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Theorem mulgass 13232

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 1005	. . 3 $/\$ $/\$ $/\$
2		elznn0 9335	. . . 4 $/\$ $\/$
3	2	simprbi 275	. . 3 $\/$
4	1, 3	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
5		simpr2 1006	. . 3 $/\$ $/\$ $/\$
6		elznn0 9335	. . . 4 $/\$ $\/$
7	6	simprbi 275	. . 3 $\/$
8	5, 7	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
9		grpmnd 13082	. . . . . 6
10	9	ad2antrr 488	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
11		simprl 529	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
12		simprr 531	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
13		simplr3 1043	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
14		mulgass.b	. . . . . 6
15		mulgass.t	. . . . . 6 ._g
16	14, 15	mulgnn0ass 13231	. . . . 5 $/\$ $/\$ $/\$
17	10, 11, 12, 13, 16	syl13anc 1251	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
19	1	zcnd 9443	. . . . . . . . 9 $/\$ $/\$ $/\$
20	5	zcnd 9443	. . . . . . . . 9 $/\$ $/\$ $/\$
21	19, 20	mulneg1d 8432	. . . . . . . 8 $/\$ $/\$ $/\$
22	21	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
23	22	oveq1d 5934	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
24	9	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
25		simprl 529	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
26		simprr 531	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
27		simpr3 1007	. . . . . . . 8 $/\$ $/\$ $/\$
28	27	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
29	14, 15	mulgnn0ass 13231	. . . . . . 7 $/\$ $/\$ $/\$
30	24, 25, 26, 28, 29	syl13anc 1251	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
31	23, 30	eqtr3d 2228	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
32		fveq2 5555	. . . . . . 7
33		simpl 109	. . . . . . . . . . 11 $/\$ $/\$ $/\$
34	1, 5	zmulcld 9448	. . . . . . . . . . 11 $/\$ $/\$ $/\$
35		eqid 2193	. . . . . . . . . . . 12
36	14, 15, 35	mulgneg 13213	. . . . . . . . . . 11 $/\$ $/\$
37	33, 34, 27, 36	syl3anc 1249	. . . . . . . . . 10 $/\$ $/\$ $/\$
38	37	fveq2d 5559	. . . . . . . . 9 $/\$ $/\$ $/\$
39	14, 15	mulgcl 13212	. . . . . . . . . . 11 $/\$ $/\$
40	33, 34, 27, 39	syl3anc 1249	. . . . . . . . . 10 $/\$ $/\$ $/\$
41	14, 35	grpinvinv 13142	. . . . . . . . . 10 $/\$
42	40, 41	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
43	38, 42	eqtrd 2226	. . . . . . . 8 $/\$ $/\$ $/\$
44	14, 15	mulgcl 13212	. . . . . . . . . . . 12 $/\$ $/\$
45	33, 5, 27, 44	syl3anc 1249	. . . . . . . . . . 11 $/\$ $/\$ $/\$
46	14, 15, 35	mulgneg 13213	. . . . . . . . . . 11 $/\$ $/\$
47	33, 1, 45, 46	syl3anc 1249	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47	fveq2d 5559	. . . . . . . . 9 $/\$ $/\$ $/\$
49	14, 15	mulgcl 13212	. . . . . . . . . . 11 $/\$ $/\$
50	33, 1, 45, 49	syl3anc 1249	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	14, 35	grpinvinv 13142	. . . . . . . . . 10 $/\$
52	50, 51	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
53	48, 52	eqtrd 2226	. . . . . . . 8 $/\$ $/\$ $/\$
54	43, 53	eqeq12d 2208	. . . . . . 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 529	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
61		simprr 531	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62	27	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
63	14, 15	mulgnn0ass 13231	. . . . . . 7 $/\$ $/\$ $/\$
64	59, 60, 61, 62, 63	syl13anc 1251	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
65	19, 20	mulneg2d 8433	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
67	66	oveq1d 5934	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
68	14, 15, 35	mulgneg 13213	. . . . . . . . . 10 $/\$ $/\$
69	33, 5, 27, 68	syl3anc 1249	. . . . . . . . 9 $/\$ $/\$ $/\$
70	69	oveq2d 5935	. . . . . . . 8 $/\$ $/\$ $/\$
71	14, 15, 35	mulgneg2 13229	. . . . . . . . 9 $/\$ $/\$
72	33, 1, 45, 71	syl3anc 1249	. . . . . . . 8 $/\$ $/\$ $/\$
73	70, 72	eqtr4d 2229	. . . . . . 7 $/\$ $/\$ $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
75	64, 67, 74	3eqtr3d 2234	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
76	75, 56	syldan 282	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
78	9	ad2antrr 488	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
79		simprl 529	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
80		simprr 531	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
81	27	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	14, 15	mulgnn0ass 13231	. . . . . 6 $/\$ $/\$ $/\$
83	78, 79, 80, 81, 82	syl13anc 1251	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
84	19, 20	mul2negd 8434	. . . . . . 7 $/\$ $/\$ $/\$
85	84	oveq1d 5934	. . . . . 6 $/\$ $/\$ $/\$
86	85	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
87	33	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
88	1	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
89		nn0z 9340	. . . . . . . . 9
90	89	ad2antll 491	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
91	14, 15	mulgcl 13212	. . . . . . . 8 $/\$ $/\$
92	87, 90, 81, 91	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
93	14, 15, 35	mulgneg2 13229	. . . . . . 7 $/\$ $/\$
94	87, 88, 92, 93	syl3anc 1249	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
95	14, 15, 35	mulgneg 13213	. . . . . . . . 9 $/\$ $/\$
96	87, 90, 81, 95	syl3anc 1249	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	20	negnegd 8323	. . . . . . . . . 10 $/\$ $/\$ $/\$
98	97	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
99	98	oveq1d 5934	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtr3d 2228	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	100	oveq2d 5935	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	94, 101	eqtrd 2226	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	83, 86, 102	3eqtr3d 2234	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
105	18, 58, 77, 104	ccased 967	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
106	4, 8, 105	mp2and 433	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 709 $/\$ w3a 980

wceq 1364

wcel 2164

cfv 5255 (class class class)co 5919

cr 7873

cmul 7879

cneg 8193

cn0 9243

cz 9320

cbs 12621

cmnd 13000

cgrp 13075

cminusg 13076 ._gcmg 13192

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-ltadd 7990

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-frec 6446 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-inn 8985 df-2 9043 df-n0 9244 df-z 9321 df-uz 9596 df-fz 10078 df-fzo 10212 df-seqfrec 10522 df-ndx 12624 df-slot 12625 df-base 12627 df-plusg 12711 df-0g 12872 df-mgm 12942 df-sgrp 12988 df-mnd 13001 df-grp 13078 df-minusg 13079 df-mulg 13193

This theorem is referenced by: mulgassr 13233 mulgrhm 14108

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