Theorem z12bdaylem1 28644

Description: Lemma for z12bday 28659. 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 28512	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0s → 0s ≤s 𝑀)
3	1, 2	syl 18	. . . . . 6 ⊢ (𝜑 → 0s ≤s 𝑀)
4		0no 27983	. . . . . . . . . . 11 ⊢ 0s ∈ No
5	4	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0s ∈ No )
6	1	n0nod 28499	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ No )
7		2no 28593	. . . . . . . . . . 11 ⊢ 2s ∈ No
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2s ∈ No )
9		2nns 28592	. . . . . . . . . . 11 ⊢ 2s ∈ ℕs
10		nnsgt0 28513	. . . . . . . . . . 11 ⊢ (2s ∈ ℕs → 0s <s 2s)
11	9, 10	mp1i 14	. . . . . . . . . 10 ⊢ (𝜑 → 0s <s 2s)
12	5, 6, 8, 11	lemuls2d 28348	. . . . . . . . 9 ⊢ (𝜑 → ( 0s ≤s 𝑀 ↔ (2s ·s 0s ) ≤s (2s ·s 𝑀)))
13		muls01 28286	. . . . . . . . . . 11 ⊢ (2s ∈ No → (2s ·s 0s ) = 0s )
14	7, 13	ax-mp 5	. . . . . . . . . 10 ⊢ (2s ·s 0s ) = 0s
15	14	breq1i 5117	. . . . . . . . 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 28309	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s 𝑀) ∈ No )
18		1no 27984	. . . . . . . . . 10 ⊢ 1s ∈ No
19	18	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1s ∈ No )
20	5, 17, 19	leadds1d 28169	. . . . . . . 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 28142	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
23	18, 22	ax-mp 5	. . . . . . . 8 ⊢ ( 0s +s 1s ) = 1s
24	23	breq1i 5117	. . . . . . 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 28154	. . . . . 6 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ No )
28		lenlts 27897	. . . . . 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 598	. . . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 = 0s ) → ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃))
33		oveq2 7420	. . . . . . . 8 ⊢ (𝑃 = 0s → (2s↑s𝑃) = (2s↑s 0s ))
34		exps0 28601	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
35	7, 34	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
36	33, 35	eqtrdi 2814	. . . . . . 7 ⊢ (𝑃 = 0s → (2s↑s𝑃) = 1s )
37	36	breq2d 5122	. . . . . 6 ⊢ (𝑃 = 0s → (((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃) ↔ ((2s ·s 𝑀) +s 1s ) <s 1s ))
38	37	adantl 486	. . . . 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 827	. . 3 ⊢ (𝜑 → ¬ 𝑃 = 0s )
41		z12bdaylem.3	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
42	27, 41	pw2divscld 28613	. . . . 5 ⊢ (𝜑 → (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) ∈ No )
43	41	n0nod 28499	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ No )
44		z12bdaylem.1	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
45	44	n0nod 28499	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ No )
46	42, 43, 45	addscan1d 28174	. . . 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 28614	. . . . 5 ⊢ (𝜑 → ((((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) = 𝑃 ↔ ((2s↑s𝑃) ·s 𝑃) = ((2s ·s 𝑀) +s 1s )))
48		breq1 5113	. . . . . . . 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 482	. . . . . . 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 28607	. . . . . . . . . . . 12 ⊢ ((2s ∈ ℕs ∧ 𝑃 ∈ ℕ0s) → (2s↑s𝑃) ∈ ℕs)
51	9, 41, 50	sylancr 598	. . . . . . . . . . 11 ⊢ (𝜑 → (2s↑s𝑃) ∈ ℕs)
52	51	nnnod 28500	. . . . . . . . . 10 ⊢ (𝜑 → (2s↑s𝑃) ∈ No )
53		nnsgt0 28513	. . . . . . . . . . 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 28346	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 <s 1s ↔ ((2s↑s𝑃) ·s 𝑃) <s ((2s↑s𝑃) ·s 1s )))
56	52	mulsridd 28288	. . . . . . . . . 10 ⊢ (𝜑 → ((2s↑s𝑃) ·s 1s ) = (2s↑s𝑃))
57	56	breq2d 5122	. . . . . . . . 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 28512	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ0s → 0s ≤s 𝑃)
60	41, 59	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 0s ≤s 𝑃)
61		lestri3 27900	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ No ∧ 0s ∈ No ) → (𝑃 = 0s ↔ (𝑃 ≤s 0s ∧ 0s ≤s 𝑃)))
62	43, 4, 61	sylancl 597	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 = 0s ↔ (𝑃 ≤s 0s ∧ 0s ≤s 𝑃)))
63	60, 62	mpbiran2d 720	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 = 0s ↔ 𝑃 ≤s 0s ))
64		0n0s 28503	. . . . . . . . . . . 12 ⊢ 0s ∈ ℕ0s
65		n0lesltp1 28540	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℕ0s ∧ 0s ∈ ℕ0s) → (𝑃 ≤s 0s ↔ 𝑃 <s ( 0s +s 1s )))
66	41, 64, 65	sylancl 597	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ≤s 0s ↔ 𝑃 <s ( 0s +s 1s )))
67	23	breq2i 5118	. . . . . . . . . . 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 706	. . . . 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 2965	1 ⊢ (𝜑 → (𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ≠ (𝑁 +s 𝑃))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   class class class wbr 5110  (class class class)co 7412   No csur 27785   <s clts 27786   ≤s cles 27889   0_s c0s 27979   1_s c1s 27980   +_s cadds 28133   ·_s cmuls 28280   /_su cdivs 28361  ℕ_0scn0s 28486  ℕ_scnns 28487  2_sc2s 28584  ↑_scexps 28586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-oadd 8458  df-nadd 8653  df-no 27788  df-lts 27789  df-bday 27790  df-les 27890  df-slts 27932  df-cuts 27934  df-0s 27981  df-1s 27982  df-made 28001  df-old 28002  df-left 28004  df-right 28005  df-norec 28112  df-norec2 28123  df-adds 28134  df-negs 28195  df-subs 28196  df-muls 28281  df-divs 28362  df-seqs 28458  df-n0s 28488  df-nns 28489  df-zs 28553  df-2s 28585  df-exps 28587

This theorem is referenced by: (None)