2lgslem1a1 - Metamath Proof Explorer

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Theorem 2lgslem1a1 27451

Description: Lemma 1 for 2lgslem1a 27453. (Contributed by AV, 16-Jun-2021.)

**Assertion**
Ref	Expression
2lgslem1a1	⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → ∀𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑖 · 2) = ((𝑖 · 2) mod 𝑃))

Distinct variable group: 𝑃,𝑖

**Proof of Theorem 2lgslem1a1**
Step	Hyp	Ref	Expression
1		nnrp 13068	. . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ⁺)
2	1	adantr 480	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ ℝ⁺)
3		elfzelz 13584	. . . . . 6 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) → 𝑖 ∈ ℤ)
4		zre 12643	. . . . . . 7 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
5		2re 12367	. . . . . . . 8 ⊢ 2 ∈ ℝ
6	5	a1i 11	. . . . . . 7 ⊢ (𝑖 ∈ ℤ → 2 ∈ ℝ)
7	4, 6	remulcld 11320	. . . . . 6 ⊢ (𝑖 ∈ ℤ → (𝑖 · 2) ∈ ℝ)
8	3, 7	syl 17	. . . . 5 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) → (𝑖 · 2) ∈ ℝ)
9	2, 8	anim12ci 613	. . . 4 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → ((𝑖 · 2) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺))
10		elfznn 13613	. . . . . 6 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) → 𝑖 ∈ ℕ)
11		nnre 12300	. . . . . . 7 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℝ)
12		nnnn0 12560	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℕ₀)
13	12	nn0ge0d 12616	. . . . . . 7 ⊢ (𝑖 ∈ ℕ → 0 ≤ 𝑖)
14		0le2 12395	. . . . . . . . 9 ⊢ 0 ≤ 2
15	5, 14	pm3.2i 470	. . . . . . . 8 ⊢ (2 ∈ ℝ ∧ 0 ≤ 2)
16	15	a1i 11	. . . . . . 7 ⊢ (𝑖 ∈ ℕ → (2 ∈ ℝ ∧ 0 ≤ 2))
17		mulge0 11808	. . . . . . 7 ⊢ (((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) ∧ (2 ∈ ℝ ∧ 0 ≤ 2)) → 0 ≤ (𝑖 · 2))
18	11, 13, 16, 17	syl21anc 837	. . . . . 6 ⊢ (𝑖 ∈ ℕ → 0 ≤ (𝑖 · 2))
19	10, 18	syl 17	. . . . 5 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) → 0 ≤ (𝑖 · 2))
20	19	adantl 481	. . . 4 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → 0 ≤ (𝑖 · 2))
21		elfz2 13574	. . . . . 6 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) ↔ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ ((𝑃 − 1) / 2))))
22	4	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 ∈ ℝ)
23		zre 12643	. . . . . . . . . . 11 ⊢ (((𝑃 − 1) / 2) ∈ ℤ → ((𝑃 − 1) / 2) ∈ ℝ)
24	23	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → ((𝑃 − 1) / 2) ∈ ℝ)
25		2pos 12396	. . . . . . . . . . . 12 ⊢ 0 < 2
26	5, 25	pm3.2i 470	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ ∧ 0 < 2)
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → (2 ∈ ℝ ∧ 0 < 2))
28		lemul1 12146	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ℝ ∧ ((𝑃 − 1) / 2) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝑖 ≤ ((𝑃 − 1) / 2) ↔ (𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2)))
29	22, 24, 27, 28	syl3anc 1371	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ ((𝑃 − 1) / 2) ↔ (𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2)))
30		nncn 12301	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℂ)
31		peano2cnm 11602	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℂ → (𝑃 − 1) ∈ ℂ)
32	30, 31	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℂ)
33		2cnd 12371	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℕ → 2 ∈ ℂ)
34		2ne0 12397	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≠ 0
35	34	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℕ → 2 ≠ 0)
36	32, 33, 35	divcan1d 12071	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℕ → (((𝑃 − 1) / 2) · 2) = (𝑃 − 1))
37	36	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (((𝑃 − 1) / 2) · 2) = (𝑃 − 1))
38	37	adantl 481	. . . . . . . . . . . . 13 ⊢ (((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → (((𝑃 − 1) / 2) · 2) = (𝑃 − 1))
39	38	breq2d 5178	. . . . . . . . . . . 12 ⊢ (((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → ((𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2) ↔ (𝑖 · 2) ≤ (𝑃 − 1)))
40		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℤ)
41		2z 12675	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
42	41	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → 2 ∈ ℤ)
43	40, 42	zmulcld 12753	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → (𝑖 · 2) ∈ ℤ)
44	43	3ad2ant3 1135	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 · 2) ∈ ℤ)
45		nnz 12660	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
46	45	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ ℤ)
47		zltlem1 12696	. . . . . . . . . . . . . 14 ⊢ (((𝑖 · 2) ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑖 · 2) < 𝑃 ↔ (𝑖 · 2) ≤ (𝑃 − 1)))
48	44, 46, 47	syl2an 595	. . . . . . . . . . . . 13 ⊢ (((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → ((𝑖 · 2) < 𝑃 ↔ (𝑖 · 2) ≤ (𝑃 − 1)))
49	48	biimprd 248	. . . . . . . . . . . 12 ⊢ (((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → ((𝑖 · 2) ≤ (𝑃 − 1) → (𝑖 · 2) < 𝑃))
50	39, 49	sylbid 240	. . . . . . . . . . 11 ⊢ (((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → ((𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2) → (𝑖 · 2) < 𝑃))
51	50	ex 412	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → ((𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2) → (𝑖 · 2) < 𝑃)))
52	51	com23 86	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → ((𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑖 · 2) < 𝑃)))
53	29, 52	sylbid 240	. . . . . . . 8 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ ((𝑃 − 1) / 2) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑖 · 2) < 𝑃)))
54	53	a1d 25	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → (1 ≤ 𝑖 → (𝑖 ≤ ((𝑃 − 1) / 2) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑖 · 2) < 𝑃))))
55	54	imp32 418	. . . . . 6 ⊢ (((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ ((𝑃 − 1) / 2))) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑖 · 2) < 𝑃))
56	21, 55	sylbi 217	. . . . 5 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑖 · 2) < 𝑃))
57	56	impcom 407	. . . 4 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → (𝑖 · 2) < 𝑃)
58		modid 13947	. . . 4 ⊢ ((((𝑖 · 2) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ (𝑖 · 2) ∧ (𝑖 · 2) < 𝑃)) → ((𝑖 · 2) mod 𝑃) = (𝑖 · 2))
59	9, 20, 57, 58	syl12anc 836	. . 3 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → ((𝑖 · 2) mod 𝑃) = (𝑖 · 2))
60	59	eqcomd 2746	. 2 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → (𝑖 · 2) = ((𝑖 · 2) mod 𝑃))
61	60	ralrimiva 3152	1 ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → ∀𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑖 · 2) = ((𝑖 · 2) mod 𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 class class class wbr 5166 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 ℤcz 12639 ℝ⁺crp 13057 ...cfz 13567 mod cmo 13920 ∥ cdvds 16302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fz 13568 df-fl 13843 df-mod 13921

This theorem is referenced by: 2lgslem1a 27453

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