Theorem z12bdaylem1 28862

Description: Lemma for z12bday 28877. Prove an inequality for birthday ordering. (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
z12bdaylem1	⊢ (𝜑 → (𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ≠ (𝑁 +s 𝑃))

Proof of Theorem z12bdaylem1

Step	Hyp	Ref	Expression
1		z12bdaylem.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0s)
2		n0sge0 28730	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0s → 0s ≤s 𝑀)
3	1, 2	syl 18	. . . . . 6 ⊢ (𝜑 → 0s ≤s 𝑀)
4		0no 28201	. . . . . . . . . . 11 ⊢ 0s ∈ No
5	4	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0s ∈ No)
6	1	n0nod 28717	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ No)
7		2no 28811	. . . . . . . . . . 11 ⊢ 2s ∈ No
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2s ∈ No)
9		2nns 28810	. . . . . . . . . . 11 ⊢ 2s ∈ ℕs
10		nnsgt0 28731	. . . . . . . . . . 11 ⊢ (2s ∈ ℕs → 0s <s 2s)
11	9, 10	mp1i 14	. . . . . . . . . 10 ⊢ (𝜑 → 0s <s 2s)
12	5, 6, 8, 11	lemuls2d 28566	. . . . . . . . 9 ⊢ (𝜑 → ( 0s ≤s 𝑀 ↔ (2s ·s 0s ) ≤s (2s ·s 𝑀)))
13		muls01 28504	. . . . . . . . . . 11 ⊢ (2s ∈ No → (2s ·s 0s ) = 0s )
14	7, 13	ax-mp 5	. . . . . . . . . 10 ⊢ (2s ·s 0s ) = 0s
15	14	breq1i 5109	. . . . . . . . 9 ⊢ ((2s ·s 0s ) ≤s (2s ·s 𝑀) ↔ 0s ≤s (2s ·s 𝑀))
16	12, 15	bitrdi 290	. . . . . . . 8 ⊢ (𝜑 → ( 0s ≤s 𝑀 ↔ 0s ≤s (2s ·s 𝑀)))
17	8, 6	mulscld 28527	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s 𝑀) ∈ No)
18		1no 28202	. . . . . . . . . 10 ⊢ 1s ∈ No
19	18	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1s ∈ No)
20	5, 17, 19	leadds1d 28387	. . . . . . . 8 ⊢ (𝜑 → ( 0s ≤s (2s ·s 𝑀) ↔ ( 0s +s 1s ) ≤s ((2s ·s 𝑀) +s 1s )))
21	16, 20	bitrd 282	. . . . . . 7 ⊢ (𝜑 → ( 0s ≤s 𝑀 ↔ ( 0s +s 1s ) ≤s ((2s ·s 𝑀) +s 1s )))
22		addslid 28360	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
23	18, 22	ax-mp 5	. . . . . . . 8 ⊢ ( 0s +s 1s ) = 1s
24	23	breq1i 5109	. . . . . . 7 ⊢ (( 0s +s 1s ) ≤s ((2s ·s 𝑀) +s 1s ) ↔ 1s ≤s ((2s ·s 𝑀) +s 1s ))
25	21, 24	bitrdi 290	. . . . . 6 ⊢ (𝜑 → ( 0s ≤s 𝑀 ↔ 1s ≤s ((2s ·s 𝑀) +s 1s )))
26	3, 25	mpbid 235	. . . . 5 ⊢ (𝜑 → 1s ≤s ((2s ·s 𝑀) +s 1s ))
27	17, 19	addscld 28372	. . . . . 6 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ No)
28		lenlts 28115	. . . . . 6 ⊢ (( 1s ∈ No ∧ ((2s ·s 𝑀) +s 1s ) ∈ No) → ( 1s ≤s ((2s ·s 𝑀) +s 1s ) ↔ ¬ ((2s ·s 𝑀) +s 1s ) <s 1s ))
29	18, 27, 28	sylancr 599	. . . . 5 ⊢ (𝜑 → ( 1s ≤s ((2s ·s 𝑀) +s 1s ) ↔ ¬ ((2s ·s 𝑀) +s 1s ) <s 1s ))
30	26, 29	mpbid 235	. . . 4 ⊢ (𝜑 → ¬ ((2s ·s 𝑀) +s 1s ) <s 1s )
31		z12bdaylem.4	. . . . . 6 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃))
