lgsdir2lem2 - Intuitionistic Logic Explorer

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Theorem lgsdir2lem2 14272

Description: Lemma for lgsdir2 14276. (Contributed by Mario Carneiro, 4-Feb-2015.)

**Hypotheses**
Ref	Expression
lgsdir2lem2.1	⊢ (𝐾 ∈ ℤ ∧ 2 ∥ (𝐾 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝐾) → (𝐴 mod 8) ∈ 𝑆)))
lgsdir2lem2.2	⊢ 𝑀 = (𝐾 + 1)
lgsdir2lem2.3	⊢ 𝑁 = (𝑀 + 1)
lgsdir2lem2.4	⊢ 𝑁 ∈ 𝑆

**Assertion**
Ref	Expression
lgsdir2lem2	⊢ (𝑁 ∈ ℤ ∧ 2 ∥ (𝑁 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝑁) → (𝐴 mod 8) ∈ 𝑆)))

**Proof of Theorem lgsdir2lem2**
Step	Hyp	Ref	Expression
1		lgsdir2lem2.3	. . 3 ⊢ 𝑁 = (𝑀 + 1)
2		lgsdir2lem2.2	. . . . 5 ⊢ 𝑀 = (𝐾 + 1)
3		lgsdir2lem2.1	. . . . . . 7 ⊢ (𝐾 ∈ ℤ ∧ 2 ∥ (𝐾 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝐾) → (𝐴 mod 8) ∈ 𝑆)))
4	3	simp1i 1006	. . . . . 6 ⊢ 𝐾 ∈ ℤ
5		peano2z 9283	. . . . . 6 ⊢ (𝐾 ∈ ℤ → (𝐾 + 1) ∈ ℤ)
6	4, 5	ax-mp 5	. . . . 5 ⊢ (𝐾 + 1) ∈ ℤ
7	2, 6	eqeltri 2250	. . . 4 ⊢ 𝑀 ∈ ℤ
8		peano2z 9283	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ)
9	7, 8	ax-mp 5	. . 3 ⊢ (𝑀 + 1) ∈ ℤ
10	1, 9	eqeltri 2250	. 2 ⊢ 𝑁 ∈ ℤ
11	3	simp2i 1007	. . . 4 ⊢ 2 ∥ (𝐾 + 1)
12		2z 9275	. . . . 5 ⊢ 2 ∈ ℤ
13		dvdsadd 11834	. . . . 5 ⊢ ((2 ∈ ℤ ∧ (𝐾 + 1) ∈ ℤ) → (2 ∥ (𝐾 + 1) ↔ 2 ∥ (2 + (𝐾 + 1))))
14	12, 6, 13	mp2an 426	. . . 4 ⊢ (2 ∥ (𝐾 + 1) ↔ 2 ∥ (2 + (𝐾 + 1)))
15	11, 14	mpbi 145	. . 3 ⊢ 2 ∥ (2 + (𝐾 + 1))
16		zcn 9252	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
17	4, 16	ax-mp 5	. . . . . . . . . 10 ⊢ 𝐾 ∈ ℂ
18		ax-1cn 7899	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
19	17, 18	addcomi 8095	. . . . . . . . 9 ⊢ (𝐾 + 1) = (1 + 𝐾)
20	2, 19	eqtri 2198	. . . . . . . 8 ⊢ 𝑀 = (1 + 𝐾)
21	20	oveq1i 5880	. . . . . . 7 ⊢ (𝑀 + 1) = ((1 + 𝐾) + 1)
22	1, 21	eqtri 2198	. . . . . 6 ⊢ 𝑁 = ((1 + 𝐾) + 1)
23		df-2 8972	. . . . . . . 8 ⊢ 2 = (1 + 1)
24	23	oveq1i 5880	. . . . . . 7 ⊢ (2 + 𝐾) = ((1 + 1) + 𝐾)
25	18, 17, 18	add32i 8115	. . . . . . 7 ⊢ ((1 + 𝐾) + 1) = ((1 + 1) + 𝐾)
26	24, 25	eqtr4i 2201	. . . . . 6 ⊢ (2 + 𝐾) = ((1 + 𝐾) + 1)
27	22, 26	eqtr4i 2201	. . . . 5 ⊢ 𝑁 = (2 + 𝐾)
28	27	oveq1i 5880	. . . 4 ⊢ (𝑁 + 1) = ((2 + 𝐾) + 1)
29		2cn 8984	. . . . 5 ⊢ 2 ∈ ℂ
30	29, 17, 18	addassi 7960	. . . 4 ⊢ ((2 + 𝐾) + 1) = (2 + (𝐾 + 1))
31	28, 30	eqtri 2198	. . 3 ⊢ (𝑁 + 1) = (2 + (𝐾 + 1))
32	15, 31	breqtrri 4028	. 2 ⊢ 2 ∥ (𝑁 + 1)
33		elfzuz2 10022	. . . . 5 ⊢ ((𝐴 mod 8) ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘0))
34		fzm1 10093	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘0) → ((𝐴 mod 8) ∈ (0...𝑁) ↔ ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁)))
35	33, 34	syl 14	. . . 4 ⊢ ((𝐴 mod 8) ∈ (0...𝑁) → ((𝐴 mod 8) ∈ (0...𝑁) ↔ ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁)))
36	35	ibi 176	. . 3 ⊢ ((𝐴 mod 8) ∈ (0...𝑁) → ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁))
37		elfzuz2 10022	. . . . . . . 8 ⊢ ((𝐴 mod 8) ∈ (0...𝑀) → 𝑀 ∈ (ℤ_≥‘0))
38		fzm1 10093	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘0) → ((𝐴 mod 8) ∈ (0...𝑀) ↔ ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀)))
39	37, 38	syl 14	. . . . . . 7 ⊢ ((𝐴 mod 8) ∈ (0...𝑀) → ((𝐴 mod 8) ∈ (0...𝑀) ↔ ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀)))
40	39	ibi 176	. . . . . 6 ⊢ ((𝐴 mod 8) ∈ (0...𝑀) → ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀))
