2lgslem1a1 - Metamath Proof Explorer

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Theorem 2lgslem1a1 27448

Description: Lemma 1 for 2lgslem1a 27450. (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 13044	. . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ⁺)
2	1	adantr 480	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ ℝ⁺)
3		elfzelz 13561	. . . . . 6 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) → 𝑖 ∈ ℤ)
4		zre 12615	. . . . . . 7 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
5		2re 12338	. . . . . . . 8 ⊢ 2 ∈ ℝ
6	5	a1i 11	. . . . . . 7 ⊢ (𝑖 ∈ ℤ → 2 ∈ ℝ)
7	4, 6	remulcld 11289	. . . . . 6 ⊢ (𝑖 ∈ ℤ → (𝑖 · 2) ∈ ℝ)
8	3, 7	syl 17	. . . . 5 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) → (𝑖 · 2) ∈ ℝ)
9	2, 8	anim12ci 614	. . . 4 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → ((𝑖 · 2) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺))
10		elfznn 13590	. . . . . 6 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) → 𝑖 ∈ ℕ)
11		nnre 12271	. . . . . . 7 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℝ)
12		nnnn0 12531	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℕ₀)
13	12	nn0ge0d 12588	. . . . . . 7 ⊢ (𝑖 ∈ ℕ → 0 ≤ 𝑖)
14		0le2 12366	. . . . . . . . 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 11779	. . . . . . 7 ⊢ (((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) ∧ (2 ∈ ℝ ∧ 0 ≤ 2)) → 0 ≤ (𝑖 · 2))
18	11, 13, 16, 17	syl21anc 838	. . . . . 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 13551	. . . . . 6 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) ↔ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ ((𝑃 − 1) / 2))))
22	4	3ad2ant3 1134	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 ∈ ℝ)
23		zre 12615	. . . . . . . . . . 11 ⊢ (((𝑃 − 1) / 2) ∈ ℤ → ((𝑃 − 1) / 2) ∈ ℝ)
24	23	3ad2ant2 1133	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → ((𝑃 − 1) / 2) ∈ ℝ)
25		2pos 12367	. . . . . . . . . . . 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 12117	. . . . . . . . . 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 1370	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ ((𝑃 − 1) / 2) ↔ (𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2)))
30		nncn 12272	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℂ)
31		peano2cnm 11573	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℂ → (𝑃 − 1) ∈ ℂ)
32	30, 31	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℂ)
33		2cnd 12342	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℕ → 2 ∈ ℂ)
34		2ne0 12368	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≠ 0
35	34	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℕ → 2 ≠ 0)
36	32, 33, 35	divcan1d 12042	. . . . . . . . . . . . . . 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 5160	. . . . . . . . . . . 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 12647	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
42	41	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → 2 ∈ ℤ)
43	40, 42	zmulcld 12726	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → (𝑖 · 2) ∈ ℤ)
44	43	3ad2ant3 1134	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 · 2) ∈ ℤ)
45		nnz 12632	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
46	45	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ ℤ)
47		zltlem1 12668	. . . . . . . . . . . . . 14 ⊢ (((𝑖 · 2) ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑖 · 2) < 𝑃 ↔ (𝑖 · 2) ≤ (𝑃 − 1)))
48	44, 46, 47	syl2an 596	. . . . . . . . . . . . 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 13933	. . . 4 ⊢ ((((𝑖 · 2) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ (𝑖 · 2) ∧ (𝑖 · 2) < 𝑃)) → ((𝑖 · 2) mod 𝑃) = (𝑖 · 2))
59	9, 20, 57, 58	syl12anc 837	. . 3 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → ((𝑖 · 2) mod 𝑃) = (𝑖 · 2))
60	59	eqcomd 2741	. 2 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → (𝑖 · 2) = ((𝑖 · 2) mod 𝑃))
61	60	ralrimiva 3144	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 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 class class class wbr 5148 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 2c2 12319 ℤcz 12611 ℝ⁺crp 13032 ...cfz 13544 mod cmo 13906 ∥ cdvds 16287

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fl 13829 df-mod 13907

This theorem is referenced by: 2lgslem1a 27450

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