32	31	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 = 0s ) → ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃))
33		oveq2 7429	. . . . . . . 8 ⊢ (𝑃 = 0s → (2s↑s𝑃) = (2s↑s 0s ))
34		exps0 28819	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
35	7, 34	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
36	33, 35	eqtrdi 2811	. . . . . . 7 ⊢ (𝑃 = 0s → (2s↑s𝑃) = 1s )
37	36	breq2d 5114	. . . . . 6 ⊢ (𝑃 = 0s → (((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃) ↔ ((2s ·s 𝑀) +s 1s ) <s 1s ))
38	37	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 = 0s ) → (((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃) ↔ ((2s ·s 𝑀) +s 1s ) <s 1s ))
39	32, 38	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ 𝑃 = 0s ) → ((2s ·s 𝑀) +s 1s ) <s 1s )
40	30, 39	mtand 828	. . 3 ⊢ (𝜑 → ¬ 𝑃 = 0s )
41		z12bdaylem.3	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
42	27, 41	pw2divscld 28831	. . . . 5 ⊢ (𝜑 → (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) ∈ No)
43	41	n0nod 28717	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ No)
44		z12bdaylem.1	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
45	44	n0nod 28717	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ No)
46	42, 43, 45	addscan1d 28392	. . . 4 ⊢ (𝜑 → ((𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) = (𝑁 +s 𝑃) ↔ (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) = 𝑃))
47	27, 43, 41	pw2divmulsd 28832	. . . . 5 ⊢ (𝜑 → ((((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) = 𝑃 ↔ ((2s↑s𝑃) ·s 𝑃) = ((2s ·s 𝑀) +s 1s )))
48		breq1 5105	. . . . . . . 8 ⊢ (((2s↑s𝑃) ·s 𝑃) = ((2s ·s 𝑀) +s 1s ) → (((2s↑s𝑃) ·s 𝑃) <s (2s↑s𝑃) ↔ ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃)))
49	48	biimpar 483	. . . . . . 7 ⊢ ((((2s↑s𝑃) ·s 𝑃) = ((2s ·s 𝑀) +s 1s ) ∧ ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃)) → ((2s↑s𝑃) ·s 𝑃) <s (2s↑s𝑃))
50		nnexpscl 28825	. . . . . . . . . . . 12 ⊢ ((2s ∈ ℕs ∧ 𝑃 ∈ ℕ0s) → (2s↑s𝑃) ∈ ℕs)
51	9, 41, 50	sylancr 599	. . . . . . . . . . 11 ⊢ (𝜑 → (2s↑s𝑃) ∈ ℕs)
52	51	nnnod 28718	. . . . . . . . . 10 ⊢ (𝜑 → (2s↑s𝑃) ∈ No)
53		nnsgt0 28731	. . . . . . . . . . 11 ⊢ ((2s↑s𝑃) ∈ ℕs → 0s <s (2s↑s𝑃))
54	51, 53	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 0s <s (2s↑s𝑃))
55	43, 19, 52, 54	ltmuls2d 28564	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 <s 1s ↔ ((2s↑s𝑃) ·s 𝑃) <s ((2s↑s𝑃) ·s 1s )))
56	52	mulsridd 28506	. . . . . . . . . 10 ⊢ (𝜑 → ((2s↑s𝑃) ·s 1s ) = (2s↑s𝑃))
57	56	breq2d 5114	. . . . . . . . 9 ⊢ (𝜑 → (((2s↑s𝑃) ·s 𝑃) <s ((2s↑s𝑃) ·s 1s ) ↔ ((2s↑s𝑃) ·s 𝑃) <s (2s↑s𝑃)))
58	55, 57	bitrd 282	. . . . . . . 8 ⊢ (𝜑 → (𝑃 <s 1s ↔ ((2s↑s𝑃) ·s 𝑃) <s (2s↑s𝑃)))
