Theorem bdaypw2bnd 28461

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 28321	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
3		bdaypw2bnd.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℕ0s)
4	3	n0nod 28321	. . . 4 ⊢ (𝜑 → 𝑌 ∈ No )
5		bdaypw2bnd.4	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
6	4, 5	pw2divscld 28435	. . 3 ⊢ (𝜑 → (𝑌 /su (2s↑s𝑃)) ∈ No )
7		addbday 28014	. . 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 584	. 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 28460	. . . . 5 ⊢ ((𝑌 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s ∧ 𝑌 <s (2s↑s𝑃)) → ( bday ‘(𝑌 /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃))
11	3, 5, 9, 10	syl3anc 1373	. . . 4 ⊢ (𝜑 → ( bday ‘(𝑌 /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃))
12		bdayon 27748	. . . . 5 ⊢ ( bday ‘(𝑌 /su (2s↑s𝑃))) ∈ On
13		bdayon 27748	. . . . . 6 ⊢ ( bday ‘𝑃) ∈ On
14	13	onsuci 7781	. . . . 5 ⊢ suc ( bday ‘𝑃) ∈ On
15		bdayon 27748	. . . . 5 ⊢ ( bday ‘𝑋) ∈ On
16		naddss2 8618	. . . . 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 1463	. . . 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 28365	. . . . . 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 7374	. . . 4 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑃 +s 1s ))) = (( bday ‘𝑋) +no suc ( bday ‘𝑃)))
22		n0on 28332	. . . . . . 7 ⊢ (𝑋 ∈ ℕ0s → 𝑋 ∈ Ons)
23	1, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Ons)
24		peano2n0s 28326	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0s → (𝑃 +s 1s ) ∈ ℕ0s)
25	5, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑃 +s 1s ) ∈ ℕ0s)
26		n0on 28332	. . . . . . 7 ⊢ ((𝑃 +s 1s ) ∈ ℕ0s → (𝑃 +s 1s ) ∈ Ons)
27	25, 26	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑃 +s 1s ) ∈ Ons)
28		addonbday 28275	. . . . . 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 584	. . . . 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 28332	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
33	31, 32	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ Ons)
34		n0addscl 28340	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℕ0s ∧ (𝑃 +s 1s ) ∈ ℕ0s) → (𝑋 +s (𝑃 +s 1s )) ∈ ℕ0s)
35	1, 25, 34	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 +s (𝑃 +s 1s )) ∈ ℕ0s)
36		n0on 28332	. . . . . . . . . 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 28263	. . . . . . . . 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 584	. . . . . . . 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 28340	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s) → (𝑋 +s 𝑃) ∈ ℕ0s)
42	1, 5, 41	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s 𝑃) ∈ ℕ0s)
43		n0ltsp1le 28361	. . . . . . . . 9 ⊢ (((𝑋 +s 𝑃) ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑋 +s 𝑃) <s 𝑁 ↔ ((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁))
44	42, 31, 43	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 +s 𝑃) <s 𝑁 ↔ ((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁))
45	5	n0nod 28321	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ No )
46		1no 27806	. . . . . . . . . . 11 ⊢ 1s ∈ No
47	46	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1s ∈ No )
48	2, 45, 47	addsassd 28002	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 +s 𝑃) +s 1s ) = (𝑋 +s (𝑃 +s 1s )))
49	48	breq1d 5108	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 +s 𝑃) +s 1s ) ≤s 𝑁 ↔ (𝑋 +s (𝑃 +s 1s )) ≤s 𝑁))
50	35	n0nod 28321	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 +s (𝑃 +s 1s )) ∈ No )
51	31	n0nod 28321	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ No )
52		lenlts 27720	. . . . . . . . 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 584	. . . . . . . 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 27748	. . . . . . . . 9 ⊢ ( bday ‘(𝑋 +s (𝑃 +s 1s ))) ∈ On
56		bdayon 27748	. . . . . . . . 9 ⊢ ( bday ‘𝑁) ∈ On
57		ontri1 6351	. . . . . . . . 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 692	. . . . . . . 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 3969	. . . 4 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑃 +s 1s ))) ⊆ ( bday ‘𝑁))
63	21, 62	eqsstrrd 3969	. . 3 ⊢ (𝜑 → (( bday ‘𝑋) +no suc ( bday ‘𝑃)) ⊆ ( bday ‘𝑁))
64	18, 63	sstrd 3944	. 2 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘(𝑌 /su (2s↑s𝑃)))) ⊆ ( bday ‘𝑁))
65	8, 64	sstrd 3944	1 ⊢ (𝜑 → ( bday ‘(𝑋 +s (𝑌 /su (2s↑s𝑃)))) ⊆ ( bday ‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113   ⊆ wss 3901   class class class wbr 5098  Oncon0 6317  suc csuc 6319  ‘cfv 6492  (class class class)co 7358   +no cnadd 8593   No csur 27607   <s clts 27608   bday cbday 27609   ≤s cles 27712   1_s c1s 27802   +_s cadds 27955   /_su cdivs 28183  On_scons 28247  ℕ_0scn0s 28308  2_sc2s 28406  ↑_scexps 28408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-dc 10356

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-nadd 8594  df-no 27610  df-lts 27611  df-bday 27612  df-les 27713  df-slts 27754  df-cuts 27756  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27934  df-norec2 27945  df-adds 27956  df-negs 28017  df-subs 28018  df-muls 28103  df-divs 28184  df-ons 28248  df-seqs 28280  df-n0s 28310  df-nns 28311  df-zs 28375  df-2s 28407  df-exps 28409

This theorem is referenced by:  bdayfinbndlem1  28463