Theorem lgsdir2lem1 16061

Description: Lemma for lgsdir2 16066. (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 9294	. . . . 5 |- 1 e. NN
2		nnq 10012	. . . . 5 |- ( 1 e. NN -> 1 e. QQ )
3	1, 2	ax-mp 5	. . . 4 |- 1 e. QQ
4		8nn 9451	. . . . 5 |- 8 e. NN
5		nnq 10012	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
6	4, 5	ax-mp 5	. . . 4 |- 8 e. QQ
7		0le1 8799	. . . 4 |- 0 <_ 1
8		1lt8 9480	. . . 4 |- 1 < 8
9		modqid 10764	. . . 4 |- ( ( ( 1 e. QQ /\ 8 e. QQ ) /\ ( 0 <_ 1 /\ 1 < 8 ) ) -> ( 1 mod 8 ) = 1 )
10	3, 6, 7, 8, 9	mp4an 431	. . 3 |- ( 1 mod 8 ) = 1
11		8cn 9369	. . . . . . . 8 |- 8 e. CC
12	11	mullidi 8319	. . . . . . 7 |- ( 1 x. 8 ) = 8
13	12	oveq2i 6086	. . . . . 6 |- ( -u 1 + ( 1 x. 8 ) ) = ( -u 1 + 8 )
14		ax-1cn 8262	. . . . . . . 8 |- 1 e. CC
15	14	negcli 8584	. . . . . . 7 |- -u 1 e. CC
16	11, 14	negsubi 8594	. . . . . . . 8 |- ( 8 + -u 1 ) = ( 8 - 1 )
17		8m1e7 9408	. . . . . . . 8 |- ( 8 - 1 ) = 7
18	16, 17	eqtri 2259	. . . . . . 7 |- ( 8 + -u 1 ) = 7
19	11, 15, 18	addcomli 8461	. . . . . 6 |- ( -u 1 + 8 ) = 7
20	13, 19	eqtri 2259	. . . . 5 |- ( -u 1 + ( 1 x. 8 ) ) = 7
21	20	oveq1i 6085	. . . 4 |- ( ( -u 1 + ( 1 x. 8 ) ) mod 8 ) = ( 7 mod 8 )
22		qnegcl 10015	. . . . . 6 |- ( 1 e. QQ -> -u 1 e. QQ )
23	3, 22	ax-mp 5	. . . . 5 |- -u 1 e. QQ
24		1z 9649	. . . . 5 |- 1 e. ZZ
25		8pos 9386	. . . . 5 |- 0 < 8
26		modqcyc 10774	. . . . 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 431	. . . 4 |- ( ( -u 1 + ( 1 x. 8 ) ) mod 8 ) = ( -u 1 mod 8 )
28		7nn 9450	. . . . . 6 |- 7 e. NN
29		nnq 10012	. . . . . 6 |- ( 7 e. NN -> 7 e. QQ )
30	28, 29	ax-mp 5	. . . . 5 |- 7 e. QQ
31		0re 8316	. . . . . 6 |- 0 e. RR
32		7re 9366	. . . . . 6 |- 7 e. RR
33		7pos 9385	. . . . . 6 |- 0 < 7
34	31, 32, 33	ltleii 8418	. . . . 5 |- 0 <_ 7
35		7lt8 9474	. . . . 5 |- 7 < 8
36		modqid 10764	. . . . 5 |- ( ( ( 7 e. QQ /\ 8 e. QQ ) /\ ( 0 <_ 7 /\ 7 < 8 ) ) -> ( 7 mod 8 ) = 7 )
37	30, 6, 34, 35, 36	mp4an 431	. . . 4 |- ( 7 mod 8 ) = 7
38	21, 27, 37	3eqtr3i 2267	. . 3 |- ( -u 1 mod 8 ) = 7
39	10, 38	pm3.2i 272	. 2 |- ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 )
40		3nn 9446	. . . . 5 |- 3 e. NN
41		nnq 10012	. . . . 5 |- ( 3 e. NN -> 3 e. QQ )