59		n0sge0 28730	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ0s → 0s ≤s 𝑃)
60	41, 59	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 0s ≤s 𝑃)
61		lestri3 28118	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ No ∧ 0s ∈ No) → (𝑃 = 0s ↔ (𝑃 ≤s 0s ∧ 0s ≤s 𝑃)))
62	43, 4, 61	sylancl 598	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 = 0s ↔ (𝑃 ≤s 0s ∧ 0s ≤s 𝑃)))
63	60, 62	mpbiran2d 721	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 = 0s ↔ 𝑃 ≤s 0s ))
64		0n0s 28721	. . . . . . . . . . . 12 ⊢ 0s ∈ ℕ0s
65		n0lesltp1 28758	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℕ0s ∧ 0s ∈ ℕ0s) → (𝑃 ≤s 0s ↔ 𝑃 <s ( 0s +s 1s )))
66	41, 64, 65	sylancl 598	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ≤s 0s ↔ 𝑃 <s ( 0s +s 1s )))
67	23	breq2i 5110	. . . . . . . . . . 11 ⊢ (𝑃 <s ( 0s +s 1s ) ↔ 𝑃 <s 1s )
68	66, 67	bitrdi 290	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ≤s 0s ↔ 𝑃 <s 1s ))
69	63, 68	bitr2d 283	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 <s 1s ↔ 𝑃 = 0s ))
70	69	biimpd 232	. . . . . . . 8 ⊢ (𝜑 → (𝑃 <s 1s → 𝑃 = 0s ))
71	58, 70	sylbird 263	. . . . . . 7 ⊢ (𝜑 → (((2s↑s𝑃) ·s 𝑃) <s (2s↑s𝑃) → 𝑃 = 0s ))
72	49, 71	syl5 35	. . . . . 6 ⊢ (𝜑 → ((((2s↑s𝑃) ·s 𝑃) = ((2s ·s 𝑀) +s 1s ) ∧ ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃)) → 𝑃 = 0s ))
73	31, 72	mpan2d 707	. . . . 5 ⊢ (𝜑 → (((2s↑s𝑃) ·s 𝑃) = ((2s ·s 𝑀) +s 1s ) → 𝑃 = 0s ))
74	47, 73	sylbid 243	. . . 4 ⊢ (𝜑 → ((((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) = 𝑃 → 𝑃 = 0s ))
75	46, 74	sylbid 243	. . 3 ⊢ (𝜑 → ((𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) = (𝑁 +s 𝑃) → 𝑃 = 0s ))
76	40, 75	mtod 201	. 2 ⊢ (𝜑 → ¬ (𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) = (𝑁 +s 𝑃))
77	76	neqned 2962	1 ⊢ (𝜑 → (𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ≠ (𝑁 +s 𝑃))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   class class class wbr 5102  (class class class)co 7421  Nocsur 28003   <s clts 28004   ≤s cles 28107   0_s c0s 28197   1_s c1s 28198   +_s cadds 28351   ·_s cmuls 28498   /_su cdivs 28579  ℕ_0scn0s 28704  ℕ_scnns 28705  2_sc2s 28802  ↑_scexps 28804

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-2o 8481  df-oadd 8484  df-nadd 8679  df-no 28006  df-lts 28007  df-bday 28008  df-les 28108  df-slts 28150  df-cuts 28152  df-0s 28199  df-1s 28200  df-made 28219  df-old 28220  df-left 28222  df-right 28223  df-norec 28330  df-norec2 28341  df-adds 28352  df-negs 28413  df-subs 28414  df-muls 28499  df-divs 28580  df-seqs 28676  df-n0s 28706  df-nns 28707  df-zs 28771  df-2s 28803  df-exps 28805

This theorem is used by: (None)