Theorem lgsdir2lem1 15830

Description: Lemma for lgsdir2 15835. (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 9196	. . . . 5 |- 1 e. NN
2		nnq 9911	. . . . 5 |- ( 1 e. NN -> 1 e. QQ )
3	1, 2	ax-mp 5	. . . 4 |- 1 e. QQ
4		8nn 9353	. . . . 5 |- 8 e. NN
5		nnq 9911	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
6	4, 5	ax-mp 5	. . . 4 |- 8 e. QQ
7		0le1 8703	. . . 4 |- 0 <_ 1
8		1lt8 9382	. . . 4 |- 1 < 8
9		modqid 10657	. . . 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 9271	. . . . . . . 8 |- 8 e. CC
12	11	mullidi 8225	. . . . . . 7 |- ( 1 x. 8 ) = 8
13	12	oveq2i 6039	. . . . . 6 |- ( -u 1 + ( 1 x. 8 ) ) = ( -u 1 + 8 )
14		ax-1cn 8168	. . . . . . . 8 |- 1 e. CC
15	14	negcli 8489	. . . . . . 7 |- -u 1 e. CC
16	11, 14	negsubi 8499	. . . . . . . 8 |- ( 8 + -u 1 ) = ( 8 - 1 )
17		8m1e7 9310	. . . . . . . 8 |- ( 8 - 1 ) = 7
18	16, 17	eqtri 2252	. . . . . . 7 |- ( 8 + -u 1 ) = 7
19	11, 15, 18	addcomli 8366	. . . . . 6 |- ( -u 1 + 8 ) = 7
20	13, 19	eqtri 2252	. . . . 5 |- ( -u 1 + ( 1 x. 8 ) ) = 7
21	20	oveq1i 6038	. . . 4 |- ( ( -u 1 + ( 1 x. 8 ) ) mod 8 ) = ( 7 mod 8 )
22		qnegcl 9914	. . . . . 6 |- ( 1 e. QQ -> -u 1 e. QQ )
23	3, 22	ax-mp 5	. . . . 5 |- -u 1 e. QQ
24		1z 9549	. . . . 5 |- 1 e. ZZ
25		8pos 9288	. . . . 5 |- 0 < 8
26		modqcyc 10667	. . . . 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 9352	. . . . . 6 |- 7 e. NN
29		nnq 9911	. . . . . 6 |- ( 7 e. NN -> 7 e. QQ )
30	28, 29	ax-mp 5	. . . . 5 |- 7 e. QQ
31		0re 8222	. . . . . 6 |- 0 e. RR
32		7re 9268	. . . . . 6 |- 7 e. RR
33		7pos 9287	. . . . . 6 |- 0 < 7
34	31, 32, 33	ltleii 8324	. . . . 5 |- 0 <_ 7
35		7lt8 9376	. . . . 5 |- 7 < 8
36		modqid 10657	. . . . 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 2260	. . 3 |- ( -u 1 mod 8 ) = 7
39	10, 38	pm3.2i 272	. 2 |- ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 )
40		3nn 9348	. . . . 5 |- 3 e. NN
41		nnq 9911	. . . . 5 |- ( 3 e. NN -> 3 e. QQ )
42	40, 41	ax-mp 5	. . . 4 |- 3 e. QQ
43		3re 9259	. . . . 5 |- 3 e. RR
44		3pos 9279	. . . . 5 |- 0 < 3
45	31, 43, 44	ltleii 8324	. . . 4 |- 0 <_ 3
46		3lt8 9380	. . . 4 |- 3 < 8
47		modqid 10657	. . . 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 6039	. . . . . 6 |- ( -u 3 + ( 1 x. 8 ) ) = ( -u 3 + 8 )
50		3cn 9260	. . . . . . . 8 |- 3 e. CC
51	50	negcli 8489	. . . . . . 7 |- -u 3 e. CC
52	11, 50	negsubi 8499	. . . . . . . 8 |- ( 8 + -u 3 ) = ( 8 - 3 )
53		5cn 9265	. . . . . . . . 9 |- 5 e. CC
54		5p3e8 9333	. . . . . . . . . 10 |- ( 5 + 3 ) = 8
55	53, 50, 54	addcomli 8366	. . . . . . . . 9 |- ( 3 + 5 ) = 8
56	11, 50, 53, 55	subaddrii 8510	. . . . . . . 8 |- ( 8 - 3 ) = 5
57	52, 56	eqtri 2252	. . . . . . 7 |- ( 8 + -u 3 ) = 5
58	11, 51, 57	addcomli 8366	. . . . . 6 |- ( -u 3 + 8 ) = 5
59	49, 58	eqtri 2252	. . . . 5 |- ( -u 3 + ( 1 x. 8 ) ) = 5
60	59	oveq1i 6038	. . . 4 |- ( ( -u 3 + ( 1 x. 8 ) ) mod 8 ) = ( 5 mod 8 )
61		qnegcl 9914	. . . . . 6 |- ( 3 e. QQ -> -u 3 e. QQ )
62	42, 61	ax-mp 5	. . . . 5 |- -u 3 e. QQ
63		modqcyc 10667	. . . . 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 9350	. . . . . 6 |- 5 e. NN
66		nnq 9911	. . . . . 6 |- ( 5 e. NN -> 5 e. QQ )
67	65, 66	ax-mp 5	. . . . 5 |- 5 e. QQ
68		5re 9264	. . . . . 6 |- 5 e. RR
69		5pos 9285	. . . . . 6 |- 0 < 5
70	31, 68, 69	ltleii 8324	. . . . 5 |- 0 <_ 5
71		5lt8 9378	. . . . 5 |- 5 < 8
72		modqid 10657	. . . . 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 2260	. . 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 1398   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   < clt 8256   <_ cle 8257   - cmin 8392   -ucneg 8393   NNcn 9185   3c3 9237   5c5 9239   7c7 9241   8c8 9242   ZZcz 9523   QQcq 9897   mod cmo 10630

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-n0 9445  df-z 9524  df-q 9898  df-rp 9933  df-fl 10576  df-mod 10631

This theorem is referenced by:  lgsdir2lem4  15833  lgsdir2lem5  15834