Theorem lgsdir2lem1 14096

Description: Lemma for lgsdir2 14101. (Contributed by Mario Carneiro, 4-Feb-2015.)

Assertion

Ref	Expression
lgsdir2lem1	|- ( ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 ) /\ ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) )

Proof of Theorem lgsdir2lem1

Step	Hyp	Ref	Expression
1		1nn 8919	. . . . 5 |- 1 e. NN
2		nnq 9622	. . . . 5 |- ( 1 e. NN -> 1 e. QQ )
3	1, 2	ax-mp 5	. . . 4 |- 1 e. QQ
4		8nn 9075	. . . . 5 |- 8 e. NN
5		nnq 9622	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
6	4, 5	ax-mp 5	. . . 4 |- 8 e. QQ
7		0le1 8428	. . . 4 |- 0 <_ 1
8		1lt8 9104	. . . 4 |- 1 < 8
9		modqid 10335	. . . 4 |- ( ( ( 1 e. QQ /\ 8 e. QQ ) /\ ( 0 <_ 1 /\ 1 < 8 ) ) -> ( 1 mod 8 ) = 1 )
10	3, 6, 7, 8, 9	mp4an 427	. . 3 |- ( 1 mod 8 ) = 1
11		8cn 8994	. . . . . . . 8 |- 8 e. CC
12	11	mulid2i 7951	. . . . . . 7 |- ( 1 x. 8 ) = 8
13	12	oveq2i 5880	. . . . . 6 |- ( -u 1 + ( 1 x. 8 ) ) = ( -u 1 + 8 )
14		ax-1cn 7895	. . . . . . . 8 |- 1 e. CC
15	14	negcli 8215	. . . . . . 7 |- -u 1 e. CC
16	11, 14	negsubi 8225	. . . . . . . 8 |- ( 8 + -u 1 ) = ( 8 - 1 )
17		8m1e7 9033	. . . . . . . 8 |- ( 8 - 1 ) = 7
18	16, 17	eqtri 2198	. . . . . . 7 |- ( 8 + -u 1 ) = 7
19	11, 15, 18	addcomli 8092	. . . . . 6 |- ( -u 1 + 8 ) = 7
20	13, 19	eqtri 2198	. . . . 5 |- ( -u 1 + ( 1 x. 8 ) ) = 7
21	20	oveq1i 5879	. . . 4 |- ( ( -u 1 + ( 1 x. 8 ) ) mod 8 ) = ( 7 mod 8 )
22		qnegcl 9625	. . . . . 6 |- ( 1 e. QQ -> -u 1 e. QQ )
23	3, 22	ax-mp 5	. . . . 5 |- -u 1 e. QQ
24		1z 9268	. . . . 5 |- 1 e. ZZ
25		8pos 9011	. . . . 5 |- 0 < 8
26		modqcyc 10345	. . . . 5 |- ( ( ( -u 1 e. QQ /\ 1 e. ZZ ) /\ ( 8 e. QQ /\ 0 < 8 ) ) -> ( ( -u 1 + ( 1 x. 8 ) ) mod 8 ) = ( -u 1 mod 8 ) )
27	23, 24, 6, 25, 26	mp4an 427	. . . 4 |- ( ( -u 1 + ( 1 x. 8 ) ) mod 8 ) = ( -u 1 mod 8 )
28		7nn 9074	. . . . . 6 |- 7 e. NN
29		nnq 9622	. . . . . 6 |- ( 7 e. NN -> 7 e. QQ )
30	28, 29	ax-mp 5	. . . . 5 |- 7 e. QQ
31		0re 7948	. . . . . 6 |- 0 e. RR
32		7re 8991	. . . . . 6 |- 7 e. RR
33		7pos 9010	. . . . . 6 |- 0 < 7
34	31, 32, 33	ltleii 8050	. . . . 5 |- 0 <_ 7
35		7lt8 9098	. . . . 5 |- 7 < 8
36		modqid 10335	. . . . 5 |- ( ( ( 7 e. QQ /\ 8 e. QQ ) /\ ( 0 <_ 7 /\ 7 < 8 ) ) -> ( 7 mod 8 ) = 7 )
37	30, 6, 34, 35, 36	mp4an 427	. . . 4 |- ( 7 mod 8 ) = 7
38	21, 27, 37	3eqtr3i 2206	. . 3 |- ( -u 1 mod 8 ) = 7
39	10, 38	pm3.2i 272	. 2 |- ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 )
40		3nn 9070	. . . . 5 |- 3 e. NN
41		nnq 9622	. . . . 5 |- ( 3 e. NN -> 3 e. QQ )
42	40, 41	ax-mp 5	. . . 4 |- 3 e. QQ
43		3re 8982	. . . . 5 |- 3 e. RR
44		3pos 9002	. . . . 5 |- 0 < 3
45	31, 43, 44	ltleii 8050	. . . 4 |- 0 <_ 3
