Theorem z12bdaylem2 28477

Description: Lemma for z12bday 28491. 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 28331	. . 3 ⊢ (𝜑 → 𝑁 ∈ No )
3		2no 28425	. . . . . . 7 ⊢ 2s ∈ No
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 2s ∈ No )
5		z12bdaylem.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0s)
6	5	n0nod 28331	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ No )
7	4, 6	mulscld 28141	. . . . 5 ⊢ (𝜑 → (2s ·s 𝑀) ∈ No )
8		1no 27816	. . . . . 6 ⊢ 1s ∈ No
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 1s ∈ No )
10	7, 9	addscld 27986	. . . 4 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ No )
11		z12bdaylem.3	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
12	10, 11	pw2divscld 28445	. . 3 ⊢ (𝜑 → (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) ∈ No )
13		addbday 28024	. . 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 585	. 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 28424	. . . . . . . 8 ⊢ 2s ∈ ℕs
16		nnn0s 28333	. . . . . . . 8 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ 2s ∈ ℕ0s
18		n0mulscl 28351	. . . . . . 7 ⊢ ((2s ∈ ℕ0s ∧ 𝑀 ∈ ℕ0s) → (2s ·s 𝑀) ∈ ℕ0s)
19	17, 5, 18	sylancr 588	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝑀) ∈ ℕ0s)
20		1n0s 28354	. . . . . 6 ⊢ 1s ∈ ℕ0s
21		n0addscl 28350	. . . . . 6 ⊢ (((2s ·s 𝑀) ∈ ℕ0s ∧ 1s ∈ ℕ0s) → ((2s ·s 𝑀) +s 1s ) ∈ ℕ0s)
22	19, 20, 21	sylancl 587	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ ℕ0s)
23		z12bdaylem.4	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃))
24		bdaypw2n0bnd 28470	. . . . 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 1374	. . . 4 ⊢ (𝜑 → ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃))
26		bdayon 27758	. . . . 5 ⊢ ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ∈ On
27		bdayon 27758	. . . . . 6 ⊢ ( bday ‘𝑃) ∈ On
28	27	onsuci 7783	. . . . 5 ⊢ suc ( bday ‘𝑃) ∈ On
29		bdayon 27758	. . . . 5 ⊢ ( bday ‘𝑁) ∈ On
30		naddss2 8619	. . . . 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 1464	. . . 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 218	. . 3 ⊢ (𝜑 → (( bday ‘𝑁) +no ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
33		n0addscl 28350	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s) → (𝑁 +s 𝑃) ∈ ℕ0s)
34	1, 11, 33	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑁 +s 𝑃) ∈ ℕ0s)
35		bdayn0p1 28375	. . . . 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 28342	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
38	1, 37	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Ons)
39		n0on 28342	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0s → 𝑃 ∈ Ons)
40	11, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ Ons)
41		addonbday 28285	. . . . . . 7 ⊢ ((𝑁 ∈ Ons ∧ 𝑃 ∈ Ons) → ( bday ‘(𝑁 +s 𝑃)) = (( bday ‘𝑁) +no ( bday ‘𝑃)))
42	38, 40, 41	syl2anc 585	. . . . . 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 8630	. . . . . 6 ⊢ ((( bday ‘𝑁) ∈ On ∧ ( bday ‘𝑃) ∈ On) → (( bday ‘𝑁) +no suc ( bday ‘𝑃)) = suc (( bday ‘𝑁) +no ( bday ‘𝑃)))
45	29, 27, 44	mp2an 693	. . . . 5 ⊢ (( bday ‘𝑁) +no suc ( bday ‘𝑃)) = suc (( bday ‘𝑁) +no ( bday ‘𝑃))
46	43, 45	eqtr4di 2790	. . . 4 ⊢ (𝜑 → suc ( bday ‘(𝑁 +s 𝑃)) = (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
47	36, 46	eqtrd 2772	. . 3 ⊢ (𝜑 → ( bday ‘((𝑁 +s 𝑃) +s 1s )) = (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
48	32, 47	sseqtrrd 3960	. 2 ⊢ (𝜑 → (( bday ‘𝑁) +no ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ ( bday ‘((𝑁 +s 𝑃) +s 1s )))
49	14, 48	sstrd 3933	1 ⊢ (𝜑 → ( bday ‘(𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ ( bday ‘((𝑁 +s 𝑃) +s 1s )))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890   class class class wbr 5086  Oncon0 6317  suc csuc 6319  ‘cfv 6492  (class class class)co 7360   +no cnadd 8594   No csur 27617   <s clts 27618   bday cbday 27619   1_s c1s 27812   +_s cadds 27965   ·_s cmuls 28112   /_su cdivs 28193  On_scons 28257  ℕ_0scn0s 28318  ℕ_scnns 28319  2_sc2s 28416  ↑_scexps 28418

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-dc 10359

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 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 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-oadd 8402  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-les 27723  df-slts 27764  df-cuts 27766  df-0s 27813  df-1s 27814  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec 27944  df-norec2 27955  df-adds 27966  df-negs 28027  df-subs 28028  df-muls 28113  df-divs 28194  df-ons 28258  df-seqs 28290  df-n0s 28320  df-nns 28321  df-zs 28385  df-2s 28417  df-exps 28419

This theorem is referenced by: (None)