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Theorem 2lgslem1a1 16422

Description: Lemma 1 for 2lgslem1a 16424. (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		elfzelz 10446	. . . . . . 7 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) → 𝑖 ∈ ℤ)
2	1	adantl 277	. . . . . 6 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → 𝑖 ∈ ℤ)
3		2z 9684	. . . . . . 7 ⊢ 2 ∈ ℤ
4	3	a1i 9	. . . . . 6 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℤ)
5	2, 4	zmulcld 9786	. . . . 5 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → (𝑖 · 2) ∈ ℤ)
6		zq 10043	. . . . 5 ⊢ ((𝑖 · 2) ∈ ℤ → (𝑖 · 2) ∈ ℚ)
7	5, 6	syl 14	. . . 4 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → (𝑖 · 2) ∈ ℚ)
8		nnq 10050	. . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
9	8	ad2antrr 492	. . . 4 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℚ)
10		elfznn 10478	. . . . . 6 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) → 𝑖 ∈ ℕ)
11		nnre 9321	. . . . . . 7 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℝ)
12		nnnn0 9582	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℕ₀)
13	12	nn0ge0d 9635	. . . . . . 7 ⊢ (𝑖 ∈ ℕ → 0 ≤ 𝑖)
14		2re 9384	. . . . . . . . 9 ⊢ 2 ∈ ℝ
15		0le2 9404	. . . . . . . . 9 ⊢ 0 ≤ 2
16	14, 15	pm3.2i 272	. . . . . . . 8 ⊢ (2 ∈ ℝ ∧ 0 ≤ 2)
17	16	a1i 9	. . . . . . 7 ⊢ (𝑖 ∈ ℕ → (2 ∈ ℝ ∧ 0 ≤ 2))
18		mulge0 8956	. . . . . . 7 ⊢ (((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) ∧ (2 ∈ ℝ ∧ 0 ≤ 2)) → 0 ≤ (𝑖 · 2))
19	11, 13, 17, 18	syl21anc 1277	. . . . . 6 ⊢ (𝑖 ∈ ℕ → 0 ≤ (𝑖 · 2))
20	10, 19	syl 14	. . . . 5 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) → 0 ≤ (𝑖 · 2))
21	20	adantl 277	. . . 4 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → 0 ≤ (𝑖 · 2))
22		elfz2 10436	. . . . . 6 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) ↔ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ ((𝑃 − 1) / 2))))
23		zre 9660	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
24	23	3ad2ant3 1051	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 ∈ ℝ)
25		zre 9660	. . . . . . . . . . 11 ⊢ (((𝑃 − 1) / 2) ∈ ℤ → ((𝑃 − 1) / 2) ∈ ℝ)
26	25	3ad2ant2 1050	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → ((𝑃 − 1) / 2) ∈ ℝ)
27		2pos 9405	. . . . . . . . . . . 12 ⊢ 0 < 2
28	14, 27	pm3.2i 272	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ ∧ 0 < 2)
29	28	a1i 9	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → (2 ∈ ℝ ∧ 0 < 2))
30		lemul1 8930	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ℝ ∧ ((𝑃 − 1) / 2) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝑖 ≤ ((𝑃 − 1) / 2) ↔ (𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2)))
31	24, 26, 29, 30	syl3anc 1278	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ ((𝑃 − 1) / 2) ↔ (𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2)))
32		nncn 9322	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℂ)
33		peano2cnm 8600	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℂ → (𝑃 − 1) ∈ ℂ)
34	32, 33	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℂ)
35		2cnd 9387	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℕ → 2 ∈ ℂ)
36		2ap0 9407	. . . . . . . . . . . . . . . . 17 ⊢ 2 # 0
37	36	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℕ → 2 # 0)
38	34, 35, 37	divcanap1d 9130	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℕ → (((𝑃 − 1) / 2) · 2) = (𝑃 − 1))
39	38	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (((𝑃 − 1) / 2) · 2) = (𝑃 − 1))
40	39	adantl 277	. . . . . . . . . . . . 13 ⊢ (((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → (((𝑃 − 1) / 2) · 2) = (𝑃 − 1))
