Theorem z12bdaylem1 28464

Description: Lemma for z12bday 28479. 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 28332	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0s → 0s ≤s 𝑀)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 0s ≤s 𝑀)
4		0no 27803	. . . . . . . . . . 11 ⊢ 0s ∈ No
5	4	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0s ∈ No )
6	1	n0nod 28319	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ No )
7		2no 28413	. . . . . . . . . . 11 ⊢ 2s ∈ No
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2s ∈ No )
9		2nns 28412	. . . . . . . . . . 11 ⊢ 2s ∈ ℕs
10		nnsgt0 28333	. . . . . . . . . . 11 ⊢ (2s ∈ ℕs → 0s <s 2s)
11	9, 10	mp1i 13	. . . . . . . . . 10 ⊢ (𝜑 → 0s <s 2s)
12	5, 6, 8, 11	lemuls2d 28168	. . . . . . . . 9 ⊢ (𝜑 → ( 0s ≤s 𝑀 ↔ (2s ·s 0s ) ≤s (2s ·s 𝑀)))
13		muls01 28106	. . . . . . . . . . 11 ⊢ (2s ∈ No → (2s ·s 0s ) = 0s )
14	7, 13	ax-mp 5	. . . . . . . . . 10 ⊢ (2s ·s 0s ) = 0s
15	14	breq1i 5093	. . . . . . . . 9 ⊢ ((2s ·s 0s ) ≤s (2s ·s 𝑀) ↔ 0s ≤s (2s ·s 𝑀))
16	12, 15	bitrdi 287	. . . . . . . 8 ⊢ (𝜑 → ( 0s ≤s 𝑀 ↔ 0s ≤s (2s ·s 𝑀)))
17	8, 6	mulscld 28129	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s 𝑀) ∈ No )
18		1no 27804	. . . . . . . . . 10 ⊢ 1s ∈ No
19	18	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1s ∈ No )
20	5, 17, 19	leadds1d 27989	. . . . . . . 8 ⊢ (𝜑 → ( 0s ≤s (2s ·s 𝑀) ↔ ( 0s +s 1s ) ≤s ((2s ·s 𝑀) +s 1s )))
21	16, 20	bitrd 279	. . . . . . 7 ⊢ (𝜑 → ( 0s ≤s 𝑀 ↔ ( 0s +s 1s ) ≤s ((2s ·s 𝑀) +s 1s )))
22		addslid 27962	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
23	18, 22	ax-mp 5	. . . . . . . 8 ⊢ ( 0s +s 1s ) = 1s
24	23	breq1i 5093	. . . . . . 7 ⊢ (( 0s +s 1s ) ≤s ((2s ·s 𝑀) +s 1s ) ↔ 1s ≤s ((2s ·s 𝑀) +s 1s ))
25	21, 24	bitrdi 287	. . . . . 6 ⊢ (𝜑 → ( 0s ≤s 𝑀 ↔ 1s ≤s ((2s ·s 𝑀) +s 1s )))
26	3, 25	mpbid 232	. . . . 5 ⊢ (𝜑 → 1s ≤s ((2s ·s 𝑀) +s 1s ))
27	17, 19	addscld 27974	. . . . . 6 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ No )
28		lenlts 27718	. . . . . 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 588	. . . . 5 ⊢ (𝜑 → ( 1s ≤s ((2s ·s 𝑀) +s 1s ) ↔ ¬ ((2s ·s 𝑀) +s 1s ) <s 1s ))
30	26, 29	mpbid 232	. . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 = 0s ) → ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃))
33		oveq2 7377	. . . . . . . 8 ⊢ (𝑃 = 0s → (2s↑s𝑃) = (2s↑s 0s ))
34		exps0 28421	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
35	7, 34	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
36	33, 35	eqtrdi 2788	. . . . . . 7 ⊢ (𝑃 = 0s → (2s↑s𝑃) = 1s )
37	36	breq2d 5098	. . . . . 6 ⊢ (𝑃 = 0s → (((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃) ↔ ((2s ·s 𝑀) +s 1s ) <s 1s ))
38	37	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 = 0s ) → (((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃) ↔ ((2s ·s 𝑀) +s 1s ) <s 1s ))
39	32, 38	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝑃 = 0s ) → ((2s ·s 𝑀) +s 1s ) <s 1s )
40	30, 39	mtand 816	. . 3 ⊢ (𝜑 → ¬ 𝑃 = 0s )
41		z12bdaylem.3	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
42	27, 41	pw2divscld 28433	. . . . 5 ⊢ (𝜑 → (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) ∈ No )
43	41	n0nod 28319	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ No )
44		z12bdaylem.1	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
45	44	n0nod 28319	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ No )
