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Theorem mulgdirlem 13803

Description: Lemma for mulgdir 13804. (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 13659	. . . . 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 13802	. . . . 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 2231	. . . . . . . . . . 11
17	6, 7, 16	mulgneg 13790	. . . . . . . . . 10 $/\$ $/\$
18	12, 14, 15, 17	syl3anc 1274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	18	oveq1d 6043	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	6, 7	mulgcl 13789	. . . . . . . . . 10 $/\$ $/\$
21	12, 14, 15, 20	syl3anc 1274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eqid 2231	. . . . . . . . . 10
23	6, 8, 22, 16	grplinv 13696	. . . . . . . . 9 $/\$
24	12, 21, 23	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	19, 24	eqtrd 2264	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 6044	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		simpl3 1029	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28		nn0z 9543	. . . . . . . . 9
29	27, 28	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	6, 7	mulgcl 13789	. . . . . . . 8 $/\$ $/\$
31	12, 29, 15, 30	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	6, 8, 22	grprid 13678	. . . . . . 7 $/\$
33	12, 31, 32	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	26, 33	eqtrd 2264	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		nn0z 9543	. . . . . . . . 9
36	35	ad2antll 491	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	6, 7	mulgcl 13789	. . . . . . . 8 $/\$ $/\$
38	12, 36, 15, 37	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	6, 8	grpass 13655	. . . . . . 7 $/\$ $/\$ $/\$
40	12, 31, 38, 21, 39	syl13anc 1276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	12	grpmndd 13659	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		simprr 533	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	6, 7, 8	mulgnn0dir 13802	. . . . . . . . 9 $/\$ $/\$ $/\$
44	41, 27, 42, 15, 43	syl13anc 1276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		simp21 1057	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
46	45	zcnd 9647	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
47	13	zcnd 9647	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
48	46, 47	addcld 8241	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
49	48	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	47	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	49, 50	negsubd 8538	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	46	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	52, 50	pncand 8533	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	51, 53	eqtrd 2264	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	54	oveq1d 6043	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	44, 55	eqtr3d 2266	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	56	oveq1d 6043	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	40, 57	eqtr3d 2266	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	34, 58	eqtr3d 2266	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	59	anassrs 400	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61		elznn0 9538	. . . . . 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 13789	. . . . 5 $/\$ $/\$
70	66, 67, 68, 69	syl3anc 1274	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
71	67	znegcld 9648	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
72	6, 7	mulgcl 13789	. . . . 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 13655	. . . 4 $/\$ $/\$ $/\$
78	66, 70, 73, 76, 77	syl13anc 1276	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
79	6, 7, 16	mulgneg 13790	. . . . . . . 8 $/\$ $/\$
80	66, 67, 68, 79	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
81	80	oveq2d 6044	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	6, 8, 22, 16	grprinv 13697	. . . . . . 7 $/\$
83	66, 70, 82	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
84	81, 83	eqtrd 2264	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
85	84	oveq1d 6043	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
86	6, 8, 22	grplid 13677	. . . . 5 $/\$
87	66, 76, 86	syl2anc 411	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
88	85, 87	eqtrd 2264	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
89	66	grpmndd 13659	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
90		simpr 110	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
91		simpl3 1029	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
92	6, 7, 8	mulgnn0dir 13802	. . . . . 6 $/\$ $/\$ $/\$
93	89, 90, 91, 68, 92	syl13anc 1276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
94	46	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
95	94	negcld 8519	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
96	48	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	95, 96	addcomd 8372	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
98	96, 94	negsubd 8538	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
99	47	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	94, 99	pncan2d 8534	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	97, 98, 100	3eqtrd 2268	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	101	oveq1d 6043	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	93, 102	eqtr3d 2266	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	oveq2d 6044	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
105	78, 88, 104	3eqtr3d 2272	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
106		elznn0 9538	. . . 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 2202

cfv 5333 (class class class)co 6028

cc 8073

cr 8074

caddc 8078

cmin 8392

cneg 8393

cn0 9444

cz 9523

cbs 13145

cplusg 13223

c0g 13402

cmnd 13562

cgrp 13646

cminusg 13647 ._gcmg 13769

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-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 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-addcom 8175 ax-addass 8177 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-0id 8183 ax-rnegex 8184 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-ltadd 8191

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 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-nul 3497 df-if 3608 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-tr 4193 df-id 4396 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 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-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-inn 9186 df-2 9244 df-n0 9445 df-z 9524 df-uz 9800 df-fz 10289 df-seqfrec 10756 df-ndx 13148 df-slot 13149 df-base 13151 df-plusg 13236 df-0g 13404 df-mgm 13502 df-sgrp 13548 df-mnd 13563 df-grp 13649 df-minusg 13650 df-mulg 13770

This theorem is referenced by: mulgdir 13804

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