Theorem z12bdaylem2 28541

Description: Lemma for z12bday 28555. 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 28395	. . 3 ⊢ (𝜑 → 𝑁 ∈ No )
3		2no 28489	. . . . . . 7 ⊢ 2s ∈ No
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 2s ∈ No )
5		z12bdaylem.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0s)
6	5	n0nod 28395	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ No )
7	4, 6	mulscld 28205	. . . . 5 ⊢ (𝜑 → (2s ·s 𝑀) ∈ No )
8		1no 27880	. . . . . 6 ⊢ 1s ∈ No
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 1s ∈ No )
10	7, 9	addscld 28050	. . . 4 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ No )
11		z12bdaylem.3	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
12	10, 11	pw2divscld 28509	. . 3 ⊢ (𝜑 → (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) ∈ No )
13		addbday 28088	. . 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 593	. 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 28488	. . . . . . . 8 ⊢ 2s ∈ ℕs
16		nnn0s 28397	. . . . . . . 8 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ 2s ∈ ℕ0s
18		n0mulscl 28415	. . . . . . 7 ⊢ ((2s ∈ ℕ0s ∧ 𝑀 ∈ ℕ0s) → (2s ·s 𝑀) ∈ ℕ0s)
19	17, 5, 18	sylancr 596	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝑀) ∈ ℕ0s)
20		1n0s 28418	. . . . . 6 ⊢ 1s ∈ ℕ0s
21		n0addscl 28414	. . . . . 6 ⊢ (((2s ·s 𝑀) ∈ ℕ0s ∧ 1s ∈ ℕ0s) → ((2s ·s 𝑀) +s 1s ) ∈ ℕ0s)
22	19, 20, 21	sylancl 595	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ ℕ0s)
23		z12bdaylem.4	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃))
24		bdaypw2n0bnd 28534	. . . . 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 1389	. . . 4 ⊢ (𝜑 → ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ⊆ suc ( bday ‘𝑃))
26		bdayon 27822	. . . . 5 ⊢ ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ∈ On
27		bdayon 27822	. . . . . 6 ⊢ ( bday ‘𝑃) ∈ On
28	27	onsuci 7815	. . . . 5 ⊢ suc ( bday ‘𝑃) ∈ On
29		bdayon 27822	. . . . 5 ⊢ ( bday ‘𝑁) ∈ On
30		naddss2 8656	. . . . 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 1481	. . . 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 220	. . 3 ⊢ (𝜑 → (( bday ‘𝑁) +no ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
33		n0addscl 28414	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s) → (𝑁 +s 𝑃) ∈ ℕ0s)
34	1, 11, 33	syl2anc 593	. . . . 5 ⊢ (𝜑 → (𝑁 +s 𝑃) ∈ ℕ0s)
35		bdayn0p1 28439	. . . . 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 28406	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
38	1, 37	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Ons)
39		n0on 28406	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0s → 𝑃 ∈ Ons)
40	11, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ Ons)
41		addonbday 28349	. . . . . . 7 ⊢ ((𝑁 ∈ Ons ∧ 𝑃 ∈ Ons) → ( bday ‘(𝑁 +s 𝑃)) = (( bday ‘𝑁) +no ( bday ‘𝑃)))
42	38, 40, 41	syl2anc 593	. . . . . 6 ⊢ (𝜑 → ( bday ‘(𝑁 +s 𝑃)) = (( bday ‘𝑁) +no ( bday ‘𝑃)))
43	42	suceqd 6409	. . . . 5 ⊢ (𝜑 → suc ( bday ‘(𝑁 +s 𝑃)) = suc (( bday ‘𝑁) +no ( bday ‘𝑃)))
44		naddsuc2 8667	. . . . . 6 ⊢ ((( bday ‘𝑁) ∈ On ∧ ( bday ‘𝑃) ∈ On) → (( bday ‘𝑁) +no suc ( bday ‘𝑃)) = suc (( bday ‘𝑁) +no ( bday ‘𝑃)))
45	29, 27, 44	mp2an 702	. . . . 5 ⊢ (( bday ‘𝑁) +no suc ( bday ‘𝑃)) = suc (( bday ‘𝑁) +no ( bday ‘𝑃))
46	43, 45	eqtr4di 2814	. . . 4 ⊢ (𝜑 → suc ( bday ‘(𝑁 +s 𝑃)) = (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
47	36, 46	eqtrd 2796	. . 3 ⊢ (𝜑 → ( bday ‘((𝑁 +s 𝑃) +s 1s )) = (( bday ‘𝑁) +no suc ( bday ‘𝑃)))
48	32, 47	sseqtrrd 3973	. 2 ⊢ (𝜑 → (( bday ‘𝑁) +no ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ ( bday ‘((𝑁 +s 𝑃) +s 1s )))
49	14, 48	sstrd 3946	1 ⊢ (𝜑 → ( bday ‘(𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ ( bday ‘((𝑁 +s 𝑃) +s 1s )))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559   ∈ wcel 2141   ⊆ wss 3904   class class class wbr 5099  Oncon0 6342  suc csuc 6344  ‘cfv 6517  (class class class)co 7392   +no cnadd 8630   No csur 27681   <s clts 27682   bday cbday 27683   1_s c1s 27876   +_s cadds 28029   ·_s cmuls 28176   /_su cdivs 28257  On_scons 28321  ℕ_0scn0s 28382  ℕ_scnns 28383  2_sc2s 28480  ↑_scexps 28482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-dc 10400

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-nadd 8631  df-no 27684  df-lts 27685  df-bday 27686  df-les 27786  df-slts 27828  df-cuts 27830  df-0s 27877  df-1s 27878  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec 28008  df-norec2 28019  df-adds 28030  df-negs 28091  df-subs 28092  df-muls 28177  df-divs 28258  df-ons 28322  df-seqs 28354  df-n0s 28384  df-nns 28385  df-zs 28449  df-2s 28481  df-exps 28483

This theorem is referenced by: (None)