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Theorem mulgass 13962

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 1034	. . 3 $/\$ $/\$ $/\$
2		elznn0 9659	. . . 4 $/\$ $\/$
3	2	simprbi 275	. . 3 $\/$
4	1, 3	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
5		simpr2 1035	. . 3 $/\$ $/\$ $/\$
6		elznn0 9659	. . . 4 $/\$ $\/$
7	6	simprbi 275	. . 3 $\/$
8	5, 7	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
9		grpmnd 13812	. . . . . 6
10	9	ad2antrr 492	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
11		simprl 535	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
12		simprr 537	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
13		simplr3 1072	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
14		mulgass.b	. . . . . 6
15		mulgass.t	. . . . . 6 ._g
16	14, 15	mulgnn0ass 13961	. . . . 5 $/\$ $/\$ $/\$
17	10, 11, 12, 13, 16	syl13anc 1280	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
19	1	zcnd 9769	. . . . . . . . 9 $/\$ $/\$ $/\$
20	5	zcnd 9769	. . . . . . . . 9 $/\$ $/\$ $/\$
21	19, 20	mulneg1d 8738	. . . . . . . 8 $/\$ $/\$ $/\$
22	21	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
23	22	oveq1d 6100	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
24	9	ad2antrr 492	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
25		simprl 535	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
26		simprr 537	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
27		simpr3 1036	. . . . . . . 8 $/\$ $/\$ $/\$
28	27	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
29	14, 15	mulgnn0ass 13961	. . . . . . 7 $/\$ $/\$ $/\$
30	24, 25, 26, 28, 29	syl13anc 1280	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
31	23, 30	eqtr3d 2273	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
32		fveq2 5695	. . . . . . 7
33		simpl 109	. . . . . . . . . . 11 $/\$ $/\$ $/\$
34	1, 5	zmulcld 9774	. . . . . . . . . . 11 $/\$ $/\$ $/\$
35		eqid 2238	. . . . . . . . . . . 12
36	14, 15, 35	mulgneg 13943	. . . . . . . . . . 11 $/\$ $/\$
37	33, 34, 27, 36	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
38	37	fveq2d 5699	. . . . . . . . 9 $/\$ $/\$ $/\$
39	14, 15	mulgcl 13942	. . . . . . . . . . 11 $/\$ $/\$
40	33, 34, 27, 39	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
41	14, 35	grpinvinv 13872	. . . . . . . . . 10 $/\$
42	40, 41	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
43	38, 42	eqtrd 2271	. . . . . . . 8 $/\$ $/\$ $/\$
44	14, 15	mulgcl 13942	. . . . . . . . . . . 12 $/\$ $/\$
45	33, 5, 27, 44	syl3anc 1278	. . . . . . . . . . 11 $/\$ $/\$ $/\$
46	14, 15, 35	mulgneg 13943	. . . . . . . . . . 11 $/\$ $/\$
47	33, 1, 45, 46	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47	fveq2d 5699	. . . . . . . . 9 $/\$ $/\$ $/\$
49	14, 15	mulgcl 13942	. . . . . . . . . . 11 $/\$ $/\$
50	33, 1, 45, 49	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	14, 35	grpinvinv 13872	. . . . . . . . . 10 $/\$
52	50, 51	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
53	48, 52	eqtrd 2271	. . . . . . . 8 $/\$ $/\$ $/\$
54	43, 53	eqeq12d 2253	. . . . . . 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 492	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
60		simprl 535	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
61		simprr 537	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62	27	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
63	14, 15	mulgnn0ass 13961	. . . . . . 7 $/\$ $/\$ $/\$
64	59, 60, 61, 62, 63	syl13anc 1280	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
65	19, 20	mulneg2d 8739	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
67	66	oveq1d 6100	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
68	14, 15, 35	mulgneg 13943	. . . . . . . . . 10 $/\$ $/\$
69	33, 5, 27, 68	syl3anc 1278	. . . . . . . . 9 $/\$ $/\$ $/\$
70	69	oveq2d 6101	. . . . . . . 8 $/\$ $/\$ $/\$
71	14, 15, 35	mulgneg2 13959	. . . . . . . . 9 $/\$ $/\$
72	33, 1, 45, 71	syl3anc 1278	. . . . . . . 8 $/\$ $/\$ $/\$
73	70, 72	eqtr4d 2274	. . . . . . 7 $/\$ $/\$ $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
75	64, 67, 74	3eqtr3d 2279	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
76	75, 56	syldan 282	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
78	9	ad2antrr 492	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
79		simprl 535	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
80		simprr 537	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
81	27	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	14, 15	mulgnn0ass 13961	. . . . . 6 $/\$ $/\$ $/\$
83	78, 79, 80, 81, 82	syl13anc 1280	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
84	19, 20	mul2negd 8740	. . . . . . 7 $/\$ $/\$ $/\$
85	84	oveq1d 6100	. . . . . 6 $/\$ $/\$ $/\$
86	85	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
87	33	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
88	1	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
89		nn0z 9664	. . . . . . . . 9
90	89	ad2antll 495	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
91	14, 15	mulgcl 13942	. . . . . . . 8 $/\$ $/\$
92	87, 90, 81, 91	syl3anc 1278	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
93	14, 15, 35	mulgneg2 13959	. . . . . . 7 $/\$ $/\$
94	87, 88, 92, 93	syl3anc 1278	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
95	14, 15, 35	mulgneg 13943	. . . . . . . . 9 $/\$ $/\$
96	87, 90, 81, 95	syl3anc 1278	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	20	negnegd 8628	. . . . . . . . . 10 $/\$ $/\$ $/\$
98	97	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
99	98	oveq1d 6100	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtr3d 2273	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	100	oveq2d 6101	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	94, 101	eqtrd 2271	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	83, 86, 102	3eqtr3d 2279	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
105	18, 58, 77, 104	ccased 978	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
106	4, 8, 105	mp2and 437	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104 $\/$ wo 720 $/\$ w3a 1009

wceq 1402

wcel 2209

cfv 5377 (class class class)co 6085

cr 8178

cmul 8184

cneg 8498

cn0 9563

cz 9644

cbs 13352

cmnd 13729

cgrp 13805

cminusg 13806 ._gcmg 13922

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-ltadd 8295

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-inn 9305 df-2 9363 df-n0 9564 df-z 9645 df-uz 9922 df-fz 10412 df-fzo 10550 df-seqfrec 10885 df-ndx 13355 df-slot 13356 df-base 13358 df-plusg 13444 df-0g 13612 df-mgm 13676 df-sgrp 13717 df-mnd 13730 df-grp 13808 df-minusg 13809 df-mulg 13923

This theorem is used by: mulgassr 13963 mulgrhm 14944

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