41		zcn 9252	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
42	7, 41	ax-mp 5	. . . . . . . 8 ⊢ 𝑀 ∈ ℂ
43	42, 18, 1	mvrraddi 8168	. . . . . . 7 ⊢ (𝑁 − 1) = 𝑀
44	43	oveq2i 5881	. . . . . 6 ⊢ (0...(𝑁 − 1)) = (0...𝑀)
45	40, 44	eleq2s 2272	. . . . 5 ⊢ ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) → ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀))
46	17, 18, 2	mvrraddi 8168	. . . . . . . . 9 ⊢ (𝑀 − 1) = 𝐾
47	46	oveq2i 5881	. . . . . . . 8 ⊢ (0...(𝑀 − 1)) = (0...𝐾)
48	47	eleq2i 2244	. . . . . . 7 ⊢ ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ↔ (𝐴 mod 8) ∈ (0...𝐾))
49	3	simp3i 1008	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝐾) → (𝐴 mod 8) ∈ 𝑆))
50	48, 49	biimtrid 152	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...(𝑀 − 1)) → (𝐴 mod 8) ∈ 𝑆))
51		2nn 9074	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
52		8nn 9080	. . . . . . . . . . 11 ⊢ 8 ∈ ℕ
53		4z 9277	. . . . . . . . . . . . . 14 ⊢ 4 ∈ ℤ
54		dvdsmul2 11812	. . . . . . . . . . . . . 14 ⊢ ((4 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (4 · 2))
55	53, 12, 54	mp2an 426	. . . . . . . . . . . . 13 ⊢ 2 ∥ (4 · 2)
56		4t2e8 9071	. . . . . . . . . . . . 13 ⊢ (4 · 2) = 8
57	55, 56	breqtri 4026	. . . . . . . . . . . 12 ⊢ 2 ∥ 8
58		dvdsmod 11858	. . . . . . . . . . . 12 ⊢ (((2 ∈ ℕ ∧ 8 ∈ ℕ ∧ 𝐴 ∈ ℤ) ∧ 2 ∥ 8) → (2 ∥ (𝐴 mod 8) ↔ 2 ∥ 𝐴))
59	57, 58	mpan2 425	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 8 ∈ ℕ ∧ 𝐴 ∈ ℤ) → (2 ∥ (𝐴 mod 8) ↔ 2 ∥ 𝐴))
60	51, 52, 59	mp3an12 1327	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → (2 ∥ (𝐴 mod 8) ↔ 2 ∥ 𝐴))
61	60	notbid 667	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → (¬ 2 ∥ (𝐴 mod 8) ↔ ¬ 2 ∥ 𝐴))
62	61	biimpar 297	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ¬ 2 ∥ (𝐴 mod 8))
63	11, 2	breqtrri 4028	. . . . . . . . 9 ⊢ 2 ∥ 𝑀
64		id 19	. . . . . . . . 9 ⊢ ((𝐴 mod 8) = 𝑀 → (𝐴 mod 8) = 𝑀)
65	63, 64	breqtrrid 4039	. . . . . . . 8 ⊢ ((𝐴 mod 8) = 𝑀 → 2 ∥ (𝐴 mod 8))
66	62, 65	nsyl 628	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ¬ (𝐴 mod 8) = 𝑀)
67	66	pm2.21d 619	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) = 𝑀 → (𝐴 mod 8) ∈ 𝑆))
68	50, 67	jaod 717	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → (((𝐴 mod 8) ∈ (0...(𝑀 − 1)) ∨ (𝐴 mod 8) = 𝑀) → (𝐴 mod 8) ∈ 𝑆))
69	45, 68	syl5 32	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...(𝑁 − 1)) → (𝐴 mod 8) ∈ 𝑆))
70		lgsdir2lem2.4	. . . . . 6 ⊢ 𝑁 ∈ 𝑆
71		eleq1 2240	. . . . . 6 ⊢ ((𝐴 mod 8) = 𝑁 → ((𝐴 mod 8) ∈ 𝑆 ↔ 𝑁 ∈ 𝑆))
72	70, 71	mpbiri 168	. . . . 5 ⊢ ((𝐴 mod 8) = 𝑁 → (𝐴 mod 8) ∈ 𝑆)
73	72	a1i 9	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) = 𝑁 → (𝐴 mod 8) ∈ 𝑆))
74	69, 73	jaod 717	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → (((𝐴 mod 8) ∈ (0...(𝑁 − 1)) ∨ (𝐴 mod 8) = 𝑁) → (𝐴 mod 8) ∈ 𝑆))
75	36, 74	syl5 32	. 2 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝑁) → (𝐴 mod 8) ∈ 𝑆))
76	10, 32, 75	3pm3.2i 1175	1 ⊢ (𝑁 ∈ ℤ ∧ 2 ∥ (𝑁 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝑁) → (𝐴 mod 8) ∈ 𝑆)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 class class class wbr 4001 ‘cfv 5213 (class class class)co 5870 ℂcc 7804 0cc0 7806 1c1 7807 + caddc 7809 · cmul 7811 − cmin 8122 ℕcn 8913 2c2 8964 4c4 8966 8c8 8970 ℤcz 9247 ℤ_≥cuz 9522 ...cfz 10002 mod cmo 10315 ∥ cdvds 11785

