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Theorem mulgass 12875

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 1001	. . 3 $/\$ $/\$ $/\$
2		elznn0 9236	. . . 4 $/\$ $\/$
3	2	simprbi 275	. . 3 $\/$
4	1, 3	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
5		simpr2 1002	. . 3 $/\$ $/\$ $/\$
6		elznn0 9236	. . . 4 $/\$ $\/$
7	6	simprbi 275	. . 3 $\/$
8	5, 7	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
9		grpmnd 12742	. . . . . 6
10	9	ad2antrr 488	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
11		simprl 529	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
12		simprr 530	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
13		simplr3 1039	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
14		mulgass.b	. . . . . 6
15		mulgass.t	. . . . . 6 ._g
16	14, 15	mulgnn0ass 12874	. . . . 5 $/\$ $/\$ $/\$
17	10, 11, 12, 13, 16	syl13anc 1238	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
19	1	zcnd 9344	. . . . . . . . 9 $/\$ $/\$ $/\$
20	5	zcnd 9344	. . . . . . . . 9 $/\$ $/\$ $/\$
21	19, 20	mulneg1d 8339	. . . . . . . 8 $/\$ $/\$ $/\$
22	21	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
23	22	oveq1d 5877	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
24	9	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
25		simprl 529	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
26		simprr 530	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
27		simpr3 1003	. . . . . . . 8 $/\$ $/\$ $/\$
28	27	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
29	14, 15	mulgnn0ass 12874	. . . . . . 7 $/\$ $/\$ $/\$
30	24, 25, 26, 28, 29	syl13anc 1238	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
31	23, 30	eqtr3d 2208	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
32		fveq2 5504	. . . . . . 7
33		simpl 109	. . . . . . . . . . 11 $/\$ $/\$ $/\$
34	1, 5	zmulcld 9349	. . . . . . . . . . 11 $/\$ $/\$ $/\$
35		eqid 2173	. . . . . . . . . . . 12
36	14, 15, 35	mulgneg 12857	. . . . . . . . . . 11 $/\$ $/\$
37	33, 34, 27, 36	syl3anc 1236	. . . . . . . . . 10 $/\$ $/\$ $/\$
38	37	fveq2d 5508	. . . . . . . . 9 $/\$ $/\$ $/\$
39	14, 15	mulgcl 12856	. . . . . . . . . . 11 $/\$ $/\$
40	33, 34, 27, 39	syl3anc 1236	. . . . . . . . . 10 $/\$ $/\$ $/\$
41	14, 35	grpinvinv 12793	. . . . . . . . . 10 $/\$
42	40, 41	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
43	38, 42	eqtrd 2206	. . . . . . . 8 $/\$ $/\$ $/\$
44	14, 15	mulgcl 12856	. . . . . . . . . . . 12 $/\$ $/\$
45	33, 5, 27, 44	syl3anc 1236	. . . . . . . . . . 11 $/\$ $/\$ $/\$
46	14, 15, 35	mulgneg 12857	. . . . . . . . . . 11 $/\$ $/\$
47	33, 1, 45, 46	syl3anc 1236	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47	fveq2d 5508	. . . . . . . . 9 $/\$ $/\$ $/\$
49	14, 15	mulgcl 12856	. . . . . . . . . . 11 $/\$ $/\$
50	33, 1, 45, 49	syl3anc 1236	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	14, 35	grpinvinv 12793	. . . . . . . . . 10 $/\$
52	50, 51	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
53	48, 52	eqtrd 2206	. . . . . . . 8 $/\$ $/\$ $/\$
54	43, 53	eqeq12d 2188	. . . . . . 7 $/\$ $/\$ $/\$
55	32, 54	syl5ib 155	. . . . . 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 530	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62	27	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
63	14, 15	mulgnn0ass 12874	. . . . . . 7 $/\$ $/\$ $/\$
64	59, 60, 61, 62, 63	syl13anc 1238	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
65	19, 20	mulneg2d 8340	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
67	66	oveq1d 5877	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
68	14, 15, 35	mulgneg 12857	. . . . . . . . . 10 $/\$ $/\$
69	33, 5, 27, 68	syl3anc 1236	. . . . . . . . 9 $/\$ $/\$ $/\$
70	69	oveq2d 5878	. . . . . . . 8 $/\$ $/\$ $/\$
71	14, 15, 35	mulgneg2 12872	. . . . . . . . 9 $/\$ $/\$
72	33, 1, 45, 71	syl3anc 1236	. . . . . . . 8 $/\$ $/\$ $/\$
73	70, 72	eqtr4d 2209	. . . . . . 7 $/\$ $/\$ $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
75	64, 67, 74	3eqtr3d 2214	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
76	75, 56	syldan 282	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
78	9	ad2antrr 488	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
79		simprl 529	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
80		simprr 530	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
81	27	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	14, 15	mulgnn0ass 12874	. . . . . 6 $/\$ $/\$ $/\$
83	78, 79, 80, 81, 82	syl13anc 1238	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
84	19, 20	mul2negd 8341	. . . . . . 7 $/\$ $/\$ $/\$
85	84	oveq1d 5877	. . . . . 6 $/\$ $/\$ $/\$
86	85	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
87	33	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
88	1	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
89		nn0z 9241	. . . . . . . . 9
90	89	ad2antll 491	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
91	14, 15	mulgcl 12856	. . . . . . . 8 $/\$ $/\$
92	87, 90, 81, 91	syl3anc 1236	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
93	14, 15, 35	mulgneg2 12872	. . . . . . 7 $/\$ $/\$
94	87, 88, 92, 93	syl3anc 1236	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
95	14, 15, 35	mulgneg 12857	. . . . . . . . 9 $/\$ $/\$
96	87, 90, 81, 95	syl3anc 1236	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	20	negnegd 8230	. . . . . . . . . 10 $/\$ $/\$ $/\$
98	97	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
99	98	oveq1d 5877	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtr3d 2208	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	100	oveq2d 5878	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	94, 101	eqtrd 2206	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	83, 86, 102	3eqtr3d 2214	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
105	18, 58, 77, 104	ccased 963	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
106	4, 8, 105	mp2and 433	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 706 $/\$ w3a 976

