Theorem lgsdir2lem2 15354

Description: Lemma for lgsdir2 15358. (Contributed by Mario Carneiro, 4-Feb-2015.)

Hypotheses

Ref	Expression
lgsdir2lem2.1	|- ( K e. ZZ /\ 2 || ( K + 1 ) /\ ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... K ) -> ( A mod 8 ) e. S ) ) )
lgsdir2lem2.2	|- M = ( K + 1 )
lgsdir2lem2.3	|- N = ( M + 1 )
lgsdir2lem2.4	|- N e. S

Assertion

Ref	Expression
lgsdir2lem2	|- ( N e. ZZ /\ 2 || ( N + 1 ) /\ ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... N ) -> ( A mod 8 ) e. 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 e. ZZ /\ 2 || ( K + 1 ) /\ ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... K ) -> ( A mod 8 ) e. S ) ) )
4	3	simp1i 1008	. . . . . 6 |- K e. ZZ
5		peano2z 9379	. . . . . 6 |- ( K e. ZZ -> ( K + 1 ) e. ZZ )
6	4, 5	ax-mp 5	. . . . 5 |- ( K + 1 ) e. ZZ
7	2, 6	eqeltri 2269	. . . 4 |- M e. ZZ
8		peano2z 9379	. . . 4 |- ( M e. ZZ -> ( M + 1 ) e. ZZ )
9	7, 8	ax-mp 5	. . 3 |- ( M + 1 ) e. ZZ
10	1, 9	eqeltri 2269	. 2 |- N e. ZZ
11	3	simp2i 1009	. . . 4 |- 2 || ( K + 1 )
12		2z 9371	. . . . 5 |- 2 e. ZZ
13		dvdsadd 12018	. . . . 5 |- ( ( 2 e. ZZ /\ ( K + 1 ) e. ZZ ) -> ( 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 9348	. . . . . . . . . . 11 |- ( K e. ZZ -> K e. CC )
17	4, 16	ax-mp 5	. . . . . . . . . 10 |- K e. CC
18		ax-1cn 7989	. . . . . . . . . 10 |- 1 e. CC
19	17, 18	addcomi 8187	. . . . . . . . 9 |- ( K + 1 ) = ( 1 + K )
20	2, 19	eqtri 2217	. . . . . . . 8 |- M = ( 1 + K )
21	20	oveq1i 5935	. . . . . . 7 |- ( M + 1 ) = ( ( 1 + K ) + 1 )
22	1, 21	eqtri 2217	. . . . . 6 |- N = ( ( 1 + K ) + 1 )
23		df-2 9066	. . . . . . . 8 |- 2 = ( 1 + 1 )
24	23	oveq1i 5935	. . . . . . 7 |- ( 2 + K ) = ( ( 1 + 1 ) + K )
25	18, 17, 18	add32i 8207	. . . . . . 7 |- ( ( 1 + K ) + 1 ) = ( ( 1 + 1 ) + K )
26	24, 25	eqtr4i 2220	. . . . . 6 |- ( 2 + K ) = ( ( 1 + K ) + 1 )
27	22, 26	eqtr4i 2220	. . . . 5 |- N = ( 2 + K )
28	27	oveq1i 5935	. . . 4 |- ( N + 1 ) = ( ( 2 + K ) + 1 )
29		2cn 9078	. . . . 5 |- 2 e. CC
30	29, 17, 18	addassi 8051	. . . 4 |- ( ( 2 + K ) + 1 ) = ( 2 + ( K + 1 ) )
31	28, 30	eqtri 2217	. . 3 |- ( N + 1 ) = ( 2 + ( K + 1 ) )
32	15, 31	breqtrri 4061	. 2 |- 2 || ( N + 1 )
33		elfzuz2 10121	. . . . 5 |- ( ( A mod 8 ) e. ( 0 ... N ) -> N e. ( ZZ>= ` 0 ) )
34		fzm1 10192	. . . . 5 |- ( N e. ( ZZ>= ` 0 ) -> ( ( A mod 8 ) e. ( 0 ... N ) <-> ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) \/ ( A mod 8 ) = N ) ) )
35	33, 34	syl 14	. . . 4 |- ( ( A mod 8 ) e. ( 0 ... N ) -> ( ( A mod 8 ) e. ( 0 ... N ) <-> ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) \/ ( A mod 8 ) = N ) ) )
36	35	ibi 176	. . 3 |- ( ( A mod 8 ) e. ( 0 ... N ) -> ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) \/ ( A mod 8 ) = N ) )
37		elfzuz2 10121	. . . . . . . 8 |- ( ( A mod 8 ) e. ( 0 ... M ) -> M e. ( ZZ>= ` 0 ) )
38		fzm1 10192	. . . . . . . 8 |- ( M e. ( ZZ>= ` 0 ) -> ( ( A mod 8 ) e. ( 0 ... M ) <-> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) ) )
39	37, 38	syl 14	. . . . . . 7 |- ( ( A mod 8 ) e. ( 0 ... M ) -> ( ( A mod 8 ) e. ( 0 ... M ) <-> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) ) )
40	39	ibi 176	. . . . . 6 |- ( ( A mod 8 ) e. ( 0 ... M ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) )
41		zcn 9348	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
42	7, 41	ax-mp 5	. . . . . . . 8 |- M e. CC
43	42, 18, 1	mvrraddi 8260	. . . . . . 7 |- ( N - 1 ) = M
