Theorem z12bdaylem2 28701

Description: Lemma for z12bday 28715. Show the first half of the equality. (Contributed by Scott Fenton, 22-Feb-2026.)

Hypotheses

Ref	Expression
z12bdaylem.1	⊢ (𝜑 → 𝑁 ∈ ℕ0s)
z12bdaylem.2	⊢ (𝜑 → 𝑀 ∈ ℕ0s)
z12bdaylem.3	⊢ (𝜑 → 𝑃 ∈ ℕ0s)
z12bdaylem.4	⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃))

Assertion

Ref	Expression
z12bdaylem2	⊢ (𝜑 → ( bday ‘(𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ ( bday ‘((𝑁 +s 𝑃) +s 1s )))

Proof of Theorem z12bdaylem2

Step	Hyp	Ref	Expression
1		z12bdaylem.1	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
2	1	n0nod 28555	. . 3 ⊢ (𝜑 → 𝑁 ∈ No )
3		2no 28649	. . . . . . 7 ⊢ 2s ∈ No
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 2s ∈ No )
5		z12bdaylem.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0s)
6	5	n0nod 28555	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ No )
7	4, 6	mulscld 28365	. . . . 5 ⊢ (𝜑 → (2s ·s 𝑀) ∈ No )
8		1no 28040	. . . . . 6 ⊢ 1s ∈ No
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 1s ∈ No )
10	7, 9	addscld 28210	. . . 4 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ No )
11		z12bdaylem.3	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
12	10, 11	pw2divscld 28669	. . 3 ⊢ (𝜑 → (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) ∈ No )
13		addbday 28248	. . 3 ⊢ ((𝑁 ∈ No ∧ (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) ∈ No ) → ( bday ‘(𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑁) +no ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))))
14	2, 12, 13	syl2anc 596	. 2 ⊢ (𝜑 → ( bday ‘(𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑁) +no ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))))
15		2nns 28648	. . . . . . . 8 ⊢ 2s ∈ ℕs
16		nnn0s 28557	. . . . . . . 8 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ 2s ∈ ℕ0s
18		n0mulscl 28575	. . . . . . 7 ⊢ ((2s ∈ ℕ0s ∧ 𝑀 ∈ ℕ0s) → (2s ·s 𝑀) ∈ ℕ0s)
19	17, 5, 18	sylancr 599	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝑀) ∈ ℕ0s)
20		1n0s 28578	. . . . . 6 ⊢ 1s ∈ ℕ0s
21		n0addscl 28574	. . . . . 6 ⊢ (((2s ·s 𝑀) ∈ ℕ0s ∧ 1s ∈ ℕ0s) → ((2s ·s 𝑀) +s 1s ) ∈ ℕ0s)
22	19, 20, 21	sylancl 598	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ ℕ0s)
23		z12bdaylem.4	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃))
24		bdaypw2n0bnd 28694	. . . . 5 ⊢ ((((2s ·s 𝑀) +s 1s ) ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s ∧ ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃)) → ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃))
25	22, 11, 23, 24	syl3anc 1398	. . . 4 ⊢ (𝜑 → ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃))
26		bdayon 27982	. . . . 5 ⊢ ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ∈ On
27		bdayon 27982	. . . . . 6 ⊢ ( bday ‘𝑃) ∈ On
28	27	onsuci 7844	. . . . 5 ⊢ suc ( bday ‘𝑃) ∈ On
29		bdayon 27982	. . . . 5 ⊢ ( bday ‘𝑁) ∈ On
30		naddss2 8686	. . . . 5 ⊢ ((( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ∈ On ∧ suc ( bday ‘𝑃) ∈ On ∧ ( bday ‘𝑁) ∈ On) → (( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃) ↔ (( bday ‘𝑁) +no ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑁) +no suc ( bday ‘𝑃))))
