Theorem lgsdir2lem2 13724

Description: Lemma for lgsdir2 13728. (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 1001	. . . . . 6 |- K e. ZZ
5		peano2z 9248	. . . . . 6 |- ( K e. ZZ -> ( K + 1 ) e. ZZ )
6	4, 5	ax-mp 5	. . . . 5 |- ( K + 1 ) e. ZZ
7	2, 6	eqeltri 2243	. . . 4 |- M e. ZZ
8		peano2z 9248	. . . 4 |- ( M e. ZZ -> ( M + 1 ) e. ZZ )
9	7, 8	ax-mp 5	. . 3 |- ( M + 1 ) e. ZZ
10	1, 9	eqeltri 2243	. 2 |- N e. ZZ
11	3	simp2i 1002	. . . 4 |- 2 || ( K + 1 )
12		2z 9240	. . . . 5 |- 2 e. ZZ
13		dvdsadd 11798	. . . . 5 |- ( ( 2 e. ZZ /\ ( K + 1 ) e. ZZ ) -> ( 2 || ( K + 1 ) <-> 2 || ( 2 + ( K + 1 ) ) ) )
14	12, 6, 13	mp2an 424	. . . 4 |- ( 2 || ( K + 1 ) <-> 2 || ( 2 + ( K + 1 ) ) )
15	11, 14	mpbi 144	. . 3 |- 2 || ( 2 + ( K + 1 ) )
16		zcn 9217	. . . . . . . . . . 11 |- ( K e. ZZ -> K e. CC )
17	4, 16	ax-mp 5	. . . . . . . . . 10 |- K e. CC
18		ax-1cn 7867	. . . . . . . . . 10 |- 1 e. CC
19	17, 18	addcomi 8063	. . . . . . . . 9 |- ( K + 1 ) = ( 1 + K )
20	2, 19	eqtri 2191	. . . . . . . 8 |- M = ( 1 + K )
21	20	oveq1i 5863	. . . . . . 7 |- ( M + 1 ) = ( ( 1 + K ) + 1 )
22	1, 21	eqtri 2191	. . . . . 6 |- N = ( ( 1 + K ) + 1 )
23		df-2 8937	. . . . . . . 8 |- 2 = ( 1 + 1 )
24	23	oveq1i 5863	. . . . . . 7 |- ( 2 + K ) = ( ( 1 + 1 ) + K )
25	18, 17, 18	add32i 8083	. . . . . . 7 |- ( ( 1 + K ) + 1 ) = ( ( 1 + 1 ) + K )
26	24, 25	eqtr4i 2194	. . . . . 6 |- ( 2 + K ) = ( ( 1 + K ) + 1 )
27	22, 26	eqtr4i 2194	. . . . 5 |- N = ( 2 + K )
28	27	oveq1i 5863	. . . 4 |- ( N + 1 ) = ( ( 2 + K ) + 1 )
29		2cn 8949	. . . . 5 |- 2 e. CC
30	29, 17, 18	addassi 7928	. . . 4 |- ( ( 2 + K ) + 1 ) = ( 2 + ( K + 1 ) )
31	28, 30	eqtri 2191	. . 3 |- ( N + 1 ) = ( 2 + ( K + 1 ) )
32	15, 31	breqtrri 4016	. 2 |- 2 || ( N + 1 )
33		elfzuz2 9985	. . . . 5 |- ( ( A mod 8 ) e. ( 0 ... N ) -> N e. ( ZZ>= ` 0 ) )
34		fzm1 10056	. . . . 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 175	. . 3 |- ( ( A mod 8 ) e. ( 0 ... N ) -> ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) \/ ( A mod 8 ) = N ) )
37		elfzuz2 9985	. . . . . . . 8 |- ( ( A mod 8 ) e. ( 0 ... M ) -> M e. ( ZZ>= ` 0 ) )
38		fzm1 10056	. . . . . . . 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 175	. . . . . 6 |- ( ( A mod 8 ) e. ( 0 ... M ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) )
41		zcn 9217	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
42	7, 41	ax-mp 5	. . . . . . . 8 |- M e. CC
43	42, 18, 1	mvrraddi 8136	. . . . . . 7 |- ( N - 1 ) = M
44	43	oveq2i 5864	. . . . . 6 |- ( 0 ... ( N - 1 ) ) = ( 0 ... M )
45	40, 44	eleq2s 2265	. . . . 5 |- ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) )
46	17, 18, 2	mvrraddi 8136	. . . . . . . . 9 |- ( M - 1 ) = K