44	43	oveq2i 5936	. . . . . 6 |- ( 0 ... ( N - 1 ) ) = ( 0 ... M )
45	40, 44	eleq2s 2291	. . . . 5 |- ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) )
46	17, 18, 2	mvrraddi 8260	. . . . . . . . 9 |- ( M - 1 ) = K
47	46	oveq2i 5936	. . . . . . . 8 |- ( 0 ... ( M - 1 ) ) = ( 0 ... K )
48	47	eleq2i 2263	. . . . . . 7 |- ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) <-> ( A mod 8 ) e. ( 0 ... K ) )
49	3	simp3i 1010	. . . . . . 7 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... K ) -> ( A mod 8 ) e. S ) )
50	48, 49	biimtrid 152	. . . . . 6 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) -> ( A mod 8 ) e. S ) )
51		2nn 9169	. . . . . . . . . . 11 |- 2 e. NN
52		8nn 9175	. . . . . . . . . . 11 |- 8 e. NN
53		4z 9373	. . . . . . . . . . . . . 14 |- 4 e. ZZ
54		dvdsmul2 11996	. . . . . . . . . . . . . 14 |- ( ( 4 e. ZZ /\ 2 e. ZZ ) -> 2 || ( 4 x. 2 ) )
55	53, 12, 54	mp2an 426	. . . . . . . . . . . . 13 |- 2 || ( 4 x. 2 )
56		4t2e8 9166	. . . . . . . . . . . . 13 |- ( 4 x. 2 ) = 8
57	55, 56	breqtri 4059	. . . . . . . . . . . 12 |- 2 || 8
58		dvdsmod 12044	. . . . . . . . . . . 12 |- ( ( ( 2 e. NN /\ 8 e. NN /\ A e. ZZ ) /\ 2 || 8 ) -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
59	57, 58	mpan2 425	. . . . . . . . . . 11 |- ( ( 2 e. NN /\ 8 e. NN /\ A e. ZZ ) -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
60	51, 52, 59	mp3an12 1338	. . . . . . . . . 10 |- ( A e. ZZ -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
61	60	notbid 668	. . . . . . . . 9 |- ( A e. ZZ -> ( -. 2 || ( A mod 8 ) <-> -. 2 || A ) )
62	61	biimpar 297	. . . . . . . 8 |- ( ( A e. ZZ /\ -. 2 || A ) -> -. 2 || ( A mod 8 ) )
63	11, 2	breqtrri 4061	. . . . . . . . 9 |- 2 || M
64		id 19	. . . . . . . . 9 |- ( ( A mod 8 ) = M -> ( A mod 8 ) = M )
65	63, 64	breqtrrid 4072	. . . . . . . 8 |- ( ( A mod 8 ) = M -> 2 || ( A mod 8 ) )
66	62, 65	nsyl 629	. . . . . . 7 |- ( ( A e. ZZ /\ -. 2 || A ) -> -. ( A mod 8 ) = M )
67	66	pm2.21d 620	. . . . . 6 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) = M -> ( A mod 8 ) e. S ) )
68	50, 67	jaod 718	. . . . 5 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) -> ( A mod 8 ) e. S ) )
69	45, 68	syl5 32	. . . 4 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) -> ( A mod 8 ) e. S ) )
70		lgsdir2lem2.4	. . . . . 6 |- N e. S
71		eleq1 2259	. . . . . 6 |- ( ( A mod 8 ) = N -> ( ( A mod 8 ) e. S <-> N e. S ) )
72	70, 71	mpbiri 168	. . . . 5 |- ( ( A mod 8 ) = N -> ( A mod 8 ) e. S )
73	72	a1i 9	. . . 4 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) = N -> ( A mod 8 ) e. S ) )
74	69, 73	jaod 718	. . 3 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) \/ ( A mod 8 ) = N ) -> ( A mod 8 ) e. S ) )
75	36, 74	syl5 32	. 2 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... N ) -> ( A mod 8 ) e. S ) )
76	10, 32, 75	3pm3.2i 1177	1 |- ( N e. ZZ /\ 2 || ( N + 1 ) /\ ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... N ) -> ( A mod 8 ) e. S ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   CCcc 7894   0cc0 7896   1c1 7897   + caddc 7899   x. cmul 7901   - cmin 8214   NNcn 9007   2c2 9058   4c4 9060   8c8 9064   ZZcz 9343   ZZ>=cuz 9618   ...cfz 10100   mod cmo 10431   || cdvds 11969

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 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fl 10377  df-mod 10432  df-dvds 11970

This theorem is referenced by:  lgsdir2lem3  15355