2lgslem1c - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 2lgslem1c		Structured version Visualization version GIF version

Theorem 2lgslem1c 27344

Description: Lemma 3 for 2lgslem1 27345. (Contributed by AV, 19-Jun-2021.)

**Assertion**
Ref	Expression
2lgslem1c	⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))

Proof of Theorem 2lgslem1c Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmnn 16650	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
2		nnnn0 12515	. . . 4 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ₀)
3		oddnn02np1 16330	. . . 4 ⊢ (𝑃 ∈ ℕ₀ → (¬ 2 ∥ 𝑃 ↔ ∃𝑛 ∈ ℕ₀ ((2 · 𝑛) + 1) = 𝑃))
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝑃 ∈ ℙ → (¬ 2 ∥ 𝑃 ↔ ∃𝑛 ∈ ℕ₀ ((2 · 𝑛) + 1) = 𝑃))
5		iftrue 4536	. . . . . . . . 9 ⊢ (2 ∥ 𝑛 → if(2 ∥ 𝑛, (𝑛 / 2), ((𝑛 − 1) / 2)) = (𝑛 / 2))
6	5	adantr 479	. . . . . . . 8 ⊢ ((2 ∥ 𝑛 ∧ 𝑛 ∈ ℕ₀) → if(2 ∥ 𝑛, (𝑛 / 2), ((𝑛 − 1) / 2)) = (𝑛 / 2))
7		2nn 12321	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
8		nn0ledivnn 13125	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ₀ ∧ 2 ∈ ℕ) → (𝑛 / 2) ≤ 𝑛)
9	7, 8	mpan2 689	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 / 2) ≤ 𝑛)
10	9	adantl 480	. . . . . . . 8 ⊢ ((2 ∥ 𝑛 ∧ 𝑛 ∈ ℕ₀) → (𝑛 / 2) ≤ 𝑛)
11	6, 10	eqbrtrd 5172	. . . . . . 7 ⊢ ((2 ∥ 𝑛 ∧ 𝑛 ∈ ℕ₀) → if(2 ∥ 𝑛, (𝑛 / 2), ((𝑛 − 1) / 2)) ≤ 𝑛)
12		iffalse 4539	. . . . . . . . 9 ⊢ (¬ 2 ∥ 𝑛 → if(2 ∥ 𝑛, (𝑛 / 2), ((𝑛 − 1) / 2)) = ((𝑛 − 1) / 2))
13	12	adantr 479	. . . . . . . 8 ⊢ ((¬ 2 ∥ 𝑛 ∧ 𝑛 ∈ ℕ₀) → if(2 ∥ 𝑛, (𝑛 / 2), ((𝑛 − 1) / 2)) = ((𝑛 − 1) / 2))
14		nn0re 12517	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℝ)
15		peano2rem 11563	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
16	15	rehalfcld 12495	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → ((𝑛 − 1) / 2) ∈ ℝ)
17	14, 16	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → ((𝑛 − 1) / 2) ∈ ℝ)
18	14	rehalfcld 12495	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 / 2) ∈ ℝ)
19	14	lem1d 12183	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 − 1) ≤ 𝑛)
20	14, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 − 1) ∈ ℝ)
21		2re 12322	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
22		2pos 12351	. . . . . . . . . . . . . 14 ⊢ 0 < 2
23	21, 22	pm3.2i 469	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℝ ∧ 0 < 2)
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → (2 ∈ ℝ ∧ 0 < 2))
25		lediv1 12115	. . . . . . . . . . . 12 ⊢ (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((𝑛 − 1) ≤ 𝑛 ↔ ((𝑛 − 1) / 2) ≤ (𝑛 / 2)))
26	20, 14, 24, 25	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → ((𝑛 − 1) ≤ 𝑛 ↔ ((𝑛 − 1) / 2) ≤ (𝑛 / 2)))
27	19, 26	mpbid 231	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → ((𝑛 − 1) / 2) ≤ (𝑛 / 2))
28	17, 18, 14, 27, 9	letrd 11407	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ((𝑛 − 1) / 2) ≤ 𝑛)
29	28	adantl 480	. . . . . . . 8 ⊢ ((¬ 2 ∥ 𝑛 ∧ 𝑛 ∈ ℕ₀) → ((𝑛 − 1) / 2) ≤ 𝑛)
30	13, 29	eqbrtrd 5172	. . . . . . 7 ⊢ ((¬ 2 ∥ 𝑛 ∧ 𝑛 ∈ ℕ₀) → if(2 ∥ 𝑛, (𝑛 / 2), ((𝑛 − 1) / 2)) ≤ 𝑛)
31	11, 30	pm2.61ian 810	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → if(2 ∥ 𝑛, (𝑛 / 2), ((𝑛 − 1) / 2)) ≤ 𝑛)
32	31	ad2antlr 725	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ₀) ∧ ((2 · 𝑛) + 1) = 𝑃) → if(2 ∥ 𝑛, (𝑛 / 2), ((𝑛 − 1) / 2)) ≤ 𝑛)
33		nn0z 12619	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
