expmulzap - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > expmulzap		Unicode version

Theorem expmulzap 9301

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 8259	. . 3 $\/$ $/\$
2		elznn0nn 8259	. . . 4 $\/$ $/\$
3		expmul 9300	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1106	. . . . . . 7 $/\$
5	4	adantlr 446	. . . . . 6 $/\$ # $/\$
6		simp2l 930	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
7	6	recnd 7054	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
8		simp3 906	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
9	8	nn0cnd 8237	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
10	7, 9	mulneg1d 7408	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
11	10	oveq2d 5528	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
12		simp1l 928	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
13		simp2r 931	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
14	13	nnnn0d 8235	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
15		expmul 9300	. . . . . . . . . . . 12 $/\$ $/\$
16	12, 14, 8, 15	syl3anc 1135	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17	11, 16	eqtr3d 2074	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
18	17	oveq2d 5528	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
19		expcl 9273	. . . . . . . . . . 11 $/\$
20	12, 14, 19	syl2anc 391	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21		simp1r 929	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ #
22	13	nnzd 8359	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
23		expap0i 9287	. . . . . . . . . . 11 $/\$ # $/\$ #
24	12, 21, 22, 23	syl3anc 1135	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
25	8	nn0zd 8358	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
26		exprecap 9296	. . . . . . . . . 10 $/\$ # $/\$
27	20, 24, 25, 26	syl3anc 1135	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
28	18, 27	eqtr4d 2075	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
29	7, 9	mulcld 7047	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
30	14, 8	nn0mulcld 8240	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
31	10, 30	eqeltrrd 2115	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
32		expineg2 9264	. . . . . . . . 9 $/\$ # $/\$ $/\$
33	12, 21, 29, 31, 32	syl22anc 1136	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
34		expineg2 9264	. . . . . . . . . 10 $/\$ # $/\$ $/\$
35	12, 21, 7, 14, 34	syl22anc 1136	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
36	35	oveq1d 5527	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
37	28, 33, 36	3eqtr4d 2082	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
38	37	3expia 1106	. . . . . 6 $/\$ # $/\$ $/\$
39	5, 38	jaodan 710	. . . . 5 $/\$ # $/\$ $\/$ $/\$
40		simp2 905	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
41	40	nn0cnd 8237	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
42		simp3l 932	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
43	42	recnd 7054	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
44	41, 43	mulneg2d 7409	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
45	44	oveq2d 5528	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
46		simp1l 928	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
47		simp3r 933	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
48	47	nnnn0d 8235	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
49		expmul 9300	. . . . . . . . . . 11 $/\$ $/\$
50	46, 40, 48, 49	syl3anc 1135	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
51	45, 50	eqtr3d 2074	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
52	51	oveq2d 5528	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
53		simp1r 929	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
54	41, 43	mulcld 7047	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
55	40, 48	nn0mulcld 8240	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
56	44, 55	eqeltrrd 2115	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
57	46, 53, 54, 56, 32	syl22anc 1136	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
58		expcl 9273	. . . . . . . . . 10 $/\$
59	46, 40, 58	syl2anc 391	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
60	40	nn0zd 8358	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
61		expap0i 9287	. . . . . . . . . 10 $/\$ # $/\$ #
62	46, 53, 60, 61	syl3anc 1135	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
63		expineg2 9264	. . . . . . . . 9 $/\$ # $/\$ $/\$
64	59, 62, 43, 48, 63	syl22anc 1136	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
65	52, 57, 64	3eqtr4d 2082	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
66	65	3expia 1106	. . . . . 6 $/\$ # $/\$ $/\$
67		simp1l 928	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
68		simp1r 929	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
69		simp2l 930	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
70	69	recnd 7054	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
71		simp2r 931	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
72	71	nnnn0d 8235	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
73	67, 68, 70, 72, 34	syl22anc 1136	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
74	73	oveq1d 5527	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
75	67, 72, 19	syl2anc 391	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
76	71	nnzd 8359	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
77	67, 68, 76, 23	syl3anc 1135	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
78	75, 77	recclapd 7757	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
79	75, 77	recap0d 7758	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$ #
80		simp3l 932	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
81	80	recnd 7054	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
82		simp3r 933	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
83	82	nnnn0d 8235	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
84		expineg2 9264	. . . . . . . . 9 $/\$ # $/\$ $/\$
85	78, 79, 81, 83, 84	syl22anc 1136	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
86	82	nnzd 8359	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
87		exprecap 9296	. . . . . . . . . . 11 $/\$ # $/\$
88	75, 77, 86, 87	syl3anc 1135	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
89	88	oveq2d 5528	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
90		expcl 9273	. . . . . . . . . . 11 $/\$
91	75, 83, 90	syl2anc 391	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
92		expap0i 9287	. . . . . . . . . . 11 $/\$ # $/\$ #
93	75, 77, 86, 92	syl3anc 1135	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
94	91, 93	recrecapd 7761	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
95		expmul 9300	. . . . . . . . . . 11 $/\$ $/\$
96	67, 72, 83, 95	syl3anc 1135	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
97	70, 81	mul2negd 7410	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
98	97	oveq2d 5528	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
99	96, 98	eqtr3d 2074	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
100	89, 94, 99	3eqtrd 2076	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
101	74, 85, 100	3eqtrrd 2077	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
102	101	3expia 1106	. . . . . 6 $/\$ # $/\$ $/\$ $/\$
103	66, 102	jaodan 710	. . . . 5 $/\$ # $/\$ $\/$ $/\$ $/\$
104	39, 103	jaod 637	. . . 4 $/\$ # $/\$ $\/$ $/\$ $\/$ $/\$
105	2, 104	sylan2b 271	. . 3 $/\$ # $/\$ $\/$ $/\$
106	1, 105	syl5bi 141	. 2 $/\$ # $/\$
107	106	impr 361	1 $/\$ # $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 97 $\/$ wo 629 $/\$ w3a 885

wceq 1243

wcel 1393 class class class wbr 3764 (class class class)co 5512

cc 6887

cr 6888

cc0 6889

c1 6890

cmul 6894

cneg 7183 # cap 7572

cdiv 7651

cn 7914

cn0 8181

cz 8245

cexp 9254

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002

This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-frec 5978 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-n0 8182 df-z 8246 df-uz 8474 df-iseq 9212 df-iexp 9255

This theorem is referenced by: (None)

W3C validator