2lgslem3 - Intuitionistic Logic Explorer

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Theorem 2lgslem3 15249

Description: Lemma 3 for 2lgs 15252. (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 9339	. . 3 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
2		lgsdir2lem3 15178	. . 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 3301	. . 3 ⊢ ((𝑃 mod 8) ∈ ({1, 7} ∪ {3, 5}) ↔ ((𝑃 mod 8) ∈ {1, 7} ∨ (𝑃 mod 8) ∈ {3, 5}))
5		elpri 3642	. . . . . . . 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 15245	. . . . . . . . . . . 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 15248	. . . . . . . . . . . 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 3563	. . . . . . 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 2229	. . . . 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 3642	. . . . 5 ⊢ ((𝑃 mod 8) ∈ {3, 5} → ((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5))
23	6	2lgslem3b1 15246	. . . . . . . . . . 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 15247	. . . . . . . . . . 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 8020	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
30		1lt3 9156	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 3
31	29, 30	ltneii 8118	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 3
32	31	nesymi 2410	. . . . . . . . . . . . . . 15 ⊢ ¬ 3 = 1
33		3re 9058	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
34		3lt7 9172	. . . . . . . . . . . . . . . . 17 ⊢ 3 < 7
35	33, 34	ltneii 8118	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 7
36	35	neii 2366	. . . . . . . . . . . . . . 15 ⊢ ¬ 3 = 7
37	32, 36	pm3.2i 272	. . . . . . . . . . . . . 14 ⊢ (¬ 3 = 1 ∧ ¬ 3 = 7)
38		eqeq1 2200	. . . . . . . . . . . . . . . 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 2200	. . . . . . . . . . . . . . . 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 9163	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 5
45	29, 44	ltneii 8118	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 5
46	45	nesymi 2410	. . . . . . . . . . . . . . 15 ⊢ ¬ 5 = 1
47		5re 9063	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℝ
48		5lt7 9170	. . . . . . . . . . . . . . . . 17 ⊢ 5 < 7
49	47, 48	ltneii 8118	. . . . . . . . . . . . . . . 16 ⊢ 5 ≠ 7
50	49	neii 2366	. . . . . . . . . . . . . . 15 ⊢ ¬ 5 = 7
51	46, 50	pm3.2i 272	. . . . . . . . . . . . . 14 ⊢ (¬ 5 = 1 ∧ ¬ 5 = 7)
52		eqeq1 2200	. . . . . . . . . . . . . . . 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 2200	. . . . . . . . . . . . . . . 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 3568	. . . . . . . 8 ⊢ ((((𝑃 mod 8) = 3 ∨ (𝑃 mod 8) = 5) ∧ 𝑃 ∈ ℕ) → if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 1)
64	28, 63	eqtr4d 2229	. . . . . . 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 2164 ∪ cun 3152 ifcif 3558 {cpr 3620 class class class wbr 4030 ‘cfv 5255 (class class class)co 5919 0cc0 7874 1c1 7875 − cmin 8192 / cdiv 8693 ℕcn 8984 2c2 9035 3c3 9036 4c4 9037 5c5 9038 7c7 9040 8c8 9041 ℤcz 9320 ⌊cfl 10340 mod cmo 10396 ∥ cdvds 11933

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-po 4328 df-iso 4329 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-7 9048 df-8 9049 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-ico 9963 df-fz 10078 df-fl 10342 df-mod 10397 df-dvds 11934

This theorem is referenced by: 2lgs 15252

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