Theorem z12bdaylem1 28743

Description: Lemma for z12bday 28758. 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 28611	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0s → 0s ≤s 𝑀)
3	1, 2	syl 18	. . . . . 6 ⊢ (𝜑 → 0s ≤s 𝑀)
4		0no 28082	. . . . . . . . . . 11 ⊢ 0s ∈ No
5	4	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0s ∈ No )
6	1	n0nod 28598	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ No )
7		2no 28692	. . . . . . . . . . 11 ⊢ 2s ∈ No
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2s ∈ No )
9		2nns 28691	. . . . . . . . . . 11 ⊢ 2s ∈ ℕs
10		nnsgt0 28612	. . . . . . . . . . 11 ⊢ (2s ∈ ℕs → 0s <s 2s)
11	9, 10	mp1i 14	. . . . . . . . . 10 ⊢ (𝜑 → 0s <s 2s)
12	5, 6, 8, 11	lemuls2d 28447	. . . . . . . . 9 ⊢ (𝜑 → ( 0s ≤s 𝑀 ↔ (2s ·s 0s ) ≤s (2s ·s 𝑀)))
13		muls01 28385	. . . . . . . . . . 11 ⊢ (2s ∈ No → (2s ·s 0s ) = 0s )
14	7, 13	ax-mp 5	. . . . . . . . . 10 ⊢ (2s ·s 0s ) = 0s
15	14	breq1i 5114	. . . . . . . . 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 28408	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s 𝑀) ∈ No )
18		1no 28083	. . . . . . . . . 10 ⊢ 1s ∈ No
19	18	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1s ∈ No )
20	5, 17, 19	leadds1d 28268	. . . . . . . 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 28241	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
23	18, 22	ax-mp 5	. . . . . . . 8 ⊢ ( 0s +s 1s ) = 1s
24	23	breq1i 5114	. . . . . . 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 28253	. . . . . 6 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ No )
28		lenlts 27996	. . . . . 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 7425	. . . . . . . 8 ⊢ (𝑃 = 0s → (2s↑s𝑃) = (2s↑s 0s ))
34		exps0 28700	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
35	7, 34	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
36	33, 35	eqtrdi 2813	. . . . . . 7 ⊢ (𝑃 = 0s → (2s↑s𝑃) = 1s )
37	36	breq2d 5119	. . . . . 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 28712	. . . . 5 ⊢ (𝜑 → (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) ∈ No )
43	41	n0nod 28598	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ No )
44		z12bdaylem.1	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
45	44	n0nod 28598	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ No )
46	42, 43, 45	addscan1d 28273	. . . 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 28713	. . . . 5 ⊢ (𝜑 → ((((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) = 𝑃 ↔ ((2s↑s𝑃) ·s 𝑃) = ((2s ·s 𝑀) +s 1s )))
48		breq1 5110	. . . . . . . 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 28706	. . . . . . . . . . . 12 ⊢ ((2s ∈ ℕs ∧ 𝑃 ∈ ℕ0s) → (2s↑s𝑃) ∈ ℕs)
51	9, 41, 50	sylancr 599	. . . . . . . . . . 11 ⊢ (𝜑 → (2s↑s𝑃) ∈ ℕs)
52	51	nnnod 28599	. . . . . . . . . 10 ⊢ (𝜑 → (2s↑s𝑃) ∈ No )
53		nnsgt0 28612	. . . . . . . . . . 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 28445	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 <s 1s ↔ ((2s↑s𝑃) ·s 𝑃) <s ((2s↑s𝑃) ·s 1s )))
56	52	mulsridd 28387	. . . . . . . . . 10 ⊢ (𝜑 → ((2s↑s𝑃) ·s 1s ) = (2s↑s𝑃))
57	56	breq2d 5119	. . . . . . . . 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 28611	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ0s → 0s ≤s 𝑃)
60	41, 59	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 0s ≤s 𝑃)
61		lestri3 27999	. . . . . . . . . . . 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 28602	. . . . . . . . . . . 12 ⊢ 0s ∈ ℕ0s
65		n0lesltp1 28639	. . . . . . . . . . . 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 5115	. . . . . . . . . . 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 2964	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 1570   ∈ wcel 2145   ≠ wne 2957   class class class wbr 5107  (class class class)co 7417   No csur 27884   <s clts 27885   ≤s cles 27988   0_s c0s 28078   1_s c1s 28079   +_s cadds 28232   ·_s cmuls 28379   /_su cdivs 28460  ℕ_0scn0s 28585  ℕ_scnns 28586  2_sc2s 28683  ↑_scexps 28685

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-nadd 8658  df-no 27887  df-lts 27888  df-bday 27889  df-les 27989  df-slts 28031  df-cuts 28033  df-0s 28080  df-1s 28081  df-made 28100  df-old 28101  df-left 28103  df-right 28104  df-norec 28211  df-norec2 28222  df-adds 28233  df-negs 28294  df-subs 28295  df-muls 28380  df-divs 28461  df-seqs 28557  df-n0s 28587  df-nns 28588  df-zs 28652  df-2s 28684  df-exps 28686

This theorem is used by: (None)