2lgslem3 - Intuitionistic Logic Explorer

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Theorem 2lgslem3 16203

Description: Lemma 3 for 2lgs 16206. (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 9646	. . 3 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
2		lgsdir2lem3 16132	. . 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 3370	. . 3 ⊢ ((𝑃 mod 8) ∈ ({1, 7} ∪ {3, 5}) ↔ ((𝑃 mod 8) ∈ {1, 7} ∨ (𝑃 mod 8) ∈ {3, 5}))
5		elpri 3731	. . . . . . . 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 16199	. . . . . . . . . . . 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 16202	. . . . . . . . . . . 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 728	. . . . . . . 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 3645	. . . . . . 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 2274	. . . . 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 3731	. . . . 5 ⊢ ((𝑃 mod 8) ∈ {3, 5} → ((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5))
23	6	2lgslem3b1 16200	. . . . . . . . . . 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 16201	. . . . . . . . . . 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 728	. . . . . . . . 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 8319	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
30		1lt3 9459	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 3
31	29, 30	ltneii 8416	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 3
32	31	nesymi 2466	. . . . . . . . . . . . . . 15 ⊢ ¬ 3 = 1
33		3re 9361	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
34		3lt7 9475	. . . . . . . . . . . . . . . . 17 ⊢ 3 < 7
35	33, 34	ltneii 8416	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 7
36	35	neii 2422	. . . . . . . . . . . . . . 15 ⊢ ¬ 3 = 7
37	32, 36	pm3.2i 272	. . . . . . . . . . . . . 14 ⊢ (¬ 3 = 1 ∧ ¬ 3 = 7)
38		eqeq1 2245	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 3 → ((𝑃 mod 8) = 1 ↔ 3 = 1))
39	38	notbid 677	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 mod 8) = 3 → (¬ (𝑃 mod 8) = 1 ↔ ¬ 3 = 1))
40		eqeq1 2245	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 3 → ((𝑃 mod 8) = 7 ↔ 3 = 7))
41	40	notbid 677	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 mod 8) = 3 → (¬ (𝑃 mod 8) = 7 ↔ ¬ 3 = 7))
42	39, 41	anbi12d 477	. . . . . . . . . . . . . 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 9466	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 5
45	29, 44	ltneii 8416	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 5
46	45	nesymi 2466	. . . . . . . . . . . . . . 15 ⊢ ¬ 5 = 1
47		5re 9366	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℝ
48		5lt7 9473	. . . . . . . . . . . . . . . . 17 ⊢ 5 < 7
49	47, 48	ltneii 8416	. . . . . . . . . . . . . . . 16 ⊢ 5 ≠ 7
50	49	neii 2422	. . . . . . . . . . . . . . 15 ⊢ ¬ 5 = 7
51	46, 50	pm3.2i 272	. . . . . . . . . . . . . 14 ⊢ (¬ 5 = 1 ∧ ¬ 5 = 7)
52		eqeq1 2245	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 5 → ((𝑃 mod 8) = 1 ↔ 5 = 1))
53	52	notbid 677	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 mod 8) = 5 → (¬ (𝑃 mod 8) = 1 ↔ ¬ 5 = 1))
54		eqeq1 2245	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 mod 8) = 5 → ((𝑃 mod 8) = 7 ↔ 5 = 7))
55	54	notbid 677	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 mod 8) = 5 → (¬ (𝑃 mod 8) = 7 ↔ ¬ 5 = 7))
56	53, 55	anbi12d 477	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . 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 764	. . . . . . . . . . 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 637	. . . . . . . . 9 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → ¬ (𝑃 mod 8) ∈ {1, 7})
63	62	iffalsed 3650	. . . . . . . 8 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 1)
64	28, 63	eqtr4d 2274	. . . . . . 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 728	. . 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 720 = wceq 1402 ∈ wcel 2209 ∪ cun 3218 ifcif 3638 {cpr 3709 class class class wbr 4128 ‘cfv 5375 (class class class)co 6079 0cc0 8173 1c1 8174 − cmin 8491 / cdiv 8996 ℕcn 9287 2c2 9338 3c3 9339 4c4 9340 5c5 9341 7c7 9343 8c8 9344 ℤcz 9627 ⌊cfl 10686 mod cmo 10742 ∥ cdvds 12537

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 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 4247 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-pre-mulext 8291 ax-arch 8292

This theorem depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 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-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-id 4436 df-po 4439 df-iso 4440 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-ap 8904 df-div 8997 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-5 9349 df-6 9350 df-7 9351 df-8 9352 df-n0 9547 df-z 9628 df-uz 9905 df-q 10003 df-rp 10038 df-ico 10279 df-fz 10395 df-fl 10688 df-mod 10743 df-dvds 12538

This theorem is referenced by: 2lgs 16206

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