42	40, 41	ax-mp 5	. . . 4 |- 3 e. QQ
43		3re 9357	. . . . 5 |- 3 e. RR
44		3pos 9377	. . . . 5 |- 0 < 3
45	31, 43, 44	ltleii 8418	. . . 4 |- 0 <_ 3
46		3lt8 9478	. . . 4 |- 3 < 8
47		modqid 10764	. . . 4 |- ( ( ( 3 e. QQ /\ 8 e. QQ ) /\ ( 0 <_ 3 /\ 3 < 8 ) ) -> ( 3 mod 8 ) = 3 )
48	42, 6, 45, 46, 47	mp4an 431	. . 3 |- ( 3 mod 8 ) = 3
49	12	oveq2i 6086	. . . . . 6 |- ( -u 3 + ( 1 x. 8 ) ) = ( -u 3 + 8 )
50		3cn 9358	. . . . . . . 8 |- 3 e. CC
51	50	negcli 8584	. . . . . . 7 |- -u 3 e. CC
52	11, 50	negsubi 8594	. . . . . . . 8 |- ( 8 + -u 3 ) = ( 8 - 3 )
53		5cn 9363	. . . . . . . . 9 |- 5 e. CC
54		5p3e8 9431	. . . . . . . . . 10 |- ( 5 + 3 ) = 8
55	53, 50, 54	addcomli 8461	. . . . . . . . 9 |- ( 3 + 5 ) = 8
56	11, 50, 53, 55	subaddrii 8605	. . . . . . . 8 |- ( 8 - 3 ) = 5
57	52, 56	eqtri 2259	. . . . . . 7 |- ( 8 + -u 3 ) = 5
58	11, 51, 57	addcomli 8461	. . . . . 6 |- ( -u 3 + 8 ) = 5
59	49, 58	eqtri 2259	. . . . 5 |- ( -u 3 + ( 1 x. 8 ) ) = 5
60	59	oveq1i 6085	. . . 4 |- ( ( -u 3 + ( 1 x. 8 ) ) mod 8 ) = ( 5 mod 8 )
61		qnegcl 10015	. . . . . 6 |- ( 3 e. QQ -> -u 3 e. QQ )
62	42, 61	ax-mp 5	. . . . 5 |- -u 3 e. QQ
63		modqcyc 10774	. . . . 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 431	. . . 4 |- ( ( -u 3 + ( 1 x. 8 ) ) mod 8 ) = ( -u 3 mod 8 )
65		5nn 9448	. . . . . 6 |- 5 e. NN
66		nnq 10012	. . . . . 6 |- ( 5 e. NN -> 5 e. QQ )
67	65, 66	ax-mp 5	. . . . 5 |- 5 e. QQ
68		5re 9362	. . . . . 6 |- 5 e. RR
69		5pos 9383	. . . . . 6 |- 0 < 5
70	31, 68, 69	ltleii 8418	. . . . 5 |- 0 <_ 5
71		5lt8 9476	. . . . 5 |- 5 < 8
72		modqid 10764	. . . . 5 |- ( ( ( 5 e. QQ /\ 8 e. QQ ) /\ ( 0 <_ 5 /\ 5 < 8 ) ) -> ( 5 mod 8 ) = 5 )
73	67, 6, 70, 71, 72	mp4an 431	. . . 4 |- ( 5 mod 8 ) = 5
74	60, 64, 73	3eqtr3i 2267	. . 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 1402   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   < clt 8350   <_ cle 8351   - cmin 8487   -ucneg 8488   NNcn 9283   3c3 9335   5c5 9337   7c7 9339   8c8 9340   ZZcz 9623   QQcq 9998   mod cmo 10737

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-n0 9543  df-z 9624  df-q 9999  df-rp 10034  df-fl 10683  df-mod 10738

This theorem is referenced by:  lgsdir2lem4  16064  lgsdir2lem5  16065