mulgass - Metamath Proof Explorer

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Theorem mulgass 17579

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 1067	. . 3 $/\$ $/\$ $/\$
2		elznn0 11392	. . . 4 $/\$ $\/$
3	2	simprbi 480	. . 3 $\/$
4	1, 3	syl 17	. 2 $/\$ $/\$ $/\$ $\/$
5		simpr2 1068	. . 3 $/\$ $/\$ $/\$
6		elznn0 11392	. . . 4 $/\$ $\/$
7	6	simprbi 480	. . 3 $\/$
8	5, 7	syl 17	. 2 $/\$ $/\$ $/\$ $\/$
9		grpmnd 17429	. . . . . 6
10	9	ad2antrr 762	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
11		simprl 794	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
12		simprr 796	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
13		simplr3 1105	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
14		mulgass.b	. . . . . 6
15		mulgass.t	. . . . . 6 ._g
16	14, 15	mulgnn0ass 17578	. . . . 5 $/\$ $/\$ $/\$
17	10, 11, 12, 13, 16	syl13anc 1328	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 450	. . 3 $/\$ $/\$ $/\$ $/\$
19	1	zcnd 11483	. . . . . . . . 9 $/\$ $/\$ $/\$
20	5	zcnd 11483	. . . . . . . . 9 $/\$ $/\$ $/\$
21	19, 20	mulneg1d 10483	. . . . . . . 8 $/\$ $/\$ $/\$
22	21	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
23	22	oveq1d 6665	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
24	9	ad2antrr 762	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
25		simprl 794	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
26		simprr 796	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
27		simpr3 1069	. . . . . . . 8 $/\$ $/\$ $/\$
28	27	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
29	14, 15	mulgnn0ass 17578	. . . . . . 7 $/\$ $/\$ $/\$
30	24, 25, 26, 28, 29	syl13anc 1328	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
31	23, 30	eqtr3d 2658	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
32		fveq2 6191	. . . . . . 7
33		simpl 473	. . . . . . . . . . 11 $/\$ $/\$ $/\$
34	1, 5	zmulcld 11488	. . . . . . . . . . 11 $/\$ $/\$ $/\$
35		eqid 2622	. . . . . . . . . . . 12
36	14, 15, 35	mulgneg 17560	. . . . . . . . . . 11 $/\$ $/\$
37	33, 34, 27, 36	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$
38	37	fveq2d 6195	. . . . . . . . 9 $/\$ $/\$ $/\$
39	14, 15	mulgcl 17559	. . . . . . . . . . 11 $/\$ $/\$
40	33, 34, 27, 39	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$
41	14, 35	grpinvinv 17482	. . . . . . . . . 10 $/\$
42	40, 41	syldan 487	. . . . . . . . 9 $/\$ $/\$ $/\$
43	38, 42	eqtrd 2656	. . . . . . . 8 $/\$ $/\$ $/\$
44	14, 15	mulgcl 17559	. . . . . . . . . . . 12 $/\$ $/\$
45	33, 5, 27, 44	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$ $/\$
46	14, 15, 35	mulgneg 17560	. . . . . . . . . . 11 $/\$ $/\$
47	33, 1, 45, 46	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47	fveq2d 6195	. . . . . . . . 9 $/\$ $/\$ $/\$
49	14, 15	mulgcl 17559	. . . . . . . . . . 11 $/\$ $/\$
50	33, 1, 45, 49	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	14, 35	grpinvinv 17482	. . . . . . . . . 10 $/\$
52	50, 51	syldan 487	. . . . . . . . 9 $/\$ $/\$ $/\$
53	48, 52	eqtrd 2656	. . . . . . . 8 $/\$ $/\$ $/\$
54	43, 53	eqeq12d 2637	. . . . . . 7 $/\$ $/\$ $/\$
55	32, 54	syl5ib 234	. . . . . 6 $/\$ $/\$ $/\$
56	55	imp 445	. . . . 5 $/\$ $/\$ $/\$ $/\$
57	31, 56	syldan 487	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
58	57	ex 450	. . 3 $/\$ $/\$ $/\$ $/\$
59	9	ad2antrr 762	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
60		simprl 794	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
61		simprr 796	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62	27	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
63	14, 15	mulgnn0ass 17578	. . . . . . 7 $/\$ $/\$ $/\$
64	59, 60, 61, 62, 63	syl13anc 1328	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
65	19, 20	mulneg2d 10484	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
67	66	oveq1d 6665	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
68	14, 15, 35	mulgneg 17560	. . . . . . . . . 10 $/\$ $/\$
69	33, 5, 27, 68	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$
70	69	oveq2d 6666	. . . . . . . 8 $/\$ $/\$ $/\$
71	14, 15, 35	mulgneg2 17575	. . . . . . . . 9 $/\$ $/\$
72	33, 1, 45, 71	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$
73	70, 72	eqtr4d 2659	. . . . . . 7 $/\$ $/\$ $/\$
74	73	adantr 481	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
75	64, 67, 74	3eqtr3d 2664	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
76	75, 56	syldan 487	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	ex 450	. . 3 $/\$ $/\$ $/\$ $/\$
78	9	ad2antrr 762	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
79		simprl 794	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
80		simprr 796	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
81	27	adantr 481	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	14, 15	mulgnn0ass 17578	. . . . . 6 $/\$ $/\$ $/\$
83	78, 79, 80, 81, 82	syl13anc 1328	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
84	19, 20	mul2negd 10485	. . . . . . 7 $/\$ $/\$ $/\$
85	84	oveq1d 6665	. . . . . 6 $/\$ $/\$ $/\$
86	85	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
87	33	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
88	1	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
89		nn0z 11400	. . . . . . . . 9
90	89	ad2antll 765	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
91	14, 15	mulgcl 17559	. . . . . . . 8 $/\$ $/\$
92	87, 90, 81, 91	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
93	14, 15, 35	mulgneg2 17575	. . . . . . 7 $/\$ $/\$
94	87, 88, 92, 93	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
95	14, 15, 35	mulgneg 17560	. . . . . . . . 9 $/\$ $/\$
96	87, 90, 81, 95	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	20	negnegd 10383	. . . . . . . . . 10 $/\$ $/\$ $/\$
98	97	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
99	98	oveq1d 6665	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtr3d 2658	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	100	oveq2d 6666	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	94, 101	eqtrd 2656	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	83, 86, 102	3eqtr3d 2664	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	ex 450	. . 3 $/\$ $/\$ $/\$ $/\$
105	18, 58, 77, 104	ccased 988	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
106	4, 8, 105	mp2and 715	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

cfv 5888 (class class class)co 6650

cr 9935

cmul 9941

cneg 10267

cn0 11292

cz 11377

cbs 15857

cmnd 17294

cgrp 17422

cminusg 17423 ._gcmg 17540

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-seq 12802 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mulg 17541

This theorem is referenced by: mulgassr 17580 odmod 17965 odmulgid 17971 odbezout 17975 gexdvdsi 17998 pgpfac1lem2 18474 pgpfac1lem3a 18475 pgpfac1lem3 18476 mulgrhm 19846 zlmlmod 19871

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