34	33	adantl 480	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℤ)
35		eqcom 2734	. . . . . . 7 ⊢ (((2 · 𝑛) + 1) = 𝑃 ↔ 𝑃 = ((2 · 𝑛) + 1))
36	35	biimpi 215	. . . . . 6 ⊢ (((2 · 𝑛) + 1) = 𝑃 → 𝑃 = ((2 · 𝑛) + 1))
37		flodddiv4 16395	. . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 𝑃 = ((2 · 𝑛) + 1)) → (⌊‘(𝑃 / 4)) = if(2 ∥ 𝑛, (𝑛 / 2), ((𝑛 − 1) / 2)))
38	34, 36, 37	syl2an 594	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ₀) ∧ ((2 · 𝑛) + 1) = 𝑃) → (⌊‘(𝑃 / 4)) = if(2 ∥ 𝑛, (𝑛 / 2), ((𝑛 − 1) / 2)))
39		oveq1 7431	. . . . . . . . . 10 ⊢ (𝑃 = ((2 · 𝑛) + 1) → (𝑃 − 1) = (((2 · 𝑛) + 1) − 1))
40	39	eqcoms 2735	. . . . . . . . 9 ⊢ (((2 · 𝑛) + 1) = 𝑃 → (𝑃 − 1) = (((2 · 𝑛) + 1) − 1))
41	40	adantl 480	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ₀) ∧ ((2 · 𝑛) + 1) = 𝑃) → (𝑃 − 1) = (((2 · 𝑛) + 1) − 1))
42		2nn0 12525	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
43	42	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℕ₀)
44		id 22	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
45	43, 44	nn0mulcld 12573	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℕ₀)
46	45	nn0cnd 12570	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℂ)
47		pncan1 11674	. . . . . . . . . 10 ⊢ ((2 · 𝑛) ∈ ℂ → (((2 · 𝑛) + 1) − 1) = (2 · 𝑛))
48	46, 47	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (((2 · 𝑛) + 1) − 1) = (2 · 𝑛))
49	48	ad2antlr 725	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ₀) ∧ ((2 · 𝑛) + 1) = 𝑃) → (((2 · 𝑛) + 1) − 1) = (2 · 𝑛))
50	41, 49	eqtrd 2767	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ₀) ∧ ((2 · 𝑛) + 1) = 𝑃) → (𝑃 − 1) = (2 · 𝑛))
51	50	oveq1d 7439	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ₀) ∧ ((2 · 𝑛) + 1) = 𝑃) → ((𝑃 − 1) / 2) = ((2 · 𝑛) / 2))
52		nn0cn 12518	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
53		2cnd 12326	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℂ)
54		2ne0 12352	. . . . . . . . 9 ⊢ 2 ≠ 0
55	54	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → 2 ≠ 0)
56	52, 53, 55	divcan3d 12031	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((2 · 𝑛) / 2) = 𝑛)
57	56	ad2antlr 725	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ₀) ∧ ((2 · 𝑛) + 1) = 𝑃) → ((2 · 𝑛) / 2) = 𝑛)
58	51, 57	eqtrd 2767	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ₀) ∧ ((2 · 𝑛) + 1) = 𝑃) → ((𝑃 − 1) / 2) = 𝑛)
59	32, 38, 58	3brtr4d 5182	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ₀) ∧ ((2 · 𝑛) + 1) = 𝑃) → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))
60	59	rexlimdva2 3153	. . 3 ⊢ (𝑃 ∈ ℙ → (∃𝑛 ∈ ℕ₀ ((2 · 𝑛) + 1) = 𝑃 → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
61	4, 60	sylbid 239	. 2 ⊢ (𝑃 ∈ ℙ → (¬ 2 ∥ 𝑃 → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
62	61	imp 405	1 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2936 ∃wrex 3066 ifcif 4530 class class class wbr 5150 ‘cfv 6551 (class class class)co 7424 ℂcc 11142 ℝcr 11143 0cc0 11144 1c1 11145 + caddc 11147 · cmul 11149 < clt 11284 ≤ cle 11285 − cmin 11480 / cdiv 11907 ℕcn 12248 2c2 12303 4c4 12305 ℕ₀cn0 12508 ℤcz 12594 ⌊cfl 13793 ∥ cdvds 16236 ℙcprime 16647

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-sup 9471 df-inf 9472 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-fl 13795 df-dvds 16237 df-prm 16648

This theorem is referenced by: 2lgslem1 27345

W3C validator

	
		OSZAR »