Theorem lgsdir2lem1 15353

Description: Lemma for lgsdir2 15358. (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 9018	. . . . 5 |- 1 e. NN
2		nnq 9724	. . . . 5 |- ( 1 e. NN -> 1 e. QQ )
3	1, 2	ax-mp 5	. . . 4 |- 1 e. QQ
4		8nn 9175	. . . . 5 |- 8 e. NN
5		nnq 9724	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
6	4, 5	ax-mp 5	. . . 4 |- 8 e. QQ
7		0le1 8525	. . . 4 |- 0 <_ 1
8		1lt8 9204	. . . 4 |- 1 < 8
9		modqid 10458	. . . 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 9093	. . . . . . . 8 |- 8 e. CC
12	11	mullidi 8046	. . . . . . 7 |- ( 1 x. 8 ) = 8
13	12	oveq2i 5936	. . . . . 6 |- ( -u 1 + ( 1 x. 8 ) ) = ( -u 1 + 8 )
14		ax-1cn 7989	. . . . . . . 8 |- 1 e. CC
15	14	negcli 8311	. . . . . . 7 |- -u 1 e. CC
16	11, 14	negsubi 8321	. . . . . . . 8 |- ( 8 + -u 1 ) = ( 8 - 1 )
17		8m1e7 9132	. . . . . . . 8 |- ( 8 - 1 ) = 7
18	16, 17	eqtri 2217	. . . . . . 7 |- ( 8 + -u 1 ) = 7
19	11, 15, 18	addcomli 8188	. . . . . 6 |- ( -u 1 + 8 ) = 7
20	13, 19	eqtri 2217	. . . . 5 |- ( -u 1 + ( 1 x. 8 ) ) = 7
21	20	oveq1i 5935	. . . 4 |- ( ( -u 1 + ( 1 x. 8 ) ) mod 8 ) = ( 7 mod 8 )
22		qnegcl 9727	. . . . . 6 |- ( 1 e. QQ -> -u 1 e. QQ )
23	3, 22	ax-mp 5	. . . . 5 |- -u 1 e. QQ
24		1z 9369	. . . . 5 |- 1 e. ZZ
25		8pos 9110	. . . . 5 |- 0 < 8
26		modqcyc 10468	. . . . 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 9174	. . . . . 6 |- 7 e. NN
29		nnq 9724	. . . . . 6 |- ( 7 e. NN -> 7 e. QQ )
30	28, 29	ax-mp 5	. . . . 5 |- 7 e. QQ
31		0re 8043	. . . . . 6 |- 0 e. RR
32		7re 9090	. . . . . 6 |- 7 e. RR
33		7pos 9109	. . . . . 6 |- 0 < 7
34	31, 32, 33	ltleii 8146	. . . . 5 |- 0 <_ 7
35		7lt8 9198	. . . . 5 |- 7 < 8
36		modqid 10458	. . . . 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 2225	. . 3 |- ( -u 1 mod 8 ) = 7
39	10, 38	pm3.2i 272	. 2 |- ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 )
40		3nn 9170	. . . . 5 |- 3 e. NN
41		nnq 9724	. . . . 5 |- ( 3 e. NN -> 3 e. QQ )
42	40, 41	ax-mp 5	. . . 4 |- 3 e. QQ
43		3re 9081	. . . . 5 |- 3 e. RR
44		3pos 9101	. . . . 5 |- 0 < 3
45	31, 43, 44	ltleii 8146	. . . 4 |- 0 <_ 3
46		3lt8 9202	. . . 4 |- 3 < 8
47		modqid 10458	. . . 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 5936	. . . . . 6 |- ( -u 3 + ( 1 x. 8 ) ) = ( -u 3 + 8 )
50		3cn 9082	. . . . . . . 8 |- 3 e. CC
51	50	negcli 8311	. . . . . . 7 |- -u 3 e. CC
52	11, 50	negsubi 8321	. . . . . . . 8 |- ( 8 + -u 3 ) = ( 8 - 3 )
53		5cn 9087	. . . . . . . . 9 |- 5 e. CC
54		5p3e8 9155	. . . . . . . . . 10 |- ( 5 + 3 ) = 8
55	53, 50, 54	addcomli 8188	. . . . . . . . 9 |- ( 3 + 5 ) = 8
56	11, 50, 53, 55	subaddrii 8332	. . . . . . . 8 |- ( 8 - 3 ) = 5
57	52, 56	eqtri 2217	. . . . . . 7 |- ( 8 + -u 3 ) = 5
58	11, 51, 57	addcomli 8188	. . . . . 6 |- ( -u 3 + 8 ) = 5
59	49, 58	eqtri 2217	. . . . 5 |- ( -u 3 + ( 1 x. 8 ) ) = 5
60	59	oveq1i 5935	. . . 4 |- ( ( -u 3 + ( 1 x. 8 ) ) mod 8 ) = ( 5 mod 8 )
61		qnegcl 9727	. . . . . 6 |- ( 3 e. QQ -> -u 3 e. QQ )
62	42, 61	ax-mp 5	. . . . 5 |- -u 3 e. QQ
63		modqcyc 10468	. . . . 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 9172	. . . . . 6 |- 5 e. NN
66		nnq 9724	. . . . . 6 |- ( 5 e. NN -> 5 e. QQ )
67	65, 66	ax-mp 5	. . . . 5 |- 5 e. QQ
68		5re 9086	. . . . . 6 |- 5 e. RR
69		5pos 9107	. . . . . 6 |- 0 < 5
70	31, 68, 69	ltleii 8146	. . . . 5 |- 0 <_ 5
71		5lt8 9200	. . . . 5 |- 5 < 8
72		modqid 10458	. . . . 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 2225	. . 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 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   0cc0 7896   1c1 7897   + caddc 7899   x. cmul 7901   < clt 8078   <_ cle 8079   - cmin 8214   -ucneg 8215   NNcn 9007   3c3 9059   5c5 9061   7c7 9063   8c8 9064   ZZcz 9343   QQcq 9710   mod cmo 10431

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-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-q 9711  df-rp 9746  df-fl 10377  df-mod 10432

This theorem is referenced by:  lgsdir2lem4  15356  lgsdir2lem5  15357