Theorem zs12bdaylem1 28447

Description: Lemma for zs12bday 28462. Prove an inequality for birthday ordering. (Contributed by Scott Fenton, 22-Feb-2026.)

Hypotheses

Ref	Expression
zs12bdaylem.1	⊢ (𝜑 → 𝑁 ∈ ℕ0s)
zs12bdaylem.2	⊢ (𝜑 → 𝑀 ∈ ℕ0s)
zs12bdaylem.3	⊢ (𝜑 → 𝑃 ∈ ℕ0s)
zs12bdaylem.4	⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃))

Assertion

Ref	Expression
zs12bdaylem1	⊢ (𝜑 → (𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ≠ (𝑁 +s 𝑃))

Proof of Theorem zs12bdaylem1

Step	Hyp	Ref	Expression
1		zs12bdaylem.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0s)
2		n0sge0 28316	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0s → 0s ≤s 𝑀)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 0s ≤s 𝑀)
4		0sno 27805	. . . . . . . . . . 11 ⊢ 0s ∈ No
5	4	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0s ∈ No )
6	1	n0snod 28304	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ No )
7		2sno 28396	. . . . . . . . . . 11 ⊢ 2s ∈ No
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2s ∈ No )
9		2nns 28395	. . . . . . . . . . 11 ⊢ 2s ∈ ℕs
10		nnsgt0 28317	. . . . . . . . . . 11 ⊢ (2s ∈ ℕs → 0s <s 2s)
11	9, 10	mp1i 13	. . . . . . . . . 10 ⊢ (𝜑 → 0s <s 2s)
12	5, 6, 8, 11	slemul2d 28154	. . . . . . . . 9 ⊢ (𝜑 → ( 0s ≤s 𝑀 ↔ (2s ·s 0s ) ≤s (2s ·s 𝑀)))
13		muls01 28092	. . . . . . . . . . 11 ⊢ (2s ∈ No → (2s ·s 0s ) = 0s )
14	7, 13	ax-mp 5	. . . . . . . . . 10 ⊢ (2s ·s 0s ) = 0s
15	14	breq1i 5104	. . . . . . . . 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 28115	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s 𝑀) ∈ No )
18		1sno 27806	. . . . . . . . . 10 ⊢ 1s ∈ No
19	18	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1s ∈ No )
20	5, 17, 19	sleadd1d 27975	. . . . . . . 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 27948	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
23	18, 22	ax-mp 5	. . . . . . . 8 ⊢ ( 0s +s 1s ) = 1s
24	23	breq1i 5104	. . . . . . 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 27960	. . . . . 6 ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) ∈ No )
28		slenlt 27722	. . . . . 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		zs12bdaylem.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 7366	. . . . . . . 8 ⊢ (𝑃 = 0s → (2s↑s𝑃) = (2s↑s 0s ))
34		exps0 28404	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
35	7, 34	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
36	33, 35	eqtrdi 2786	. . . . . . 7 ⊢ (𝑃 = 0s → (2s↑s𝑃) = 1s )
37	36	breq2d 5109	. . . . . 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		zs12bdaylem.3	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ0s)
42	27, 41	pw2divscld 28416	. . . . 5 ⊢ (𝜑 → (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) ∈ No )
43	41	n0snod 28304	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ No )
44		zs12bdaylem.1	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
45	44	n0snod 28304	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ No )
46	42, 43, 45	addscan1d 27980	. . . 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	pw2divsmuld 28417	. . . . 5 ⊢ (𝜑 → ((((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)) = 𝑃 ↔ ((2s↑s𝑃) ·s 𝑃) = ((2s ·s 𝑀) +s 1s )))
48		breq1 5100	. . . . . . . 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 28410	. . . . . . . . . . . 12 ⊢ ((2s ∈ ℕs ∧ 𝑃 ∈ ℕ0s) → (2s↑s𝑃) ∈ ℕs)
51	9, 41, 50	sylancr 588	. . . . . . . . . . 11 ⊢ (𝜑 → (2s↑s𝑃) ∈ ℕs)
52	51	nnsnod 28305	. . . . . . . . . 10 ⊢ (𝜑 → (2s↑s𝑃) ∈ No )
53		nnsgt0 28317	. . . . . . . . . . 11 ⊢ ((2s↑s𝑃) ∈ ℕs → 0s <s (2s↑s𝑃))
54	51, 53	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0s <s (2s↑s𝑃))
55	43, 19, 52, 54	sltmul2d 28152	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 <s 1s ↔ ((2s↑s𝑃) ·s 𝑃) <s ((2s↑s𝑃) ·s 1s )))
56	52	mulsridd 28094	. . . . . . . . . 10 ⊢ (𝜑 → ((2s↑s𝑃) ·s 1s ) = (2s↑s𝑃))
57	56	breq2d 5109	. . . . . . . . 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 28316	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ0s → 0s ≤s 𝑃)
60	41, 59	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 0s ≤s 𝑃)
61		sletri3 27725	. . . . . . . . . . . 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 28308	. . . . . . . . . . . 12 ⊢ 0s ∈ ℕ0s
65		n0sleltp1 28343	. . . . . . . . . . . 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 5105	. . . . . . . . . . 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 2938	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 2931   class class class wbr 5097  (class class class)co 7358   No csur 27609   <s cslt 27610   ≤s csle 27714   0_s c0s 27801   1_s c1s 27802   +_s cadds 27939   ·_s cmuls 28086   /_su cdivs 28167  ℕ_0scnn0s 28291  ℕ_scnns 28292  2_sc2s 28387  ↑_scexps 28389

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-nadd 8594  df-no 27612  df-slt 27613  df-bday 27614  df-sle 27715  df-sslt 27756  df-scut 27758  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27918  df-norec2 27929  df-adds 27940  df-negs 28001  df-subs 28002  df-muls 28087  df-divs 28168  df-seqs 28263  df-n0s 28293  df-nns 28294  df-zs 28356  df-2s 28388  df-exps 28390

This theorem is referenced by: (None)