47	46	oveq2i 5864	. . . . . . . 8 |- ( 0 ... ( M - 1 ) ) = ( 0 ... K )
48	47	eleq2i 2237	. . . . . . 7 |- ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) <-> ( A mod 8 ) e. ( 0 ... K ) )
49	3	simp3i 1003	. . . . . . 7 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... K ) -> ( A mod 8 ) e. S ) )
50	48, 49	syl5bi 151	. . . . . 6 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) -> ( A mod 8 ) e. S ) )
51		2nn 9039	. . . . . . . . . . 11 |- 2 e. NN
52		8nn 9045	. . . . . . . . . . 11 |- 8 e. NN
53		4z 9242	. . . . . . . . . . . . . 14 |- 4 e. ZZ
54		dvdsmul2 11776	. . . . . . . . . . . . . 14 |- ( ( 4 e. ZZ /\ 2 e. ZZ ) -> 2 || ( 4 x. 2 ) )
55	53, 12, 54	mp2an 424	. . . . . . . . . . . . 13 |- 2 || ( 4 x. 2 )
56		4t2e8 9036	. . . . . . . . . . . . 13 |- ( 4 x. 2 ) = 8
57	55, 56	breqtri 4014	. . . . . . . . . . . 12 |- 2 || 8
58		dvdsmod 11822	. . . . . . . . . . . 12 |- ( ( ( 2 e. NN /\ 8 e. NN /\ A e. ZZ ) /\ 2 || 8 ) -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
59	57, 58	mpan2 423	. . . . . . . . . . 11 |- ( ( 2 e. NN /\ 8 e. NN /\ A e. ZZ ) -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
60	51, 52, 59	mp3an12 1322	. . . . . . . . . 10 |- ( A e. ZZ -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
61	60	notbid 662	. . . . . . . . 9 |- ( A e. ZZ -> ( -. 2 || ( A mod 8 ) <-> -. 2 || A ) )
62	61	biimpar 295	. . . . . . . 8 |- ( ( A e. ZZ /\ -. 2 || A ) -> -. 2 || ( A mod 8 ) )
63	11, 2	breqtrri 4016	. . . . . . . . 9 |- 2 || M
64		id 19	. . . . . . . . 9 |- ( ( A mod 8 ) = M -> ( A mod 8 ) = M )
65	63, 64	breqtrrid 4027	. . . . . . . 8 |- ( ( A mod 8 ) = M -> 2 || ( A mod 8 ) )
66	62, 65	nsyl 623	. . . . . . 7 |- ( ( A e. ZZ /\ -. 2 || A ) -> -. ( A mod 8 ) = M )
67	66	pm2.21d 614	. . . . . 6 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) = M -> ( A mod 8 ) e. S ) )
68	50, 67	jaod 712	. . . . 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 2233	. . . . . 6 |- ( ( A mod 8 ) = N -> ( ( A mod 8 ) e. S <-> N e. S ) )
72	70, 71	mpbiri 167	. . . . 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 712	. . 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 1170	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 103   <-> wb 104   \/ wo 703   /\ w3a 973   = wceq 1348   e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   - cmin 8090   NNcn 8878   2c2 8929   4c4 8931   8c8 8935   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   mod cmo 10278   || cdvds 11749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-7 8942  df-8 8943  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fl 10226  df-mod 10279  df-dvds 11750

This theorem is referenced by:  lgsdir2lem3  13725