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Theorem expmulzap 10339

Description: Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)

**Assertion**
Ref	Expression
expmulzap	$/\$ # $/\$ $/\$

**Proof of Theorem expmulzap**
Step	Hyp	Ref	Expression
1		elznn0nn 9068	. . 3 $\/$ $/\$
2		elznn0nn 9068	. . . 4 $\/$ $/\$
3		expmul 10338	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1183	. . . . . . 7 $/\$
5	4	adantlr 468	. . . . . 6 $/\$ # $/\$
6		simp2l 1007	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
7	6	recnd 7794	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
8		simp3 983	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
9	8	nn0cnd 9032	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
10	7, 9	mulneg1d 8173	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
11	10	oveq2d 5790	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
12		simp1l 1005	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
13		simp2r 1008	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
14	13	nnnn0d 9030	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
15		expmul 10338	. . . . . . . . . . . 12 $/\$ $/\$
16	12, 14, 8, 15	syl3anc 1216	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17	11, 16	eqtr3d 2174	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
18	17	oveq2d 5790	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
19		expcl 10311	. . . . . . . . . . 11 $/\$
20	12, 14, 19	syl2anc 408	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21		simp1r 1006	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ #
22	13	nnzd 9172	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
23		expap0i 10325	. . . . . . . . . . 11 $/\$ # $/\$ #
24	12, 21, 22, 23	syl3anc 1216	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
25	8	nn0zd 9171	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
26		exprecap 10334	. . . . . . . . . 10 $/\$ # $/\$
27	20, 24, 25, 26	syl3anc 1216	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
28	18, 27	eqtr4d 2175	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
29	7, 9	mulcld 7786	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
30	14, 8	nn0mulcld 9035	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
31	10, 30	eqeltrrd 2217	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
32		expineg2 10302	. . . . . . . . 9 $/\$ # $/\$ $/\$
33	12, 21, 29, 31, 32	syl22anc 1217	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
34		expineg2 10302	. . . . . . . . . 10 $/\$ # $/\$ $/\$
35	12, 21, 7, 14, 34	syl22anc 1217	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
36	35	oveq1d 5789	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
37	28, 33, 36	3eqtr4d 2182	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
38	37	3expia 1183	. . . . . 6 $/\$ # $/\$ $/\$
39	5, 38	jaodan 786	. . . . 5 $/\$ # $/\$ $\/$ $/\$
40		simp2 982	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
41	40	nn0cnd 9032	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
42		simp3l 1009	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
43	42	recnd 7794	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
44	41, 43	mulneg2d 8174	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
45	44	oveq2d 5790	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
46		simp1l 1005	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
47		simp3r 1010	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
48	47	nnnn0d 9030	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
49		expmul 10338	. . . . . . . . . . 11 $/\$ $/\$
50	46, 40, 48, 49	syl3anc 1216	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
51	45, 50	eqtr3d 2174	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
52	51	oveq2d 5790	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
53		simp1r 1006	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
54	41, 43	mulcld 7786	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
55	40, 48	nn0mulcld 9035	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
56	44, 55	eqeltrrd 2217	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
57	46, 53, 54, 56, 32	syl22anc 1217	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
58		expcl 10311	. . . . . . . . . 10 $/\$
59	46, 40, 58	syl2anc 408	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
60	40	nn0zd 9171	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
61		expap0i 10325	. . . . . . . . . 10 $/\$ # $/\$ #
62	46, 53, 60, 61	syl3anc 1216	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
63		expineg2 10302	. . . . . . . . 9 $/\$ # $/\$ $/\$
64	59, 62, 43, 48, 63	syl22anc 1217	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
65	52, 57, 64	3eqtr4d 2182	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
66	65	3expia 1183	. . . . . 6 $/\$ # $/\$ $/\$
67		simp1l 1005	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
68		simp1r 1006	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
69		simp2l 1007	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
70	69	recnd 7794	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
71		simp2r 1008	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
72	71	nnnn0d 9030	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
73	67, 68, 70, 72, 34	syl22anc 1217	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
74	73	oveq1d 5789	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
75	67, 72, 19	syl2anc 408	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
76	71	nnzd 9172	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
77	67, 68, 76, 23	syl3anc 1216	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
78	75, 77	recclapd 8541	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
79	75, 77	recap0d 8542	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$ #
80		simp3l 1009	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
81	80	recnd 7794	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
82		simp3r 1010	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
83	82	nnnn0d 9030	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
84		expineg2 10302	. . . . . . . . 9 $/\$ # $/\$ $/\$
85	78, 79, 81, 83, 84	syl22anc 1217	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
86	82	nnzd 9172	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
87		exprecap 10334	. . . . . . . . . . 11 $/\$ # $/\$
88	75, 77, 86, 87	syl3anc 1216	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
89	88	oveq2d 5790	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
90		expcl 10311	. . . . . . . . . . 11 $/\$
91	75, 83, 90	syl2anc 408	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
92		expap0i 10325	. . . . . . . . . . 11 $/\$ # $/\$ #
93	75, 77, 86, 92	syl3anc 1216	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
94	91, 93	recrecapd 8545	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
95		expmul 10338	. . . . . . . . . . 11 $/\$ $/\$
96	67, 72, 83, 95	syl3anc 1216	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
97	70, 81	mul2negd 8175	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
98	97	oveq2d 5790	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
99	96, 98	eqtr3d 2174	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
100	89, 94, 99	3eqtrd 2176	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
101	74, 85, 100	3eqtrrd 2177	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
102	101	3expia 1183	. . . . . 6 $/\$ # $/\$ $/\$ $/\$
103	66, 102	jaodan 786	. . . . 5 $/\$ # $/\$ $\/$ $/\$ $/\$
104	39, 103	jaod 706	. . . 4 $/\$ # $/\$ $\/$ $/\$ $\/$ $/\$
105	2, 104	sylan2b 285	. . 3 $/\$ # $/\$ $\/$ $/\$
106	1, 105	syl5bi 151	. 2 $/\$ # $/\$
107	106	impr 376	1 $/\$ # $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $\/$ wo 697 $/\$ w3a 962

wceq 1331

wcel 1480 class class class wbr 3929 (class class class)co 5774

cc 7618

cr 7619

cc0 7620

c1 7621

cmul 7625

cneg 7934 # cap 8343

cdiv 8432

cn 8720

cn0 8977

cz 9054

cexp 10292

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-seqfrec 10219 df-exp 10293

This theorem is referenced by: iexpcyc 10397

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