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Theorem mulgass 13745

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 1029	. . 3 $/\$ $/\$ $/\$
2		elznn0 9493	. . . 4 $/\$ $\/$
3	2	simprbi 275	. . 3 $\/$
4	1, 3	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
5		simpr2 1030	. . 3 $/\$ $/\$ $/\$
6		elznn0 9493	. . . 4 $/\$ $\/$
7	6	simprbi 275	. . 3 $\/$
8	5, 7	syl 14	. 2 $/\$ $/\$ $/\$ $\/$
9		grpmnd 13589	. . . . . 6
10	9	ad2antrr 488	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
11		simprl 531	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
12		simprr 533	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
13		simplr3 1067	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
14		mulgass.b	. . . . . 6
15		mulgass.t	. . . . . 6 ._g
16	14, 15	mulgnn0ass 13744	. . . . 5 $/\$ $/\$ $/\$
17	10, 11, 12, 13, 16	syl13anc 1275	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
19	1	zcnd 9602	. . . . . . . . 9 $/\$ $/\$ $/\$
20	5	zcnd 9602	. . . . . . . . 9 $/\$ $/\$ $/\$
21	19, 20	mulneg1d 8589	. . . . . . . 8 $/\$ $/\$ $/\$
22	21	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
23	22	oveq1d 6032	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
24	9	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
25		simprl 531	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
26		simprr 533	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
27		simpr3 1031	. . . . . . . 8 $/\$ $/\$ $/\$
28	27	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
29	14, 15	mulgnn0ass 13744	. . . . . . 7 $/\$ $/\$ $/\$
30	24, 25, 26, 28, 29	syl13anc 1275	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
31	23, 30	eqtr3d 2266	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
32		fveq2 5639	. . . . . . 7
33		simpl 109	. . . . . . . . . . 11 $/\$ $/\$ $/\$
34	1, 5	zmulcld 9607	. . . . . . . . . . 11 $/\$ $/\$ $/\$
35		eqid 2231	. . . . . . . . . . . 12
36	14, 15, 35	mulgneg 13726	. . . . . . . . . . 11 $/\$ $/\$
37	33, 34, 27, 36	syl3anc 1273	. . . . . . . . . 10 $/\$ $/\$ $/\$
38	37	fveq2d 5643	. . . . . . . . 9 $/\$ $/\$ $/\$
39	14, 15	mulgcl 13725	. . . . . . . . . . 11 $/\$ $/\$
40	33, 34, 27, 39	syl3anc 1273	. . . . . . . . . 10 $/\$ $/\$ $/\$
41	14, 35	grpinvinv 13649	. . . . . . . . . 10 $/\$
42	40, 41	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
43	38, 42	eqtrd 2264	. . . . . . . 8 $/\$ $/\$ $/\$
44	14, 15	mulgcl 13725	. . . . . . . . . . . 12 $/\$ $/\$
45	33, 5, 27, 44	syl3anc 1273	. . . . . . . . . . 11 $/\$ $/\$ $/\$
46	14, 15, 35	mulgneg 13726	. . . . . . . . . . 11 $/\$ $/\$
47	33, 1, 45, 46	syl3anc 1273	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47	fveq2d 5643	. . . . . . . . 9 $/\$ $/\$ $/\$
49	14, 15	mulgcl 13725	. . . . . . . . . . 11 $/\$ $/\$
50	33, 1, 45, 49	syl3anc 1273	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	14, 35	grpinvinv 13649	. . . . . . . . . 10 $/\$
52	50, 51	syldan 282	. . . . . . . . 9 $/\$ $/\$ $/\$
53	48, 52	eqtrd 2264	. . . . . . . 8 $/\$ $/\$ $/\$
54	43, 53	eqeq12d 2246	. . . . . . 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 531	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
61		simprr 533	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62	27	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
63	14, 15	mulgnn0ass 13744	. . . . . . 7 $/\$ $/\$ $/\$
64	59, 60, 61, 62, 63	syl13anc 1275	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
65	19, 20	mulneg2d 8590	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
67	66	oveq1d 6032	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
68	14, 15, 35	mulgneg 13726	. . . . . . . . . 10 $/\$ $/\$
69	33, 5, 27, 68	syl3anc 1273	. . . . . . . . 9 $/\$ $/\$ $/\$
70	69	oveq2d 6033	. . . . . . . 8 $/\$ $/\$ $/\$
71	14, 15, 35	mulgneg2 13742	. . . . . . . . 9 $/\$ $/\$
72	33, 1, 45, 71	syl3anc 1273	. . . . . . . 8 $/\$ $/\$ $/\$
73	70, 72	eqtr4d 2267	. . . . . . 7 $/\$ $/\$ $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
75	64, 67, 74	3eqtr3d 2272	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
76	75, 56	syldan 282	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
78	9	ad2antrr 488	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
79		simprl 531	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
80		simprr 533	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
81	27	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	14, 15	mulgnn0ass 13744	. . . . . 6 $/\$ $/\$ $/\$
83	78, 79, 80, 81, 82	syl13anc 1275	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
84	19, 20	mul2negd 8591	. . . . . . 7 $/\$ $/\$ $/\$
85	84	oveq1d 6032	. . . . . 6 $/\$ $/\$ $/\$
86	85	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
87	33	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
88	1	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
89		nn0z 9498	. . . . . . . . 9
90	89	ad2antll 491	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
91	14, 15	mulgcl 13725	. . . . . . . 8 $/\$ $/\$
92	87, 90, 81, 91	syl3anc 1273	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
93	14, 15, 35	mulgneg2 13742	. . . . . . 7 $/\$ $/\$
94	87, 88, 92, 93	syl3anc 1273	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
95	14, 15, 35	mulgneg 13726	. . . . . . . . 9 $/\$ $/\$
96	87, 90, 81, 95	syl3anc 1273	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	20	negnegd 8480	. . . . . . . . . 10 $/\$ $/\$ $/\$
98	97	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
99	98	oveq1d 6032	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtr3d 2266	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	100	oveq2d 6033	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	94, 101	eqtrd 2264	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	83, 86, 102	3eqtr3d 2272	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	ex 115	. . 3 $/\$ $/\$ $/\$ $/\$
105	18, 58, 77, 104	ccased 973	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
106	4, 8, 105	mp2and 433	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 715 $/\$ w3a 1004

wceq 1397

wcel 2202

cfv 5326 (class class class)co 6017

cr 8030

cmul 8036

cneg 8350

cn0 9401

cz 9478

cbs 13081

cmnd 13498

cgrp 13582

cminusg 13583 ._gcmg 13705

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-ltadd 8147

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-frec 6556 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-inn 9143 df-2 9201 df-n0 9402 df-z 9479 df-uz 9755 df-fz 10243 df-fzo 10377 df-seqfrec 10709 df-ndx 13084 df-slot 13085 df-base 13087 df-plusg 13172 df-0g 13340 df-mgm 13438 df-sgrp 13484 df-mnd 13499 df-grp 13585 df-minusg 13586 df-mulg 13706

This theorem is referenced by: mulgassr 13746 mulgrhm 14622

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