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Theorem mulgdirlem 13906

Description: Lemma for mulgdir 13907. (Contributed by Mario Carneiro, 13-Dec-2014.)

**Hypotheses**
Ref	Expression
mulgnndir.b
mulgnndir.t	._g
mulgnndir.p

**Assertion**
Ref	Expression
mulgdirlem	$/\$ $/\$ $/\$ $/\$

**Proof of Theorem mulgdirlem**
Step	Hyp	Ref	Expression
1		simpl1 1027	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2	1	grpmndd 13768	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
3		simprl 531	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 533	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5		simpl23 1104	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6		mulgnndir.b	. . . . . 6
7		mulgnndir.t	. . . . . 6 ._g
8		mulgnndir.p	. . . . . 6
9	6, 7, 8	mulgnn0dir 13905	. . . . 5 $/\$ $/\$ $/\$
10	2, 3, 4, 5, 9	syl13anc 1276	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11	10	anassrs 400	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		simpl1 1027	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13		simp22 1058	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
14	13	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpl23 1104	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		eqid 2234	. . . . . . . . . . 11
17	6, 7, 16	mulgneg 13893	. . . . . . . . . 10 $/\$ $/\$
18	12, 14, 15, 17	syl3anc 1274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	18	oveq1d 6073	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	6, 7	mulgcl 13892	. . . . . . . . . 10 $/\$ $/\$
21	12, 14, 15, 20	syl3anc 1274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eqid 2234	. . . . . . . . . 10
23	6, 8, 22, 16	grplinv 13805	. . . . . . . . 9 $/\$
24	12, 21, 23	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	19, 24	eqtrd 2267	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 6074	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		simpl3 1029	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28		nn0z 9614	. . . . . . . . 9
29	27, 28	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	6, 7	mulgcl 13892	. . . . . . . 8 $/\$ $/\$
31	12, 29, 15, 30	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	6, 8, 22	grprid 13787	. . . . . . 7 $/\$
33	12, 31, 32	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	26, 33	eqtrd 2267	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		nn0z 9614	. . . . . . . . 9
36	35	ad2antll 491	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	6, 7	mulgcl 13892	. . . . . . . 8 $/\$ $/\$
38	12, 36, 15, 37	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	6, 8	grpass 13764	. . . . . . 7 $/\$ $/\$ $/\$
40	12, 31, 38, 21, 39	syl13anc 1276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	12	grpmndd 13768	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		simprr 533	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	6, 7, 8	mulgnn0dir 13905	. . . . . . . . 9 $/\$ $/\$ $/\$
44	41, 27, 42, 15, 43	syl13anc 1276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		simp21 1057	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
46	45	zcnd 9719	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
47	13	zcnd 9719	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
48	46, 47	addcld 8309	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
49	48	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	47	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	49, 50	negsubd 8606	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	46	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	52, 50	pncand 8601	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	51, 53	eqtrd 2267	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	54	oveq1d 6073	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	44, 55	eqtr3d 2269	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	56	oveq1d 6073	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	40, 57	eqtr3d 2269	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	34, 58	eqtr3d 2269	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	59	anassrs 400	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61		elznn0 9609	. . . . . 6 $/\$ $\/$
62	61	simprbi 275	. . . . 5 $\/$
63	13, 62	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$
64	63	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
65	11, 60, 64	mpjaodan 806	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
66		simpl1 1027	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
67	45	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
68		simpl23 1104	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
69	6, 7	mulgcl 13892	. . . . 5 $/\$ $/\$
70	66, 67, 68, 69	syl3anc 1274	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
71	67	znegcld 9720	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
72	6, 7	mulgcl 13892	. . . . 5 $/\$ $/\$
73	66, 71, 68, 72	syl3anc 1274	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
74	28	3ad2ant3 1047	. . . . . 6 $/\$ $/\$ $/\$ $/\$
75	74	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
76	66, 75, 68, 30	syl3anc 1274	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
77	6, 8	grpass 13764	. . . 4 $/\$ $/\$ $/\$
78	66, 70, 73, 76, 77	syl13anc 1276	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
79	6, 7, 16	mulgneg 13893	. . . . . . . 8 $/\$ $/\$
80	66, 67, 68, 79	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
81	80	oveq2d 6074	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	6, 8, 22, 16	grprinv 13806	. . . . . . 7 $/\$
83	66, 70, 82	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
84	81, 83	eqtrd 2267	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
85	84	oveq1d 6073	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
86	6, 8, 22	grplid 13786	. . . . 5 $/\$
87	66, 76, 86	syl2anc 411	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
88	85, 87	eqtrd 2267	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
89	66	grpmndd 13768	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
90		simpr 110	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
91		simpl3 1029	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
92	6, 7, 8	mulgnn0dir 13905	. . . . . 6 $/\$ $/\$ $/\$
93	89, 90, 91, 68, 92	syl13anc 1276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
94	46	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
95	94	negcld 8587	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
96	48	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	95, 96	addcomd 8440	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
98	96, 94	negsubd 8606	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
99	47	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	94, 99	pncan2d 8602	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	97, 98, 100	3eqtrd 2271	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	101	oveq1d 6073	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	93, 102	eqtr3d 2269	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	oveq2d 6074	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
105	78, 88, 104	3eqtr3d 2275	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
106		elznn0 9609	. . . 4 $/\$ $\/$
107	106	simprbi 275	. . 3 $\/$
108	45, 107	syl 14	. 2 $/\$ $/\$ $/\$ $/\$ $\/$
109	65, 105, 108	mpjaodan 806	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 716 $/\$ w3a 1005

wceq 1398

wcel 2205

cfv 5357 (class class class)co 6058

cc 8141

cr 8142

caddc 8146

cmin 8460

cneg 8461

cn0 9513

cz 9594

cbs 13296

cplusg 13374

c0g 13553

cmnd 13677

cgrp 13755

cminusg 13756 ._gcmg 13872

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-addcom 8243 ax-addass 8245 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-0id 8251 ax-rnegex 8252 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-ltadd 8259

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-inn 9255 df-2 9313 df-n0 9514 df-z 9595 df-uz 9872 df-fz 10362 df-seqfrec 10834 df-ndx 13299 df-slot 13300 df-base 13302 df-plusg 13387 df-0g 13555 df-mgm 13619 df-sgrp 13665 df-mnd 13678 df-grp 13758 df-minusg 13759 df-mulg 13873

This theorem is referenced by: mulgdir 13907

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