27948  df-norec2 27959  df-adds 27970  df-negs 28031  df-subs 28032  df-muls 28117  df-divs 28198  df-ons 28262  df-seqs 28294  df-n0s 28324  df-nns 28325  df-zs 28389  df-2s 28421  df-exps 28423

This theorem is referenced by:  bdayfinbndlem1  28477