46	42, 43, 45	addscan1d 27994	. . . 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 28434	. . . . 5 ⊢ (𝜑 → ((((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) = 𝑃 ↔ ((2s↑s𝑃) ·s 𝑃) = ((2s ·s 𝑀) +s 1s )))
48		breq1 5089	. . . . . . . 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 477	. . . . . . 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 28427	. . . . . . . . . . . 12 ⊢ ((2s ∈ ℕs ∧ 𝑃 ∈ ℕ0s) → (2s↑s𝑃) ∈ ℕs)
51	9, 41, 50	sylancr 588	. . . . . . . . . . 11 ⊢ (𝜑 → (2s↑s𝑃) ∈ ℕs)
52	51	nnnod 28320	. . . . . . . . . 10 ⊢ (𝜑 → (2s↑s𝑃) ∈ No )
53		nnsgt0 28333	. . . . . . . . . . 11 ⊢ ((2s↑s𝑃) ∈ ℕs → 0s <s (2s↑s𝑃))
54	51, 53	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0s <s (2s↑s𝑃))
55	43, 19, 52, 54	ltmuls2d 28166	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 <s 1s ↔ ((2s↑s𝑃) ·s 𝑃) <s ((2s↑s𝑃) ·s 1s )))
56	52	mulsridd 28108	. . . . . . . . . 10 ⊢ (𝜑 → ((2s↑s𝑃) ·s 1s ) = (2s↑s𝑃))
57	56	breq2d 5098	. . . . . . . . 9 ⊢ (𝜑 → (((2s↑s𝑃) ·s 𝑃) <s ((2s↑s𝑃) ·s 1s ) ↔ ((2s↑s𝑃) ·s 𝑃) <s (2s↑s𝑃)))
58	55, 57	bitrd 279	. . . . . . . 8 ⊢ (𝜑 → (𝑃 <s 1s ↔ ((2s↑s𝑃) ·s 𝑃) <s (2s↑s𝑃)))
59		n0sge0 28332	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ0s → 0s ≤s 𝑃)
60	41, 59	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 0s ≤s 𝑃)
61		lestri3 27721	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ No ∧ 0s ∈ No ) → (𝑃 = 0s ↔ (𝑃 ≤s 0s ∧ 0s ≤s 𝑃)))
62	43, 4, 61	sylancl 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 = 0s ↔ (𝑃 ≤s 0s ∧ 0s ≤s 𝑃)))
63	60, 62	mpbiran2d 709	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 = 0s ↔ 𝑃 ≤s 0s ))
64		0n0s 28323	. . . . . . . . . . . 12 ⊢ 0s ∈ ℕ0s
65		n0lesltp1 28360	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℕ0s ∧ 0s ∈ ℕ0s) → (𝑃 ≤s 0s ↔ 𝑃 <s ( 0s +s 1s )))
66	41, 64, 65	sylancl 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ≤s 0s ↔ 𝑃 <s ( 0s +s 1s )))
67	23	breq2i 5094	. . . . . . . . . . 11 ⊢ (𝑃 <s ( 0s +s 1s ) ↔ 𝑃 <s 1s )
68	66, 67	bitrdi 287	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ≤s 0s ↔ 𝑃 <s 1s ))
69	63, 68	bitr2d 280	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 <s 1s ↔ 𝑃 = 0s ))
70	69	biimpd 229	. . . . . . . 8 ⊢ (𝜑 → (𝑃 <s 1s → 𝑃 = 0s ))
71	58, 70	sylbird 260	. . . . . . 7 ⊢ (𝜑 → (((2s↑s𝑃) ·s 𝑃) <s (2s↑s𝑃) → 𝑃 = 0s ))
72	49, 71	syl5 34	. . . . . 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 695	. . . . 5 ⊢ (𝜑 → (((2s↑s𝑃) ·s 𝑃) = ((2s ·s 𝑀) +s 1s ) → 𝑃 = 0s ))
74	47, 73	sylbid 240	. . . 4 ⊢ (𝜑 → ((((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) = 𝑃 → 𝑃 = 0s ))
75	46, 74	sylbid 240	. . 3 ⊢ (𝜑 → ((𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) = (𝑁 +s 𝑃) → 𝑃 = 0s ))
76	40, 75	mtod 198	. 2 ⊢ (𝜑 → ¬ (𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) = (𝑁 +s 𝑃))
77	76	neqned 2940	1 ⊢ (𝜑 → (𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ≠ (𝑁 +s 𝑃))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   class class class wbr 5086  (class class class)co 7369   No csur 27605   <s clts 27606   ≤s cles 27710   0_s c0s 27799   1_s c1s 27800   +_s cadds 27953   ·_s cmuls 28100   /_su cdivs 28181  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

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-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-seqs 28278  df-n0s 28308  df-nns 28309  df-zs 28373  df-2s 28405  df-exps 28407

This theorem is referenced by: (None)