41	40	breq2d 4143	. . . . . . . . . . . 12 ⊢ (((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → ((𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2) ↔ (𝑖 · 2) ≤ (𝑃 − 1)))
42		id 19	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℤ)
43	3	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → 2 ∈ ℤ)
44	42, 43	zmulcld 9786	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → (𝑖 · 2) ∈ ℤ)
45	44	3ad2ant3 1051	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 · 2) ∈ ℤ)
46		nnz 9675	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
47	46	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ ℤ)
48		zltlem1 9714	. . . . . . . . . . . . . 14 ⊢ (((𝑖 · 2) ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑖 · 2) < 𝑃 ↔ (𝑖 · 2) ≤ (𝑃 − 1)))
49	45, 47, 48	syl2an 289	. . . . . . . . . . . . 13 ⊢ (((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → ((𝑖 · 2) < 𝑃 ↔ (𝑖 · 2) ≤ (𝑃 − 1)))
50	49	biimprd 158	. . . . . . . . . . . 12 ⊢ (((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → ((𝑖 · 2) ≤ (𝑃 − 1) → (𝑖 · 2) < 𝑃))
51	41, 50	sylbid 150	. . . . . . . . . . 11 ⊢ (((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → ((𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2) → (𝑖 · 2) < 𝑃))
52	51	ex 115	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → ((𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2) → (𝑖 · 2) < 𝑃)))
53	52	com23 78	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → ((𝑖 · 2) ≤ (((𝑃 − 1) / 2) · 2) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑖 · 2) < 𝑃)))
54	31, 53	sylbid 150	. . . . . . . 8 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ ((𝑃 − 1) / 2) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑖 · 2) < 𝑃)))
55	54	a1d 22	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) → (1 ≤ 𝑖 → (𝑖 ≤ ((𝑃 − 1) / 2) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑖 · 2) < 𝑃))))
56	55	imp32 257	. . . . . 6 ⊢ (((1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ ((𝑃 − 1) / 2))) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑖 · 2) < 𝑃))
57	22, 56	sylbi 121	. . . . 5 ⊢ (𝑖 ∈ (1...((𝑃 − 1) / 2)) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑖 · 2) < 𝑃))
58	57	impcom 125	. . . 4 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → (𝑖 · 2) < 𝑃)
59		modqid 10808	. . . 4 ⊢ ((((𝑖 · 2) ∈ ℚ ∧ 𝑃 ∈ ℚ) ∧ (0 ≤ (𝑖 · 2) ∧ (𝑖 · 2) < 𝑃)) → ((𝑖 · 2) mod 𝑃) = (𝑖 · 2))
60	7, 9, 21, 58, 59	syl22anc 1279	. . 3 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → ((𝑖 · 2) mod 𝑃) = (𝑖 · 2))
61	60	eqcomd 2244	. 2 ⊢ (((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) ∧ 𝑖 ∈ (1...((𝑃 − 1) / 2))) → (𝑖 · 2) = ((𝑖 · 2) mod 𝑃))
62	61	ralrimiva 2623	1 ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → ∀𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑖 · 2) = ((𝑖 · 2) mod 𝑃))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ∀wral 2528 class class class wbr 4131 (class class class)co 6088 ℂcc 8184 ℝcr 8185 0cc0 8186 1c1 8187 · cmul 8191 < clt 8367 ≤ cle 8368 − cmin 8505 # cap 8918 / cdiv 9011 ℕcn 9314 2c2 9365 ℤcz 9656 ℚcq 10036 ...cfz 10429 mod cmo 10781 ∥ cdvds 12580

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4251 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305

This proof depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-id 4440 df-po 4443 df-iso 4444 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-n0 9576 df-z 9657 df-uz 9939 df-q 10037 df-rp 10073 df-fz 10430 df-fl 10723 df-mod 10782

This theorem is used by: 2lgslem1a 16424

W3C validator