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Theorem mulgass 13696

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 1027	. . 3 $/\$ $/\$ $/\$
2		elznn0 9461	. . . 4 $/\$ $\/$
3	2	simprbi 275	. . 3 $\/$
4	1, 3	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
5		simpr2 1028	. . 3 $/\$ $/\$ $/\$
6		elznn0 9461	. . . 4 $/\$ $\/$
7	6	simprbi 275	. . 3 $\/$
8	5, 7	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
9		grpmnd 13540	. . . . . 6
10	9	ad2antrr 488	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
11		simprl 529	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
12		simprr 531	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
13		simplr3 1065	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
14		mulgass.b	. . . . . 6
15		mulgass.t	. . . . . 6 ._g
16	14, 15	mulgnn0ass 13695	. . . . 5 $/\$ $/\$ $/\$
17	10, 11, 12, 13, 16	syl13anc 1273	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
19	1	zcnd 9570	. . . . . . . . 9 $/\$ $/\$ $/\$
20	5	zcnd 9570	. . . . . . . . 9 $/\$ $/\$ $/\$
21	19, 20	mulneg1d 8557	. . . . . . . 8 $/\$ $/\$ $/\$
22	21	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
23	22	oveq1d 6016	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
24	9	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
25		simprl 529	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
26		simprr 531	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
27		simpr3 1029	. . . . . . . 8 $/\$ $/\$ $/\$
28	27	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
29	14, 15	mulgnn0ass 13695	. . . . . . 7 $/\$ $/\$ $/\$
30	24, 25, 26, 28, 29	syl13anc 1273	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
31	23, 30	eqtr3d 2264	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
32		fveq2 5627	. . . . . . 7
33		simpl 109	. . . . . . . . . . 11 $/\$ $/\$ $/\$
34	1, 5	zmulcld 9575	. . . . . . . . . . 11 $/\$ $/\$ $/\$
35		eqid 2229	. . . . . . . . . . . 12
36	14, 15, 35	mulgneg 13677	. . . . . . . . . . 11 $/\$ $/\$
37	33, 34, 27, 36	syl3anc 1271	. . . . . . . . . 10 $/\$ $/\$ $/\$
38	37	fveq2d 5631	. . . . . . . . 9 $/\$ $/\$ $/\$
39	14, 15	mulgcl 13676	. . . . . . . . . . 11 $/\$ $/\$
40	33, 34, 27, 39	syl3anc 1271	. . . . . . . . . 10 $/\$ $/\$ $/\$
41	14, 35	grpinvinv 13600	. . . . . . . . . 10 $/\$
42	40, 41	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
43	38, 42	eqtrd 2262	. . . . . . . 8 $/\$ $/\$ $/\$
44	14, 15	mulgcl 13676	. . . . . . . . . . . 12 $/\$ $/\$
45	33, 5, 27, 44	syl3anc 1271	. . . . . . . . . . 11 $/\$ $/\$ $/\$
46	14, 15, 35	mulgneg 13677	. . . . . . . . . . 11 $/\$ $/\$
47	33, 1, 45, 46	syl3anc 1271	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47	fveq2d 5631	. . . . . . . . 9 $/\$ $/\$ $/\$
49	14, 15	mulgcl 13676	. . . . . . . . . . 11 $/\$ $/\$
50	33, 1, 45, 49	syl3anc 1271	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	14, 35	grpinvinv 13600	. . . . . . . . . 10 $/\$
52	50, 51	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
53	48, 52	eqtrd 2262	. . . . . . . 8 $/\$ $/\$ $/\$
54	43, 53	eqeq12d 2244	. . . . . . 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 13695	. . . . . . 7 $/\$ $/\$ $/\$
64	59, 60, 61, 62, 63	syl13anc 1273	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
65	19, 20	mulneg2d 8558	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
67	66	oveq1d 6016	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
68	14, 15, 35	mulgneg 13677	. . . . . . . . . 10 $/\$ $/\$
69	33, 5, 27, 68	syl3anc 1271	. . . . . . . . 9 $/\$ $/\$ $/\$
70	69	oveq2d 6017	. . . . . . . 8 $/\$ $/\$ $/\$
71	14, 15, 35	mulgneg2 13693	. . . . . . . . 9 $/\$ $/\$
72	33, 1, 45, 71	syl3anc 1271	. . . . . . . 8 $/\$ $/\$ $/\$
73	70, 72	eqtr4d 2265	. . . . . . 7 $/\$ $/\$ $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
75	64, 67, 74	3eqtr3d 2270	. . . . 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 13695	. . . . . 6 $/\$ $/\$ $/\$
83	78, 79, 80, 81, 82	syl13anc 1273	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
84	19, 20	mul2negd 8559	. . . . . . 7 $/\$ $/\$ $/\$
85	84	oveq1d 6016	. . . . . 6 $/\$ $/\$ $/\$
86	85	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
87	33	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
88	1	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
89		nn0z 9466	. . . . . . . . 9
90	89	ad2antll 491	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
91	14, 15	mulgcl 13676	. . . . . . . 8 $/\$ $/\$
92	87, 90, 81, 91	syl3anc 1271	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
93	14, 15, 35	mulgneg2 13693	. . . . . . 7 $/\$ $/\$
94	87, 88, 92, 93	syl3anc 1271	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
95	14, 15, 35	mulgneg 13677	. . . . . . . . 9 $/\$ $/\$
96	87, 90, 81, 95	syl3anc 1271	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	20	negnegd 8448	. . . . . . . . . 10 $/\$ $/\$ $/\$
98	97	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
99	98	oveq1d 6016	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtr3d 2264	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	100	oveq2d 6017	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	94, 101	eqtrd 2262	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	83, 86, 102	3eqtr3d 2270	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
105	18, 58, 77, 104	ccased 971	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
106	4, 8, 105	mp2and 433	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 713 $/\$ w3a 1002

wceq 1395

wcel 2200

cfv 5318 (class class class)co 6001

cr 7998

cmul 8004

cneg 8318

cn0 9369

cz 9446

cbs 13032

cmnd 13449

cgrp 13533

cminusg 13534 ._gcmg 13656

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-ltadd 8115

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-frec 6537 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-inn 9111 df-2 9169 df-n0 9370 df-z 9447 df-uz 9723 df-fz 10205 df-fzo 10339 df-seqfrec 10670 df-ndx 13035 df-slot 13036 df-base 13038 df-plusg 13123 df-0g 13291 df-mgm 13389 df-sgrp 13435 df-mnd 13450 df-grp 13536 df-minusg 13537 df-mulg 13657

This theorem is referenced by: mulgassr 13697 mulgrhm 14573

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