2lgslem3 - Intuitionistic Logic Explorer

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Theorem 2lgslem3 15788

Description: Lemma 3 for 2lgs 15791. (Contributed by AV, 16-Jul-2021.)

**Hypothesis**
Ref	Expression
2lgslem2.n	⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))

**Assertion**
Ref	Expression
2lgslem3	⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1))

**Proof of Theorem 2lgslem3**
Step	Hyp	Ref	Expression
1		nnz 9473	. . 3 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
2		lgsdir2lem3 15717	. . 3 ⊢ ((𝑃 ∈ ℤ ∧ ¬ 2 ∥ 𝑃) → (𝑃 mod 8) ∈ ({1, 7} ∪ {3, 5}))
3	1, 2	sylan 283	. 2 ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑃 mod 8) ∈ ({1, 7} ∪ {3, 5}))
4		elun 3345	. . 3 ⊢ ((𝑃 mod 8) ∈ ({1, 7} ∪ {3, 5}) ↔ ((𝑃 mod 8) ∈ {1, 7} ∨ (𝑃 mod 8) ∈ {3, 5}))
5		elpri 3689	. . . . . . . 8 ⊢ ((𝑃 mod 8) ∈ {1, 7} → ((𝑃 mod 8) = 1 ∨ (𝑃 mod 8) = 7))
6		2lgslem2.n	. . . . . . . . . . . . 13 ⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))
7	6	2lgslem3a1 15784	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 1) → (𝑁 mod 2) = 0)
8	7	a1d 22	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 1) → (¬ 2 ∥ 𝑃 → (𝑁 mod 2) = 0))
9	8	expcom 116	. . . . . . . . . 10 ⊢ ((𝑃 mod 8) = 1 → (𝑃 ∈ ℕ → (¬ 2 ∥ 𝑃 → (𝑁 mod 2) = 0)))
10	9	impd 254	. . . . . . . . 9 ⊢ ((𝑃 mod 8) = 1 → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = 0))
11	6	2lgslem3d1 15787	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 7) → (𝑁 mod 2) = 0)
12	11	a1d 22	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 7) → (¬ 2 ∥ 𝑃 → (𝑁 mod 2) = 0))
13	12	expcom 116	. . . . . . . . . 10 ⊢ ((𝑃 mod 8) = 7 → (𝑃 ∈ ℕ → (¬ 2 ∥ 𝑃 → (𝑁 mod 2) = 0)))
14	13	impd 254	. . . . . . . . 9 ⊢ ((𝑃 mod 8) = 7 → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = 0))
15	10, 14	jaoi 721	. . . . . . . 8 ⊢ (((𝑃 mod 8) = 1 ∨ (𝑃 mod 8) = 7) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = 0))
16	5, 15	syl 14	. . . . . . 7 ⊢ ((𝑃 mod 8) ∈ {1, 7} → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = 0))
17	16	imp 124	. . . . . 6 ⊢ (((𝑃 mod 8) ∈ {1, 7} ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → (𝑁 mod 2) = 0)
18		iftrue 3607	. . . . . . 7 ⊢ ((𝑃 mod 8) ∈ {1, 7} → if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0)
19	18	adantr 276	. . . . . 6 ⊢ (((𝑃 mod 8) ∈ {1, 7} ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0)
20	17, 19	eqtr4d 2265	. . . . 5 ⊢ (((𝑃 mod 8) ∈ {1, 7} ∧ (𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃)) → (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1))
21	20	ex 115	. . . 4 ⊢ ((𝑃 mod 8) ∈ {1, 7} → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1)))
22		elpri 3689	. . . . 5 ⊢ ((𝑃 mod 8) ∈ {3, 5} → ((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5))
23	6	2lgslem3b1 15785	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 3) → (𝑁 mod 2) = 1)
24	23	expcom 116	. . . . . . . . . 10 ⊢ ((𝑃 mod 8) = 3 → (𝑃 ∈ ℕ → (𝑁 mod 2) = 1))
25	6	2lgslem3c1 15786	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 5) → (𝑁 mod 2) = 1)
26	25	expcom 116	. . . . . . . . . 10 ⊢ ((𝑃 mod 8) = 5 → (𝑃 ∈ ℕ → (𝑁 mod 2) = 1))
27	24, 26	jaoi 721	. . . . . . . . 9 ⊢ (((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) → (𝑃 ∈ ℕ → (𝑁 mod 2) = 1))
28	27	imp 124	. . . . . . . 8 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → (𝑁 mod 2) = 1)
29		1re 8153	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
30		1lt3 9290	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 3
31	29, 30	ltneii 8251	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 3
32	31	nesymi 2446	. . . . . . . . . . . . . . 15 ⊢ ¬ 3 = 1
33		3re 9192	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
34		3lt7 9306	. . . . . . . . . . . . . . . . 17 ⊢ 3 < 7
35	33, 34	ltneii 8251	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 7
36	35	neii 2402	. . . . . . . . . . . . . . 15 ⊢ ¬ 3 = 7
37	32, 36	pm3.2i 272	. . . . . . . . . . . . . 14 ⊢ (¬ 3 = 1 ∧ ¬ 3 = 7)
