expmulz - Metamath Proof Explorer

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Theorem expmulz 12906

Description: Product of exponents law for integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 7-Jul-2014.)

**Assertion**
Ref	Expression
expmulz	$/\$ $/\$ $/\$

**Proof of Theorem expmulz**
Step	Hyp	Ref	Expression
1		elznn0nn 11391	. . 3 $\/$ $/\$
2		elznn0nn 11391	. . . 4 $\/$ $/\$
3		expmul 12905	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1267	. . . . . . 7 $/\$
5	4	adantlr 751	. . . . . 6 $/\$ $/\$
6		simp2l 1087	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
7	6	recnd 10068	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		simp3 1063	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9	8	nn0cnd 11353	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
10	7, 9	mulneg1d 10483	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
11	10	oveq2d 6666	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
12		simp1l 1085	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
13		simp2r 1088	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
14	13	nnnn0d 11351	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
15		expmul 12905	. . . . . . . . . . . 12 $/\$ $/\$
16	12, 14, 8, 15	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
17	11, 16	eqtr3d 2658	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	17	oveq2d 6666	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
19		expcl 12878	. . . . . . . . . . 11 $/\$
20	12, 14, 19	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		simp1r 1086	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
22	13	nnzd 11481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
23		expne0i 12892	. . . . . . . . . . 11 $/\$ $/\$
24	12, 21, 22, 23	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
25	8	nn0zd 11480	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
26		exprec 12901	. . . . . . . . . 10 $/\$ $/\$
27	20, 24, 25, 26	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28	18, 27	eqtr4d 2659	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
29	7, 9	mulcld 10060	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
30	14, 8	nn0mulcld 11356	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
31	10, 30	eqeltrrd 2702	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
32		expneg2 12869	. . . . . . . . 9 $/\$ $/\$
33	12, 29, 31, 32	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
34		expneg2 12869	. . . . . . . . . 10 $/\$ $/\$
35	12, 7, 14, 34	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
36	35	oveq1d 6665	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
37	28, 33, 36	3eqtr4d 2666	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	3expia 1267	. . . . . 6 $/\$ $/\$ $/\$
39	5, 38	jaodan 826	. . . . 5 $/\$ $/\$ $\/$ $/\$
40		simp2 1062	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
41	40	nn0cnd 11353	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
42		simp3l 1089	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
43	42	recnd 10068	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
44	41, 43	mulneg2d 10484	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
45	44	oveq2d 6666	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
46		simp1l 1085	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
47		simp3r 1090	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
48	47	nnnn0d 11351	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
49		expmul 12905	. . . . . . . . . . 11 $/\$ $/\$
50	46, 40, 48, 49	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
51	45, 50	eqtr3d 2658	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
52	51	oveq2d 6666	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
53	41, 43	mulcld 10060	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
54	40, 48	nn0mulcld 11356	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
55	44, 54	eqeltrrd 2702	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
56	46, 53, 55, 32	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
57		expcl 12878	. . . . . . . . . 10 $/\$
58	46, 40, 57	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
59		expneg2 12869	. . . . . . . . 9 $/\$ $/\$
60	58, 43, 48, 59	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
61	52, 56, 60	3eqtr4d 2666	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
62	61	3expia 1267	. . . . . 6 $/\$ $/\$ $/\$
63		simp1l 1085	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
64		simp2l 1087	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
65	64	recnd 10068	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
66		simp2r 1088	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
67	66	nnnn0d 11351	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
68	63, 65, 67, 34	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
69	68	oveq1d 6665	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
70	63, 67, 19	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
71		simp1r 1086	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
72	66	nnzd 11481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
73	63, 71, 72, 23	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
74	70, 73	reccld 10794	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
75		simp3l 1089	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
76	75	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
77		simp3r 1090	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
78	77	nnnn0d 11351	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
79		expneg2 12869	. . . . . . . . 9 $/\$ $/\$
80	74, 76, 78, 79	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
81	77	nnzd 11481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
82		exprec 12901	. . . . . . . . . . 11 $/\$ $/\$
83	70, 73, 81, 82	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
84	83	oveq2d 6666	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
85		expcl 12878	. . . . . . . . . . 11 $/\$
86	70, 78, 85	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
87		expne0i 12892	. . . . . . . . . . 11 $/\$ $/\$
88	70, 73, 81, 87	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
89	86, 88	recrecd 10798	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
90		expmul 12905	. . . . . . . . . . 11 $/\$ $/\$
91	63, 67, 78, 90	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
92	65, 76	mul2negd 10485	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
93	92	oveq2d 6666	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
94	91, 93	eqtr3d 2658	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
95	84, 89, 94	3eqtrd 2660	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
96	69, 80, 95	3eqtrrd 2661	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
97	96	3expia 1267	. . . . . 6 $/\$ $/\$ $/\$ $/\$
98	62, 97	jaodan 826	. . . . 5 $/\$ $/\$ $\/$ $/\$ $/\$
99	39, 98	jaod 395	. . . 4 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
100	2, 99	sylan2b 492	. . 3 $/\$ $/\$ $\/$ $/\$
101	1, 100	syl5bi 232	. 2 $/\$ $/\$
102	101	impr 649	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

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

wceq 1483

wcel 1990

wne 2794 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

cmul 9941

cneg 10267

cdiv 10684

cn 11020

cn0 11292

cz 11377

cexp 12860

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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 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-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-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-exp 12861

This theorem is referenced by: iexpcyc 12969 iseraltlem2 14413 iseraltlem3 14414 dvexp3 23741 cxpeq 24498 atantayl2 24665 basellem3 24809 lgseisenlem1 25100 lgseisenlem4 25103 lgsquadlem1 25105 lgsquad2lem1 25109 m1lgs 25113 jm2.21 37561 fmtnorec1 41449 m1expevenALTV 41560 oexpnegnz 41589

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