2lgslem3 - Intuitionistic Logic Explorer

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Theorem 2lgslem3 15859

Description: Lemma 3 for 2lgs 15862. (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 9503	. . 3 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
2		lgsdir2lem3 15788	. . 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 3347	. . 3 ⊢ ((𝑃 mod 8) ∈ ({1, 7} ∪ {3, 5}) ↔ ((𝑃 mod 8) ∈ {1, 7} ∨ (𝑃 mod 8) ∈ {3, 5}))
5		elpri 3693	. . . . . . . 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 15855	. . . . . . . . . . . 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 15858	. . . . . . . . . . . 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 723	. . . . . . . 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 3611	. . . . . . 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 2266	. . . . 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 3693	. . . . 5 ⊢ ((𝑃 mod 8) ∈ {3, 5} → ((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5))
23	6	2lgslem3b1 15856	. . . . . . . . . . 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 15857	. . . . . . . . . . 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 723	. . . . . . . . 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 8183	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
30		1lt3 9320	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 3
31	29, 30	ltneii 8281	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 3
32	31	nesymi 2447	. . . . . . . . . . . . . . 15 ⊢ ¬ 3 = 1
33		3re 9222	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
34		3lt7 9336	. . . . . . . . . . . . . . . . 17 ⊢ 3 < 7
35	33, 34	ltneii 8281	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 7
36	35	neii 2403	. . . . . . . . . . . . . . 15 ⊢ ¬ 3 = 7
37	32, 36	pm3.2i 272	. . . . . . . . . . . . . 14 ⊢ (¬ 3 = 1 ∧ ¬ 3 = 7)
38		eqeq1 2237	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 3 → ((𝑃 mod 8) = 1 ↔ 3 = 1))
39	38	notbid 673	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 mod 8) = 3 → (¬ (𝑃 mod 8) = 1 ↔ ¬ 3 = 1))
40		eqeq1 2237	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 3 → ((𝑃 mod 8) = 7 ↔ 3 = 7))
41	40	notbid 673	. . . . . . . . . . . . . . 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 9327	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 5
45	29, 44	ltneii 8281	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 5
46	45	nesymi 2447	. . . . . . . . . . . . . . 15 ⊢ ¬ 5 = 1
47		5re 9227	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℝ
48		5lt7 9334	. . . . . . . . . . . . . . . . 17 ⊢ 5 < 7
49	47, 48	ltneii 8281	. . . . . . . . . . . . . . . 16 ⊢ 5 ≠ 7
50	49	neii 2403	. . . . . . . . . . . . . . 15 ⊢ ¬ 5 = 7
51	46, 50	pm3.2i 272	. . . . . . . . . . . . . 14 ⊢ (¬ 5 = 1 ∧ ¬ 5 = 7)
52		eqeq1 2237	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 5 → ((𝑃 mod 8) = 1 ↔ 5 = 1))
53	52	notbid 673	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 mod 8) = 5 → (¬ (𝑃 mod 8) = 1 ↔ ¬ 5 = 1))
54		eqeq1 2237	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 5 → ((𝑃 mod 8) = 7 ↔ 5 = 7))
55	54	notbid 673	. . . . . . . . . . . . . . 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 723	. . . . . . . . . . . 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 759	. . . . . . . . . . 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 633	. . . . . . . . 9 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → ¬ (𝑃 mod 8) ∈ {1, 7})
63	62	iffalsed 3616	. . . . . . . 8 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 1)
64	28, 63	eqtr4d 2266	. . . . . . 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 723	. . 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 715 = wceq 1397 ∈ wcel 2201 ∪ cun 3197 ifcif 3604 {cpr 3671 class class class wbr 4089 ‘cfv 5328 (class class class)co 6023 0cc0 8037 1c1 8038 − cmin 8355 / cdiv 8857 ℕcn 9148 2c2 9199 3c3 9200 4c4 9201 5c5 9202 7c7 9204 8c8 9205 ℤcz 9484 ⌊cfl 10534 mod cmo 10590 ∥ cdvds 12371

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2203 ax-14 2204 ax-ext 2212 ax-sep 4208 ax-pow 4266 ax-pr 4301 ax-un 4532 ax-setind 4637 ax-cnex 8128 ax-resscn 8129 ax-1cn 8130 ax-1re 8131 ax-icn 8132 ax-addcl 8133 ax-addrcl 8134 ax-mulcl 8135 ax-mulrcl 8136 ax-addcom 8137 ax-mulcom 8138 ax-addass 8139 ax-mulass 8140 ax-distr 8141 ax-i2m1 8142 ax-0lt1 8143 ax-1rid 8144 ax-0id 8145 ax-rnegex 8146 ax-precex 8147 ax-cnre 8148 ax-pre-ltirr 8149 ax-pre-ltwlin 8150 ax-pre-lttrn 8151 ax-pre-apti 8152 ax-pre-ltadd 8153 ax-pre-mulgt0 8154 ax-pre-mulext 8155 ax-arch 8156

This theorem depends on definitions: df-bi 117 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-xor 1420 df-nf 1509 df-sb 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ne 2402 df-nel 2497 df-ral 2514 df-rex 2515 df-reu 2516 df-rmo 2517 df-rab 2518 df-v 2803 df-sbc 3031 df-csb 3127 df-dif 3201 df-un 3203 df-in 3205 df-ss 3212 df-nul 3494 df-if 3605 df-pw 3655 df-sn 3676 df-pr 3677 df-op 3679 df-uni 3895 df-int 3930 df-iun 3973 df-br 4090 df-opab 4152 df-mpt 4153 df-id 4392 df-po 4395 df-iso 4396 df-xp 4733 df-rel 4734 df-cnv 4735 df-co 4736 df-dm 4737 df-rn 4738 df-res 4739 df-ima 4740 df-iota 5288 df-fun 5330 df-fn 5331 df-f 5332 df-fv 5336 df-riota 5976 df-ov 6026 df-oprab 6027 df-mpo 6028 df-1st 6308 df-2nd 6309 df-pnf 8221 df-mnf 8222 df-xr 8223 df-ltxr 8224 df-le 8225 df-sub 8357 df-neg 8358 df-reap 8760 df-ap 8767 df-div 8858 df-inn 9149 df-2 9207 df-3 9208 df-4 9209 df-5 9210 df-6 9211 df-7 9212 df-8 9213 df-n0 9408 df-z 9485 df-uz 9761 df-q 9859 df-rp 9894 df-ico 10134 df-fz 10249 df-fl 10536 df-mod 10591 df-dvds 12372

This theorem is referenced by: 2lgs 15862

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