46		3lt8 9102	. . . 4 |- 3 < 8
47		modqid 10335	. . . 4 |- ( ( ( 3 e. QQ /\ 8 e. QQ ) /\ ( 0 <_ 3 /\ 3 < 8 ) ) -> ( 3 mod 8 ) = 3 )
48	42, 6, 45, 46, 47	mp4an 427	. . 3 |- ( 3 mod 8 ) = 3
49	12	oveq2i 5880	. . . . . 6 |- ( -u 3 + ( 1 x. 8 ) ) = ( -u 3 + 8 )
50		3cn 8983	. . . . . . . 8 |- 3 e. CC
51	50	negcli 8215	. . . . . . 7 |- -u 3 e. CC
52	11, 50	negsubi 8225	. . . . . . . 8 |- ( 8 + -u 3 ) = ( 8 - 3 )
53		5cn 8988	. . . . . . . . 9 |- 5 e. CC
54		5p3e8 9055	. . . . . . . . . 10 |- ( 5 + 3 ) = 8
55	53, 50, 54	addcomli 8092	. . . . . . . . 9 |- ( 3 + 5 ) = 8
56	11, 50, 53, 55	subaddrii 8236	. . . . . . . 8 |- ( 8 - 3 ) = 5
57	52, 56	eqtri 2198	. . . . . . 7 |- ( 8 + -u 3 ) = 5
58	11, 51, 57	addcomli 8092	. . . . . 6 |- ( -u 3 + 8 ) = 5
59	49, 58	eqtri 2198	. . . . 5 |- ( -u 3 + ( 1 x. 8 ) ) = 5
60	59	oveq1i 5879	. . . 4 |- ( ( -u 3 + ( 1 x. 8 ) ) mod 8 ) = ( 5 mod 8 )
61		qnegcl 9625	. . . . . 6 |- ( 3 e. QQ -> -u 3 e. QQ )
62	42, 61	ax-mp 5	. . . . 5 |- -u 3 e. QQ
63		modqcyc 10345	. . . . 5 |- ( ( ( -u 3 e. QQ /\ 1 e. ZZ ) /\ ( 8 e. QQ /\ 0 < 8 ) ) -> ( ( -u 3 + ( 1 x. 8 ) ) mod 8 ) = ( -u 3 mod 8 ) )
64	62, 24, 6, 25, 63	mp4an 427	. . . 4 |- ( ( -u 3 + ( 1 x. 8 ) ) mod 8 ) = ( -u 3 mod 8 )
65		5nn 9072	. . . . . 6 |- 5 e. NN
66		nnq 9622	. . . . . 6 |- ( 5 e. NN -> 5 e. QQ )
67	65, 66	ax-mp 5	. . . . 5 |- 5 e. QQ
68		5re 8987	. . . . . 6 |- 5 e. RR
69		5pos 9008	. . . . . 6 |- 0 < 5
70	31, 68, 69	ltleii 8050	. . . . 5 |- 0 <_ 5
71		5lt8 9100	. . . . 5 |- 5 < 8
72		modqid 10335	. . . . 5 |- ( ( ( 5 e. QQ /\ 8 e. QQ ) /\ ( 0 <_ 5 /\ 5 < 8 ) ) -> ( 5 mod 8 ) = 5 )
73	67, 6, 70, 71, 72	mp4an 427	. . . 4 |- ( 5 mod 8 ) = 5
74	60, 64, 73	3eqtr3i 2206	. . 3 |- ( -u 3 mod 8 ) = 5
75	48, 74	pm3.2i 272	. 2 |- ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 )
76	39, 75	pm3.2i 272	1 |- ( ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 ) /\ ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) )

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2148   class class class wbr 4000  (class class class)co 5869   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   < clt 7982   <_ cle 7983   - cmin 8118   -ucneg 8119   NNcn 8908   3c3 8960   5c5 8962   7c7 8964   8c8 8965   ZZcz 9242   QQcq 9608   mod cmo 10308

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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921

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-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-n0 9166  df-z 9243  df-q 9609  df-rp 9641  df-fl 10256  df-mod 10309

This theorem is referenced by:  lgsdir2lem4  14099  lgsdir2lem5  14100