Theorem lgsdir2lem2 13999

Description: Lemma for lgsdir2 14003. (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 1006	. . . . . 6 |- K e. ZZ
5		peano2z 9260	. . . . . 6 |- ( K e. ZZ -> ( K + 1 ) e. ZZ )
6	4, 5	ax-mp 5	. . . . 5 |- ( K + 1 ) e. ZZ
7	2, 6	eqeltri 2248	. . . 4 |- M e. ZZ
8		peano2z 9260	. . . 4 |- ( M e. ZZ -> ( M + 1 ) e. ZZ )
9	7, 8	ax-mp 5	. . 3 |- ( M + 1 ) e. ZZ
10	1, 9	eqeltri 2248	. 2 |- N e. ZZ
11	3	simp2i 1007	. . . 4 |- 2 || ( K + 1 )
12		2z 9252	. . . . 5 |- 2 e. ZZ
13		dvdsadd 11809	. . . . 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 9229	. . . . . . . . . . 11 |- ( K e. ZZ -> K e. CC )
17	4, 16	ax-mp 5	. . . . . . . . . 10 |- K e. CC
18		ax-1cn 7879	. . . . . . . . . 10 |- 1 e. CC
19	17, 18	addcomi 8075	. . . . . . . . 9 |- ( K + 1 ) = ( 1 + K )
20	2, 19	eqtri 2196	. . . . . . . 8 |- M = ( 1 + K )
21	20	oveq1i 5875	. . . . . . 7 |- ( M + 1 ) = ( ( 1 + K ) + 1 )
22	1, 21	eqtri 2196	. . . . . 6 |- N = ( ( 1 + K ) + 1 )
23		df-2 8949	. . . . . . . 8 |- 2 = ( 1 + 1 )
24	23	oveq1i 5875	. . . . . . 7 |- ( 2 + K ) = ( ( 1 + 1 ) + K )
25	18, 17, 18	add32i 8095	. . . . . . 7 |- ( ( 1 + K ) + 1 ) = ( ( 1 + 1 ) + K )
26	24, 25	eqtr4i 2199	. . . . . 6 |- ( 2 + K ) = ( ( 1 + K ) + 1 )
27	22, 26	eqtr4i 2199	. . . . 5 |- N = ( 2 + K )
28	27	oveq1i 5875	. . . 4 |- ( N + 1 ) = ( ( 2 + K ) + 1 )
29		2cn 8961	. . . . 5 |- 2 e. CC
30	29, 17, 18	addassi 7940	. . . 4 |- ( ( 2 + K ) + 1 ) = ( 2 + ( K + 1 ) )
31	28, 30	eqtri 2196	. . 3 |- ( N + 1 ) = ( 2 + ( K + 1 ) )
32	15, 31	breqtrri 4025	. 2 |- 2 || ( N + 1 )
33		elfzuz2 9997	. . . . 5 |- ( ( A mod 8 ) e. ( 0 ... N ) -> N e. ( ZZ>= ` 0 ) )
34		fzm1 10068	. . . . 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 9997	. . . . . . . 8 |- ( ( A mod 8 ) e. ( 0 ... M ) -> M e. ( ZZ>= ` 0 ) )
38		fzm1 10068	. . . . . . . 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 9229	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
42	7, 41	ax-mp 5	. . . . . . . 8 |- M e. CC
43	42, 18, 1	mvrraddi 8148	. . . . . . 7 |- ( N - 1 ) = M
44	43	oveq2i 5876	. . . . . 6 |- ( 0 ... ( N - 1 ) ) = ( 0 ... M )
45	40, 44	eleq2s 2270	. . . . 5 |- ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) )
46	17, 18, 2	mvrraddi 8148	. . . . . . . . 9 |- ( M - 1 ) = K
47	46	oveq2i 5876	. . . . . . . 8 |- ( 0 ... ( M - 1 ) ) = ( 0 ... K )
48	47	eleq2i 2242	. . . . . . 7 |- ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) <-> ( A mod 8 ) e. ( 0 ... K ) )
49	3	simp3i 1008	. . . . . . 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 9051	. . . . . . . . . . 11 |- 2 e. NN
52		8nn 9057	. . . . . . . . . . 11 |- 8 e. NN
53		4z 9254	. . . . . . . . . . . . . 14 |- 4 e. ZZ
54		dvdsmul2 11787	. . . . . . . . . . . . . 14 |- ( ( 4 e. ZZ /\ 2 e. ZZ ) -> 2 || ( 4 x. 2 ) )
55	53, 12, 54	mp2an 426	. . . . . . . . . . . . 13 |- 2 || ( 4 x. 2 )
56		4t2e8 9048	. . . . . . . . . . . . 13 |- ( 4 x. 2 ) = 8
57	55, 56	breqtri 4023	. . . . . . . . . . . 12 |- 2 || 8
58		dvdsmod 11833	. . . . . . . . . . . 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 1327	. . . . . . . . . 10 |- ( A e. ZZ -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
61	60	notbid 667	. . . . . . . . 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 4025	. . . . . . . . 9 |- 2 || M
64		id 19	. . . . . . . . 9 |- ( ( A mod 8 ) = M -> ( A mod 8 ) = M )
65	63, 64	breqtrrid 4036	. . . . . . . 8 |- ( ( A mod 8 ) = M -> 2 || ( A mod 8 ) )
66	62, 65	nsyl 628	. . . . . . 7 |- ( ( A e. ZZ /\ -. 2 || A ) -> -. ( A mod 8 ) = M )
67	66	pm2.21d 619	. . . . . 6 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) = M -> ( A mod 8 ) e. S ) )
68	50, 67	jaod 717	. . . . 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 2238	. . . . . 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 717	. . 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 1175	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 708   /\ w3a 978   = wceq 1353   e. wcel 2146   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   CCcc 7784   0cc0 7786   1c1 7787   + caddc 7789   x. cmul 7791   - cmin 8102   NNcn 8890   2c2 8941   4c4 8943   8c8 8947   ZZcz 9224   ZZ>=cuz 9499   ...cfz 9977   mod cmo 10290   || cdvds 11760

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-5 8952  df-6 8953  df-7 8954  df-8 8955  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fl 10238  df-mod 10291  df-dvds 11761

This theorem is referenced by:  lgsdir2lem3  14000