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4119 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-setind 4534 ax-cnex 7897 ax-resscn 7898 ax-1cn 7899 ax-1re 7900 ax-icn 7901 ax-addcl 7902 ax-addrcl 7903 ax-mulcl 7904 ax-mulrcl 7905 ax-addcom 7906 ax-mulcom 7907 ax-addass 7908 ax-mulass 7909 ax-distr 7910 ax-i2m1 7911 ax-0lt1 7912 ax-1rid 7913 ax-0id 7914 ax-rnegex 7915 ax-precex 7916 ax-cnre 7917 ax-pre-ltirr 7918 ax-pre-ltwlin 7919 ax-pre-lttrn 7920 ax-pre-apti 7921 ax-pre-ltadd 7922 ax-pre-mulgt0 7923 ax-pre-mulext 7924 ax-arch 7925

This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-int 3844 df-iun 3887 df-br 4002 df-opab 4063 df-mpt 4064 df-id 4291 df-po 4294 df-iso 4295 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-rn 4635 df-res 4636 df-ima 4637 df-iota 5175 df-fun 5215 df-fn 5216 df-f 5217 df-fv 5221 df-riota 5826 df-ov 5873 df-oprab 5874 df-mpo 5875 df-1st 6136 df-2nd 6137 df-pnf 7988 df-mnf 7989 df-xr 7990 df-ltxr 7991 df-le 7992 df-sub 8124 df-neg 8125 df-reap 8526 df-ap 8533 df-div 8624 df-inn 8914 df-2 8972 df-3 8973 df-4 8974 df-5 8975 df-6 8976 df-7 8977 df-8 8978 df-n0 9171 df-z 9248 df-uz 9523 df-q 9614 df-rp 9648 df-fz 10003 df-fl 10263 df-mod 10316 df-dvds 11786

This theorem is referenced by: lgsdir2lem3 14273

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