wceq 1351

wcel 2144

cfv 5205 (class class class)co 5862

cr 7782

cmul 7788

cneg 8100

cn0 9144

cz 9221

cbs 12425

cmnd 12679

cgrp 12735

cminusg 12736 ._gcmg 12839

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 612 ax-in2 613 ax-io 707 ax-5 1443 ax-7 1444 ax-gen 1445 ax-ie1 1489 ax-ie2 1490 ax-8 1500 ax-10 1501 ax-11 1502 ax-i12 1503 ax-bndl 1505 ax-4 1506 ax-17 1522 ax-i9 1526 ax-ial 1530 ax-i5r 1531 ax-13 2146 ax-14 2147 ax-ext 2155 ax-coll 4110 ax-sep 4113 ax-nul 4121 ax-pow 4166 ax-pr 4200 ax-un 4424 ax-setind 4527 ax-iinf 4578 ax-cnex 7874 ax-resscn 7875 ax-1cn 7876 ax-1re 7877 ax-icn 7878 ax-addcl 7879 ax-addrcl 7880 ax-mulcl 7881 ax-mulrcl 7882 ax-addcom 7883 ax-mulcom 7884 ax-addass 7885 ax-mulass 7886 ax-distr 7887 ax-i2m1 7888 ax-0lt1 7889 ax-1rid 7890 ax-0id 7891 ax-rnegex 7892 ax-cnre 7894 ax-pre-ltirr 7895 ax-pre-ltwlin 7896 ax-pre-lttrn 7897 ax-pre-ltadd 7899

This theorem depends on definitions: df-bi 117 df-dc 833 df-3or 977 df-3an 978 df-tru 1354 df-fal 1357 df-nf 1457 df-sb 1759 df-eu 2025 df-mo 2026 df-clab 2160 df-cleq 2166 df-clel 2169 df-nfc 2304 df-ne 2344 df-nel 2439 df-ral 2456 df-rex 2457 df-reu 2458 df-rmo 2459 df-rab 2460 df-v 2735 df-sbc 2959 df-csb 3053 df-dif 3126 df-un 3128 df-in 3130 df-ss 3137 df-nul 3418 df-if 3530 df-pw 3571 df-sn 3592 df-pr 3593 df-op 3595 df-uni 3803 df-int 3838 df-iun 3881 df-br 3996 df-opab 4057 df-mpt 4058 df-tr 4094 df-id 4284 df-iord 4357 df-on 4359 df-ilim 4360 df-suc 4362 df-iom 4581 df-xp 4623 df-rel 4624 df-cnv 4625 df-co 4626 df-dm 4627 df-rn 4628 df-res 4629 df-ima 4630 df-iota 5167 df-fun 5207 df-fn 5208 df-f 5209 df-f1 5210 df-fo 5211 df-f1o 5212 df-fv 5213 df-riota 5818 df-ov 5865 df-oprab 5866 df-mpo 5867 df-1st 6128 df-2nd 6129 df-recs 6293 df-frec 6379 df-pnf 7965 df-mnf 7966 df-xr 7967 df-ltxr 7968 df-le 7969 df-sub 8101 df-neg 8102 df-inn 8888 df-2 8946 df-n0 9145 df-z 9222 df-uz 9497 df-fz 9975 df-fzo 10108 df-seqfrec 10411 df-ndx 12428 df-slot 12429 df-base 12431 df-plusg 12502 df-0g 12625 df-mgm 12637 df-sgrp 12670 df-mnd 12680 df-grp 12738 df-minusg 12739 df-mulg 12840

This theorem is referenced by: mulgassr 12876

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