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Theorem mulgass 14021

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 9670	. . . 4 $/\$ $\/$
3	2	simprbi 275	. . 3 $\/$
4	1, 3	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
5		simpr2 1035	. . 3 $/\$ $/\$ $/\$
6		elznn0 9670	. . . 4 $/\$ $\/$
7	6	simprbi 275	. . 3 $\/$
8	5, 7	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
9		grpmnd 13871	. . . . . 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 14020	. . . . 5 $/\$ $/\$ $/\$
17	10, 11, 12, 13, 16	syl13anc 1280	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
19	1	zcnd 9780	. . . . . . . . 9 $/\$ $/\$ $/\$
20	5	zcnd 9780	. . . . . . . . 9 $/\$ $/\$ $/\$
21	19, 20	mulneg1d 8745	. . . . . . . 8 $/\$ $/\$ $/\$
22	21	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
23	22	oveq1d 6103	. . . . . 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 14020	. . . . . . 7 $/\$ $/\$ $/\$
30	24, 25, 26, 28, 29	syl13anc 1280	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
31	23, 30	eqtr3d 2273	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
32		fveq2 5698	. . . . . . 7
33		simpl 109	. . . . . . . . . . 11 $/\$ $/\$ $/\$
34	1, 5	zmulcld 9785	. . . . . . . . . . 11 $/\$ $/\$ $/\$
35		eqid 2238	. . . . . . . . . . . 12
36	14, 15, 35	mulgneg 14002	. . . . . . . . . . 11 $/\$ $/\$
37	33, 34, 27, 36	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
38	37	fveq2d 5702	. . . . . . . . 9 $/\$ $/\$ $/\$
39	14, 15	mulgcl 14001	. . . . . . . . . . 11 $/\$ $/\$
40	33, 34, 27, 39	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
41	14, 35	grpinvinv 13931	. . . . . . . . . 10 $/\$
42	40, 41	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
43	38, 42	eqtrd 2271	. . . . . . . 8 $/\$ $/\$ $/\$
44	14, 15	mulgcl 14001	. . . . . . . . . . . 12 $/\$ $/\$
45	33, 5, 27, 44	syl3anc 1278	. . . . . . . . . . 11 $/\$ $/\$ $/\$
46	14, 15, 35	mulgneg 14002	. . . . . . . . . . 11 $/\$ $/\$
47	33, 1, 45, 46	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47	fveq2d 5702	. . . . . . . . 9 $/\$ $/\$ $/\$
49	14, 15	mulgcl 14001	. . . . . . . . . . 11 $/\$ $/\$
50	33, 1, 45, 49	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	14, 35	grpinvinv 13931	. . . . . . . . . 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 14020	. . . . . . 7 $/\$ $/\$ $/\$
64	59, 60, 61, 62, 63	syl13anc 1280	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
65	19, 20	mulneg2d 8746	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
67	66	oveq1d 6103	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
68	14, 15, 35	mulgneg 14002	. . . . . . . . . 10 $/\$ $/\$
69	33, 5, 27, 68	syl3anc 1278	. . . . . . . . 9 $/\$ $/\$ $/\$
70	69	oveq2d 6104	. . . . . . . 8 $/\$ $/\$ $/\$
71	14, 15, 35	mulgneg2 14018	. . . . . . . . 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 14020	. . . . . 6 $/\$ $/\$ $/\$
83	78, 79, 80, 81, 82	syl13anc 1280	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
84	19, 20	mul2negd 8747	. . . . . . 7 $/\$ $/\$ $/\$
85	84	oveq1d 6103	. . . . . 6 $/\$ $/\$ $/\$
86	85	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
87	33	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
88	1	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
89		nn0z 9675	. . . . . . . . 9
90	89	ad2antll 495	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
91	14, 15	mulgcl 14001	. . . . . . . 8 $/\$ $/\$
92	87, 90, 81, 91	syl3anc 1278	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
93	14, 15, 35	mulgneg2 14018	. . . . . . 7 $/\$ $/\$
94	87, 88, 92, 93	syl3anc 1278	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
95	14, 15, 35	mulgneg 14002	. . . . . . . . 9 $/\$ $/\$
96	87, 90, 81, 95	syl3anc 1278	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	20	negnegd 8635	. . . . . . . . . 10 $/\$ $/\$ $/\$
98	97	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
99	98	oveq1d 6103	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtr3d 2273	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	100	oveq2d 6104	. . . . . 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 5380 (class class class)co 6088

cr 8184

cmul 8190

cneg 8505

cn0 9574

cz 9655

cbs 13410

cmnd 13788

cgrp 13864

cminusg 13865 ._gcmg 13981

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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-ltadd 8301

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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-inn 9314 df-2 9372 df-n0 9575 df-z 9656 df-uz 9938 df-fz 10429 df-fzo 10567 df-seqfrec 10906 df-ndx 13413 df-slot 13414 df-base 13416 df-plusg 13503 df-0g 13671 df-mgm 13735 df-sgrp 13776 df-mnd 13789 df-grp 13867 df-minusg 13868 df-mulg 13982

This theorem is used by: mulgassr 14022 mulgrhm 15053

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