Theorem z12bdaylem2 28488

Description: Lemma for z12bday 28502. 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 28342	. . 3 ⊢ (𝜑 → 𝑁 ∈ No )
3		2no 28436	. . . . . . 7 ⊢ 2s ∈ No
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 2s ∈ No )
5		z12bdaylem.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0s)
6	5	n0nod 28342	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ No )
7	4, 6	mulscld 28152	. . . . 5 ⊢ (𝜑 → (2s ·s 𝑀) ∈ No )
8		1no 27827	. . . . . 6 ⊢ 1s ∈ No
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 1s ∈ No )
10	7, 9	addscld 27997	. . . 4 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ No )
11		z12bdaylem.3	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
12	10, 11	pw2divscld 28456	. . 3 ⊢ (𝜑 → (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) ∈ No )
13		addbday 28035	. . 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 590	. 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 28435	. . . . . . . 8 ⊢ 2s ∈ ℕs
16		nnn0s 28344	. . . . . . . 8 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ 2s ∈ ℕ0s
18		n0mulscl 28362	. . . . . . 7 ⊢ ((2s ∈ ℕ0s ∧ 𝑀 ∈ ℕ0s) → (2s ·s 𝑀) ∈ ℕ0s)
19	17, 5, 18	sylancr 593	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝑀) ∈ ℕ0s)
20		1n0s 28365	. . . . . 6 ⊢ 1s ∈ ℕ0s
21		n0addscl 28361	. . . . . 6 ⊢ (((2s ·s 𝑀) ∈ ℕ0s ∧ 1s ∈ ℕ0s) → ((2s ·s 𝑀) +s 1s ) ∈ ℕ0s)
22	19, 20, 21	sylancl 592	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ ℕ0s)
23		z12bdaylem.4	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃))
24		bdaypw2n0bnd 28481	. . . . 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 1379	. . . 4 ⊢ (𝜑 → ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃))
26		bdayon 27769	. . . . 5 ⊢ ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ∈ On
27		bdayon 27769	. . . . . 6 ⊢ ( bday ‘𝑃) ∈ On
28	27	onsuci 7786	. . . . 5 ⊢ suc ( bday ‘𝑃) ∈ On
29		bdayon 27769	. . . . 5 ⊢ ( bday ‘𝑁) ∈ On
30		naddss2 8623	. . . . 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 1469	. . . 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 219	. . 3 ⊢ (𝜑 → (( bday ‘𝑁) +no ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
33		n0addscl 28361	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s) → (𝑁 +s 𝑃) ∈ ℕ0s)
34	1, 11, 33	syl2anc 590	. . . . 5 ⊢ (𝜑 → (𝑁 +s 𝑃) ∈ ℕ0s)
35		bdayn0p1 28386	. . . . 5 ⊢ ((𝑁 +s 𝑃) ∈ ℕ0s → ( bday ‘((𝑁 +s 𝑃) +s 1s )) = suc ( bday ‘(𝑁 +s 𝑃)))
36	34, 35	syl 17	. . . 4 ⊢ (𝜑 → ( bday ‘((𝑁 +s 𝑃) +s 1s )) = suc ( bday ‘(𝑁 +s 𝑃)))
37		n0on 28353	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
38	1, 37	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Ons)
39		n0on 28353	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0s → 𝑃 ∈ Ons)
40	11, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ Ons)
41		addonbday 28296	. . . . . . 7 ⊢ ((𝑁 ∈ Ons ∧ 𝑃 ∈ Ons) → ( bday ‘(𝑁 +s 𝑃)) = (( bday ‘𝑁) +no ( bday ‘𝑃)))
42	38, 40, 41	syl2anc 590	. . . . . 6 ⊢ (𝜑 → ( bday ‘(𝑁 +s 𝑃)) = (( bday ‘𝑁) +no ( bday ‘𝑃)))
43	42	suceqd 6384	. . . . 5 ⊢ (𝜑 → suc ( bday ‘(𝑁 +s 𝑃)) = suc (( bday ‘𝑁) +no ( bday ‘𝑃)))
44		naddsuc2 8634	. . . . . 6 ⊢ ((( bday ‘𝑁) ∈ On ∧ ( bday ‘𝑃) ∈ On) → (( bday ‘𝑁) +no suc ( bday ‘𝑃)) = suc (( bday ‘𝑁) +no ( bday ‘𝑃)))
45	29, 27, 44	mp2an 698	. . . . 5 ⊢ (( bday ‘𝑁) +no suc ( bday ‘𝑃)) = suc (( bday ‘𝑁) +no ( bday ‘𝑃))
46	43, 45	eqtr4di 2793	. . . 4 ⊢ (𝜑 → suc ( bday ‘(𝑁 +s 𝑃)) = (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
47	36, 46	eqtrd 2775	. . 3 ⊢ (𝜑 → ( bday ‘((𝑁 +s 𝑃) +s 1s )) = (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
48	32, 47	sseqtrrd 3959	. 2 ⊢ (𝜑 → (( bday ‘𝑁) +no ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ ( bday ‘((𝑁 +s 𝑃) +s 1s )))
49	14, 48	sstrd 3932	1 ⊢ (𝜑 → ( bday ‘(𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ ( bday ‘((𝑁 +s 𝑃) +s 1s )))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119   ⊆ wss 3890   class class class wbr 5079  Oncon0 6317  suc csuc 6319  ‘cfv 6492  (class class class)co 7363   +no cnadd 8598   No csur 27628   <s clts 27629   bday cbday 27630   1_s c1s 27823   +_s cadds 27976   ·_s cmuls 28123   /_su cdivs 28204  On_scons 28268  ℕ_0scn0s 28329  ℕ_scnns 28330  2_sc2s 28427  ↑_scexps 28429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-dc 10366

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-1s 27825  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038  df-subs 28039  df-muls 28124  df-divs 28205  df-ons 28269  df-seqs 28301  df-n0s 28331  df-nns 28332  df-zs 28396  df-2s 28428  df-exps 28430

This theorem is referenced by: (None)