expmulzap - Intuitionistic Logic Explorer

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Theorem expmulzap 10501

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 9205	. . 3 $\/$ $/\$
2		elznn0nn 9205	. . . 4 $\/$ $/\$
3		expmul 10500	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1195	. . . . . . 7 $/\$
5	4	adantlr 469	. . . . . 6 $/\$ # $/\$
6		simp2l 1013	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
7	6	recnd 7927	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
8		simp3 989	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
9	8	nn0cnd 9169	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
10	7, 9	mulneg1d 8309	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
11	10	oveq2d 5858	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
12		simp1l 1011	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
13		simp2r 1014	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
14	13	nnnn0d 9167	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
15		expmul 10500	. . . . . . . . . . . 12 $/\$ $/\$
16	12, 14, 8, 15	syl3anc 1228	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17	11, 16	eqtr3d 2200	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
18	17	oveq2d 5858	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
19		expcl 10473	. . . . . . . . . . 11 $/\$
20	12, 14, 19	syl2anc 409	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21		simp1r 1012	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ #
22	13	nnzd 9312	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
23		expap0i 10487	. . . . . . . . . . 11 $/\$ # $/\$ #
24	12, 21, 22, 23	syl3anc 1228	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
25	8	nn0zd 9311	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
26		exprecap 10496	. . . . . . . . . 10 $/\$ # $/\$
27	20, 24, 25, 26	syl3anc 1228	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
28	18, 27	eqtr4d 2201	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
29	7, 9	mulcld 7919	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
30	14, 8	nn0mulcld 9172	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
31	10, 30	eqeltrrd 2244	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
32		expineg2 10464	. . . . . . . . 9 $/\$ # $/\$ $/\$
33	12, 21, 29, 31, 32	syl22anc 1229	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
34		expineg2 10464	. . . . . . . . . 10 $/\$ # $/\$ $/\$
35	12, 21, 7, 14, 34	syl22anc 1229	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
36	35	oveq1d 5857	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
37	28, 33, 36	3eqtr4d 2208	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
38	37	3expia 1195	. . . . . 6 $/\$ # $/\$ $/\$
39	5, 38	jaodan 787	. . . . 5 $/\$ # $/\$ $\/$ $/\$
40		simp2 988	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
41	40	nn0cnd 9169	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
42		simp3l 1015	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
43	42	recnd 7927	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
44	41, 43	mulneg2d 8310	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
45	44	oveq2d 5858	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
46		simp1l 1011	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
47		simp3r 1016	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
48	47	nnnn0d 9167	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
49		expmul 10500	. . . . . . . . . . 11 $/\$ $/\$
50	46, 40, 48, 49	syl3anc 1228	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
51	45, 50	eqtr3d 2200	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
52	51	oveq2d 5858	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
53		simp1r 1012	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
54	41, 43	mulcld 7919	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
55	40, 48	nn0mulcld 9172	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
56	44, 55	eqeltrrd 2244	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
57	46, 53, 54, 56, 32	syl22anc 1229	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
58		expcl 10473	. . . . . . . . . 10 $/\$
59	46, 40, 58	syl2anc 409	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
60	40	nn0zd 9311	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
61		expap0i 10487	. . . . . . . . . 10 $/\$ # $/\$ #
62	46, 53, 60, 61	syl3anc 1228	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
63		expineg2 10464	. . . . . . . . 9 $/\$ # $/\$ $/\$
64	59, 62, 43, 48, 63	syl22anc 1229	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
65	52, 57, 64	3eqtr4d 2208	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
66	65	3expia 1195	. . . . . 6 $/\$ # $/\$ $/\$
67		simp1l 1011	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
68		simp1r 1012	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
69		simp2l 1013	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
70	69	recnd 7927	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
71		simp2r 1014	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
72	71	nnnn0d 9167	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
73	67, 68, 70, 72, 34	syl22anc 1229	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
74	73	oveq1d 5857	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
75	67, 72, 19	syl2anc 409	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
76	71	nnzd 9312	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
77	67, 68, 76, 23	syl3anc 1228	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
78	75, 77	recclapd 8677	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
79	75, 77	recap0d 8678	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$ #
80		simp3l 1015	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
81	80	recnd 7927	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
82		simp3r 1016	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
83	82	nnnn0d 9167	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
84		expineg2 10464	. . . . . . . . 9 $/\$ # $/\$ $/\$
85	78, 79, 81, 83, 84	syl22anc 1229	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
86	82	nnzd 9312	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
87		exprecap 10496	. . . . . . . . . . 11 $/\$ # $/\$
88	75, 77, 86, 87	syl3anc 1228	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
89	88	oveq2d 5858	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
90		expcl 10473	. . . . . . . . . . 11 $/\$
91	75, 83, 90	syl2anc 409	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
92		expap0i 10487	. . . . . . . . . . 11 $/\$ # $/\$ #
93	75, 77, 86, 92	syl3anc 1228	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
94	91, 93	recrecapd 8681	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
95		expmul 10500	. . . . . . . . . . 11 $/\$ $/\$
96	67, 72, 83, 95	syl3anc 1228	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
97	70, 81	mul2negd 8311	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
98	97	oveq2d 5858	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
99	96, 98	eqtr3d 2200	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
100	89, 94, 99	3eqtrd 2202	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
101	74, 85, 100	3eqtrrd 2203	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
102	101	3expia 1195	. . . . . 6 $/\$ # $/\$ $/\$ $/\$
103	66, 102	jaodan 787	. . . . 5 $/\$ # $/\$ $\/$ $/\$ $/\$
104	39, 103	jaod 707	. . . 4 $/\$ # $/\$ $\/$ $/\$ $\/$ $/\$
105	2, 104	sylan2b 285	. . 3 $/\$ # $/\$ $\/$ $/\$
106	1, 105	syl5bi 151	. 2 $/\$ # $/\$
107	106	impr 377	1 $/\$ # $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $\/$ wo 698 $/\$ w3a 968

wceq 1343

wcel 2136 class class class wbr 3982 (class class class)co 5842

cc 7751

cr 7752

cc0 7753

c1 7754

cmul 7758

cneg 8070 # cap 8479

cdiv 8568

cn 8857

cn0 9114

cz 9191

cexp 10454

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-n0 9115 df-z 9192 df-uz 9467 df-seqfrec 10381 df-exp 10455

This theorem is referenced by: iexpcyc 10559

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