2lgslem3 - Intuitionistic Logic Explorer

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Theorem 2lgslem3 15342

Description: Lemma 3 for 2lgs 15345. (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 9345	. . 3 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
2		lgsdir2lem3 15271	. . 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 3304	. . 3 ⊢ ((𝑃 mod 8) ∈ ({1, 7} ∪ {3, 5}) ↔ ((𝑃 mod 8) ∈ {1, 7} ∨ (𝑃 mod 8) ∈ {3, 5}))
5		elpri 3645	. . . . . . . 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 15338	. . . . . . . . . . . 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 15341	. . . . . . . . . . . 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 717	. . . . . . . 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 3566	. . . . . . 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 2232	. . . . 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 3645	. . . . 5 ⊢ ((𝑃 mod 8) ∈ {3, 5} → ((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5))
23	6	2lgslem3b1 15339	. . . . . . . . . . 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 15340	. . . . . . . . . . 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 717	. . . . . . . . 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 8025	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
30		1lt3 9162	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 3
31	29, 30	ltneii 8123	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 3
32	31	nesymi 2413	. . . . . . . . . . . . . . 15 ⊢ ¬ 3 = 1
33		3re 9064	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
34		3lt7 9178	. . . . . . . . . . . . . . . . 17 ⊢ 3 < 7
35	33, 34	ltneii 8123	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 7
36	35	neii 2369	. . . . . . . . . . . . . . 15 ⊢ ¬ 3 = 7
37	32, 36	pm3.2i 272	. . . . . . . . . . . . . 14 ⊢ (¬ 3 = 1 ∧ ¬ 3 = 7)
38		eqeq1 2203	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 3 → ((𝑃 mod 8) = 1 ↔ 3 = 1))
39	38	notbid 668	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 mod 8) = 3 → (¬ (𝑃 mod 8) = 1 ↔ ¬ 3 = 1))
40		eqeq1 2203	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 3 → ((𝑃 mod 8) = 7 ↔ 3 = 7))
41	40	notbid 668	. . . . . . . . . . . . . . 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 9169	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 5
45	29, 44	ltneii 8123	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 5
46	45	nesymi 2413	. . . . . . . . . . . . . . 15 ⊢ ¬ 5 = 1
47		5re 9069	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℝ
48		5lt7 9176	. . . . . . . . . . . . . . . . 17 ⊢ 5 < 7
49	47, 48	ltneii 8123	. . . . . . . . . . . . . . . 16 ⊢ 5 ≠ 7
50	49	neii 2369	. . . . . . . . . . . . . . 15 ⊢ ¬ 5 = 7
51	46, 50	pm3.2i 272	. . . . . . . . . . . . . 14 ⊢ (¬ 5 = 1 ∧ ¬ 5 = 7)
52		eqeq1 2203	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 5 → ((𝑃 mod 8) = 1 ↔ 5 = 1))
53	52	notbid 668	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 mod 8) = 5 → (¬ (𝑃 mod 8) = 1 ↔ ¬ 5 = 1))
54		eqeq1 2203	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 5 → ((𝑃 mod 8) = 7 ↔ 5 = 7))
55	54	notbid 668	. . . . . . . . . . . . . . 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 717	. . . . . . . . . . . 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 753	. . . . . . . . . . 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 629	. . . . . . . . 9 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → ¬ (𝑃 mod 8) ∈ {1, 7})
63	62	iffalsed 3571	. . . . . . . 8 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 1)
64	28, 63	eqtr4d 2232	. . . . . . 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 717	. . 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 709 = wceq 1364 ∈ wcel 2167 ∪ cun 3155 ifcif 3561 {cpr 3623 class class class wbr 4033 ‘cfv 5258 (class class class)co 5922 0cc0 7879 1c1 7880 − cmin 8197 / cdiv 8699 ℕcn 8990 2c2 9041 3c3 9042 4c4 9043 5c5 9044 7c7 9046 8c8 9047 ℤcz 9326 ⌊cfl 10358 mod cmo 10414 ∥ cdvds 11952

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-po 4331 df-iso 4332 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-5 9052 df-6 9053 df-7 9054 df-8 9055 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-ico 9969 df-fz 10084 df-fl 10360 df-mod 10415 df-dvds 11953

This theorem is referenced by: 2lgs 15345

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