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Theorem mulgass 13826

Description: Product of group multiples, generalized to

. (Contributed by Mario Carneiro, 13-Dec-2014.)

**Hypotheses**
Ref	Expression
mulgass.b
mulgass.t	._g

**Assertion**
Ref	Expression
mulgass	$/\$ $/\$ $/\$

**Proof of Theorem mulgass**
Step	Hyp	Ref	Expression
1		simpr1 1030	. . 3 $/\$ $/\$ $/\$
2		elznn0 9555	. . . 4 $/\$ $\/$
3	2	simprbi 275	. . 3 $\/$
4	1, 3	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
5		simpr2 1031	. . 3 $/\$ $/\$ $/\$
6		elznn0 9555	. . . 4 $/\$ $\/$
7	6	simprbi 275	. . 3 $\/$
8	5, 7	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
9		grpmnd 13670	. . . . . 6
10	9	ad2antrr 488	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
11		simprl 531	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
12		simprr 533	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
13		simplr3 1068	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
14		mulgass.b	. . . . . 6
15		mulgass.t	. . . . . 6 ._g
16	14, 15	mulgnn0ass 13825	. . . . 5 $/\$ $/\$ $/\$
17	10, 11, 12, 13, 16	syl13anc 1276	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
19	1	zcnd 9664	. . . . . . . . 9 $/\$ $/\$ $/\$
20	5	zcnd 9664	. . . . . . . . 9 $/\$ $/\$ $/\$
21	19, 20	mulneg1d 8649	. . . . . . . 8 $/\$ $/\$ $/\$
22	21	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
23	22	oveq1d 6043	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
24	9	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
25		simprl 531	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
26		simprr 533	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
27		simpr3 1032	. . . . . . . 8 $/\$ $/\$ $/\$
28	27	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
29	14, 15	mulgnn0ass 13825	. . . . . . 7 $/\$ $/\$ $/\$
30	24, 25, 26, 28, 29	syl13anc 1276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
31	23, 30	eqtr3d 2266	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
32		fveq2 5648	. . . . . . 7
33		simpl 109	. . . . . . . . . . 11 $/\$ $/\$ $/\$
34	1, 5	zmulcld 9669	. . . . . . . . . . 11 $/\$ $/\$ $/\$
35		eqid 2231	. . . . . . . . . . . 12
36	14, 15, 35	mulgneg 13807	. . . . . . . . . . 11 $/\$ $/\$
37	33, 34, 27, 36	syl3anc 1274	. . . . . . . . . 10 $/\$ $/\$ $/\$
38	37	fveq2d 5652	. . . . . . . . 9 $/\$ $/\$ $/\$
39	14, 15	mulgcl 13806	. . . . . . . . . . 11 $/\$ $/\$
40	33, 34, 27, 39	syl3anc 1274	. . . . . . . . . 10 $/\$ $/\$ $/\$
41	14, 35	grpinvinv 13730	. . . . . . . . . 10 $/\$
42	40, 41	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
43	38, 42	eqtrd 2264	. . . . . . . 8 $/\$ $/\$ $/\$
44	14, 15	mulgcl 13806	. . . . . . . . . . . 12 $/\$ $/\$
45	33, 5, 27, 44	syl3anc 1274	. . . . . . . . . . 11 $/\$ $/\$ $/\$
46	14, 15, 35	mulgneg 13807	. . . . . . . . . . 11 $/\$ $/\$
47	33, 1, 45, 46	syl3anc 1274	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47	fveq2d 5652	. . . . . . . . 9 $/\$ $/\$ $/\$
49	14, 15	mulgcl 13806	. . . . . . . . . . 11 $/\$ $/\$
50	33, 1, 45, 49	syl3anc 1274	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	14, 35	grpinvinv 13730	. . . . . . . . . 10 $/\$
52	50, 51	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
53	48, 52	eqtrd 2264	. . . . . . . 8 $/\$ $/\$ $/\$
54	43, 53	eqeq12d 2246	. . . . . . 7 $/\$ $/\$ $/\$
55	32, 54	imbitrid 154	. . . . . 6 $/\$ $/\$ $/\$
56	55	imp 124	. . . . 5 $/\$ $/\$ $/\$ $/\$
57	31, 56	syldan 282	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
58	57	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
59	9	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
60		simprl 531	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
61		simprr 533	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62	27	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
63	14, 15	mulgnn0ass 13825	. . . . . . 7 $/\$ $/\$ $/\$
64	59, 60, 61, 62, 63	syl13anc 1276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
65	19, 20	mulneg2d 8650	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
67	66	oveq1d 6043	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
68	14, 15, 35	mulgneg 13807	. . . . . . . . . 10 $/\$ $/\$
69	33, 5, 27, 68	syl3anc 1274	. . . . . . . . 9 $/\$ $/\$ $/\$
70	69	oveq2d 6044	. . . . . . . 8 $/\$ $/\$ $/\$
71	14, 15, 35	mulgneg2 13823	. . . . . . . . 9 $/\$ $/\$
72	33, 1, 45, 71	syl3anc 1274	. . . . . . . 8 $/\$ $/\$ $/\$
73	70, 72	eqtr4d 2267	. . . . . . 7 $/\$ $/\$ $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
75	64, 67, 74	3eqtr3d 2272	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
76	75, 56	syldan 282	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
78	9	ad2antrr 488	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
79		simprl 531	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
80		simprr 533	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
81	27	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	14, 15	mulgnn0ass 13825	. . . . . 6 $/\$ $/\$ $/\$
83	78, 79, 80, 81, 82	syl13anc 1276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
84	19, 20	mul2negd 8651	. . . . . . 7 $/\$ $/\$ $/\$
85	84	oveq1d 6043	. . . . . 6 $/\$ $/\$ $/\$
86	85	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
87	33	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
88	1	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
89		nn0z 9560	. . . . . . . . 9
90	89	ad2antll 491	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
91	14, 15	mulgcl 13806	. . . . . . . 8 $/\$ $/\$
92	87, 90, 81, 91	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
93	14, 15, 35	mulgneg2 13823	. . . . . . 7 $/\$ $/\$
94	87, 88, 92, 93	syl3anc 1274	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
95	14, 15, 35	mulgneg 13807	. . . . . . . . 9 $/\$ $/\$
96	87, 90, 81, 95	syl3anc 1274	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	20	negnegd 8540	. . . . . . . . . 10 $/\$ $/\$ $/\$
98	97	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
99	98	oveq1d 6043	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtr3d 2266	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	100	oveq2d 6044	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	94, 101	eqtrd 2264	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	83, 86, 102	3eqtr3d 2272	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
105	18, 58, 77, 104	ccased 974	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
106	4, 8, 105	mp2and 433	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

cr 8091

cmul 8097

cneg 8410

cn0 9461

cz 9540

cbs 13162

cmnd 13579

cgrp 13663

cminusg 13664 ._gcmg 13786

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 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-ltadd 8208

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 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-sub 8411 df-neg 8412 df-inn 9203 df-2 9261 df-n0 9462 df-z 9541 df-uz 9817 df-fz 10306 df-fzo 10440 df-seqfrec 10773 df-ndx 13165 df-slot 13166 df-base 13168 df-plusg 13253 df-0g 13421 df-mgm 13519 df-sgrp 13565 df-mnd 13580 df-grp 13666 df-minusg 13667 df-mulg 13787

This theorem is referenced by: mulgassr 13827 mulgrhm 14705

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