31	26, 28, 29, 30	mp3an 1490	. . . 4 ⊢ (( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃) ↔ (( bday ‘𝑁) +no ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
32	25, 31	sylib 221	. . 3 ⊢ (𝜑 → (( bday ‘𝑁) +no ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
33		n0addscl 28574	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s) → (𝑁 +s 𝑃) ∈ ℕ0s)
34	1, 11, 33	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑁 +s 𝑃) ∈ ℕ0s)
35		bdayn0p1 28599	. . . . 5 ⊢ ((𝑁 +s 𝑃) ∈ ℕ0s → ( bday ‘((𝑁 +s 𝑃) +s 1s )) = suc ( bday ‘(𝑁 +s 𝑃)))
36	34, 35	syl 18	. . . 4 ⊢ (𝜑 → ( bday ‘((𝑁 +s 𝑃) +s 1s )) = suc ( bday ‘(𝑁 +s 𝑃)))
37		n0on 28566	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
38	1, 37	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Ons)
39		n0on 28566	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0s → 𝑃 ∈ Ons)
40	11, 39	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ Ons)
41		addonbday 28509	. . . . . . 7 ⊢ ((𝑁 ∈ Ons ∧ 𝑃 ∈ Ons) → ( bday ‘(𝑁 +s 𝑃)) = (( bday ‘𝑁) +no ( bday ‘𝑃)))
42	38, 40, 41	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ( bday ‘(𝑁 +s 𝑃)) = (( bday ‘𝑁) +no ( bday ‘𝑃)))
43	42	suceqd 6435	. . . . 5 ⊢ (𝜑 → suc ( bday ‘(𝑁 +s 𝑃)) = suc (( bday ‘𝑁) +no ( bday ‘𝑃)))
44		naddsuc2 8697	. . . . . 6 ⊢ ((( bday ‘𝑁) ∈ On ∧ ( bday ‘𝑃) ∈ On) → (( bday ‘𝑁) +no suc ( bday ‘𝑃)) = suc (( bday ‘𝑁) +no ( bday ‘𝑃)))
45	29, 27, 44	mp2an 705	. . . . 5 ⊢ (( bday ‘𝑁) +no suc ( bday ‘𝑃)) = suc (( bday ‘𝑁) +no ( bday ‘𝑃))
46	43, 45	eqtr4di 2819	. . . 4 ⊢ (𝜑 → suc ( bday ‘(𝑁 +s 𝑃)) = (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
47	36, 46	eqtrd 2801	. . 3 ⊢ (𝜑 → ( bday ‘((𝑁 +s 𝑃) +s 1s )) = (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
48	32, 47	sseqtrrd 3977	. 2 ⊢ (𝜑 → (( bday ‘𝑁) +no ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ ( bday ‘((𝑁 +s 𝑃) +s 1s )))
49	14, 48	sstrd 3950	1 ⊢ (𝜑 → ( bday ‘(𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ ( bday ‘((𝑁 +s 𝑃) +s 1s )))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146   ⊆ wss 3908   class class class wbr 5114  Oncon0 6367  suc csuc 6369  ‘cfv 6543  (class class class)co 7423   +no cnadd 8660   No csur 27841   <s clts 27842   bday cbday 27843   1_s c1s 28036   +_s cadds 28189   ·_s cmuls 28336   /_su cdivs 28417  On_scons 28481  ℕ_0scn0s 28542  ℕ_scnns 28543  2_sc2s 28640  ↑_scexps 28642

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-dc 10448

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-nadd 8661  df-no 27844  df-lts 27845  df-bday 27846  df-les 27946  df-slts 27988  df-cuts 27990  df-0s 28037  df-1s 28038  df-made 28057  df-old 28058  df-left 28060  df-right 28061  df-norec 28168  df-norec2 28179  df-adds 28190  df-negs 28251  df-subs 28252  df-muls 28337  df-divs 28418  df-ons 28482  df-seqs 28514  df-n0s 28544  df-nns 28545  df-zs 28609  df-2s 28641  df-exps 28643

This theorem is used by: (None)