Theorem z12bdaylem2 28465

Description: Lemma for z12bday 28479. 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 28319	. . 3 ⊢ (𝜑 → 𝑁 ∈ No )
3		2no 28413	. . . . . . 7 ⊢ 2s ∈ No
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 2s ∈ No )
5		z12bdaylem.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0s)
6	5	n0nod 28319	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ No )
7	4, 6	mulscld 28129	. . . . 5 ⊢ (𝜑 → (2s ·s 𝑀) ∈ No )
8		1no 27804	. . . . . 6 ⊢ 1s ∈ No
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 1s ∈ No )
10	7, 9	addscld 27974	. . . 4 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ No )
11		z12bdaylem.3	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
12	10, 11	pw2divscld 28433	. . 3 ⊢ (𝜑 → (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) ∈ No )
13		addbday 28012	. . 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 28412	. . . . . . . 8 ⊢ 2s ∈ ℕs
16		nnn0s 28321	. . . . . . . 8 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ 2s ∈ ℕ0s
18		n0mulscl 28339	. . . . . . 7 ⊢ ((2s ∈ ℕ0s ∧ 𝑀 ∈ ℕ0s) → (2s ·s 𝑀) ∈ ℕ0s)
19	17, 5, 18	sylancr 588	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝑀) ∈ ℕ0s)
20		1n0s 28342	. . . . . 6 ⊢ 1s ∈ ℕ0s
21		n0addscl 28338	. . . . . 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 28458	. . . . 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 27746	. . . . 5 ⊢ ( bday ‘(((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ∈ On
27		bdayon 27746	. . . . . 6 ⊢ ( bday ‘𝑃) ∈ On
28	27	onsuci 7792	. . . . 5 ⊢ suc ( bday ‘𝑃) ∈ On
29		bdayon 27746	. . . . 5 ⊢ ( bday ‘𝑁) ∈ On
30		naddss2 8628	. . . . 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 28338	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝑃 ∈ ℕ0s) → (𝑁 +s 𝑃) ∈ ℕ0s)
34	1, 11, 33	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑁 +s 𝑃) ∈ ℕ0s)
35		bdayn0p1 28363	. . . . 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 28330	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
38	1, 37	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Ons)
39		n0on 28330	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0s → 𝑃 ∈ Ons)
40	11, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ Ons)
41		addonbday 28273	. . . . . . 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 6392	. . . . 5 ⊢ (𝜑 → suc ( bday ‘(𝑁 +s 𝑃)) = suc (( bday ‘𝑁) +no ( bday ‘𝑃)))
44		naddsuc2 8639	. . . . . 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 6325  suc csuc 6327  ‘cfv 6500  (class class class)co 7369   +no cnadd 8603   No csur 27605   <s clts 27606   bday cbday 27607   1_s c1s 27800   +_s cadds 27953   ·_s cmuls 28100   /_su cdivs 28181  On_scons 28245  ℕ_0scn0s 28306  ℕ_scnns 28307  2_sc2s 28404  ↑_scexps 28406

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7691  ax-dc 10370

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 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7820  df-1st 7944  df-2nd 7945  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-nadd 8604  df-no 27608  df-lts 27609  df-bday 27610  df-les 27711  df-slts 27752  df-cuts 27754  df-0s 27801  df-1s 27802  df-made 27821  df-old 27822  df-left 27824  df-right 27825  df-norec 27932  df-norec2 27943  df-adds 27954  df-negs 28015  df-subs 28016  df-muls 28101  df-divs 28182  df-ons 28246  df-seqs 28278  df-n0s 28308  df-nns 28309  df-zs 28373  df-2s 28405  df-exps 28407

This theorem is referenced by: (None)