38		eqeq1 2236	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 3 → ((𝑃 mod 8) = 1 ↔ 3 = 1))
39	38	notbid 671	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 mod 8) = 3 → (¬ (𝑃 mod 8) = 1 ↔ ¬ 3 = 1))
40		eqeq1 2236	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 3 → ((𝑃 mod 8) = 7 ↔ 3 = 7))
41	40	notbid 671	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 mod 8) = 3 → (¬ (𝑃 mod 8) = 7 ↔ ¬ 3 = 7))
42	39, 41	anbi12d 473	. . . . . . . . . . . . . 14 ⊢ ((𝑃 mod 8) = 3 → ((¬ (𝑃 mod 8) = 1 ∧ ¬ (𝑃 mod 8) = 7) ↔ (¬ 3 = 1 ∧ ¬ 3 = 7)))
43	37, 42	mpbiri 168	. . . . . . . . . . . . 13 ⊢ ((𝑃 mod 8) = 3 → (¬ (𝑃 mod 8) = 1 ∧ ¬ (𝑃 mod 8) = 7))
44		1lt5 9297	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 5
45	29, 44	ltneii 8251	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 5
46	45	nesymi 2446	. . . . . . . . . . . . . . 15 ⊢ ¬ 5 = 1
47		5re 9197	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℝ
48		5lt7 9304	. . . . . . . . . . . . . . . . 17 ⊢ 5 < 7
49	47, 48	ltneii 8251	. . . . . . . . . . . . . . . 16 ⊢ 5 ≠ 7
50	49	neii 2402	. . . . . . . . . . . . . . 15 ⊢ ¬ 5 = 7
51	46, 50	pm3.2i 272	. . . . . . . . . . . . . 14 ⊢ (¬ 5 = 1 ∧ ¬ 5 = 7)
52		eqeq1 2236	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 5 → ((𝑃 mod 8) = 1 ↔ 5 = 1))
53	52	notbid 671	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 mod 8) = 5 → (¬ (𝑃 mod 8) = 1 ↔ ¬ 5 = 1))
54		eqeq1 2236	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 5 → ((𝑃 mod 8) = 7 ↔ 5 = 7))
55	54	notbid 671	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 mod 8) = 5 → (¬ (𝑃 mod 8) = 7 ↔ ¬ 5 = 7))
56	53, 55	anbi12d 473	. . . . . . . . . . . . . 14 ⊢ ((𝑃 mod 8) = 5 → ((¬ (𝑃 mod 8) = 1 ∧ ¬ (𝑃 mod 8) = 7) ↔ (¬ 5 = 1 ∧ ¬ 5 = 7)))
57	51, 56	mpbiri 168	. . . . . . . . . . . . 13 ⊢ ((𝑃 mod 8) = 5 → (¬ (𝑃 mod 8) = 1 ∧ ¬ (𝑃 mod 8) = 7))
58	43, 57	jaoi 721	. . . . . . . . . . . 12 ⊢ (((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) → (¬ (𝑃 mod 8) = 1 ∧ ¬ (𝑃 mod 8) = 7))
59	58	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → (¬ (𝑃 mod 8) = 1 ∧ ¬ (𝑃 mod 8) = 7))
60		ioran 757	. . . . . . . . . . 11 ⊢ (¬ ((𝑃 mod 8) = 1 ∨ (𝑃 mod 8) = 7) ↔ (¬ (𝑃 mod 8) = 1 ∧ ¬ (𝑃 mod 8) = 7))
61	59, 60	sylibr 134	. . . . . . . . . 10 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → ¬ ((𝑃 mod 8) = 1 ∨ (𝑃 mod 8) = 7))
62	61, 5	nsyl 631	. . . . . . . . 9 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → ¬ (𝑃 mod 8) ∈ {1, 7})
63	62	iffalsed 3612	. . . . . . . 8 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 1)
64	28, 63	eqtr4d 2265	. . . . . . 7 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1))
65	64	a1d 22	. . . . . 6 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → (¬ 2 ∥ 𝑃 → (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1)))
66	65	expimpd 363	. . . . 5 ⊢ (((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1)))
67	22, 66	syl 14	. . . 4 ⊢ ((𝑃 mod 8) ∈ {3, 5} → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1)))
68	21, 67	jaoi 721	. . 3 ⊢ (((𝑃 mod 8) ∈ {1, 7} ∨ (𝑃 mod 8) ∈ {3, 5}) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1)))
69	4, 68	sylbi 121	. 2 ⊢ ((𝑃 mod 8) ∈ ({1, 7} ∪ {3, 5}) → ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1)))
70	3, 69	mpcom 36	1 ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 713 = wceq 1395 ∈ wcel 2200 ∪ cun 3195 ifcif 3602 {cpr 3667 class class class wbr 4083 ‘cfv 5318 (class class class)co 6007 0cc0 8007 1c1 8008 − cmin 8325 / cdiv 8827 ℕcn 9118 2c2 9169 3c3 9170 4c4 9171 5c5 9172 7c7 9174 8c8 9175 ℤcz 9454 ⌊cfl 10496 mod cmo 10552 ∥ cdvds 12306

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-xor 1418 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-id 4384 df-po 4387 df-iso 4388 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-5 9180 df-6 9181 df-7 9182 df-8 9183 df-n0 9378 df-z 9455 df-uz 9731 df-q 9823 df-rp 9858 df-ico 10098 df-fz 10213 df-fl 10498 df-mod 10553 df-dvds 12307

This theorem is referenced by: 2lgs 15791

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