expmulzap - Intuitionistic Logic Explorer

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Theorem expmulzap 9522

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 8365	. . 3 $\/$ $/\$
2		elznn0nn 8365	. . . 4 $\/$ $/\$
3		expmul 9521	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1140	. . . . . . 7 $/\$
5	4	adantlr 460	. . . . . 6 $/\$ # $/\$
6		simp2l 964	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
7	6	recnd 7147	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
8		simp3 940	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
9	8	nn0cnd 8343	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
10	7, 9	mulneg1d 7515	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
11	10	oveq2d 5548	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
12		simp1l 962	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
13		simp2r 965	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
14	13	nnnn0d 8341	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
15		expmul 9521	. . . . . . . . . . . 12 $/\$ $/\$
16	12, 14, 8, 15	syl3anc 1169	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17	11, 16	eqtr3d 2115	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
18	17	oveq2d 5548	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
19		expcl 9494	. . . . . . . . . . 11 $/\$
20	12, 14, 19	syl2anc 403	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21		simp1r 963	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ #
22	13	nnzd 8468	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
23		expap0i 9508	. . . . . . . . . . 11 $/\$ # $/\$ #
24	12, 21, 22, 23	syl3anc 1169	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
25	8	nn0zd 8467	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
26		exprecap 9517	. . . . . . . . . 10 $/\$ # $/\$
27	20, 24, 25, 26	syl3anc 1169	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
28	18, 27	eqtr4d 2116	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
29	7, 9	mulcld 7139	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
30	14, 8	nn0mulcld 8346	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
31	10, 30	eqeltrrd 2156	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
32		expineg2 9485	. . . . . . . . 9 $/\$ # $/\$ $/\$
33	12, 21, 29, 31, 32	syl22anc 1170	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
34		expineg2 9485	. . . . . . . . . 10 $/\$ # $/\$ $/\$
35	12, 21, 7, 14, 34	syl22anc 1170	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
36	35	oveq1d 5547	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
37	28, 33, 36	3eqtr4d 2123	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
38	37	3expia 1140	. . . . . 6 $/\$ # $/\$ $/\$
39	5, 38	jaodan 743	. . . . 5 $/\$ # $/\$ $\/$ $/\$
40		simp2 939	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
41	40	nn0cnd 8343	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
42		simp3l 966	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
43	42	recnd 7147	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
44	41, 43	mulneg2d 7516	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
45	44	oveq2d 5548	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
46		simp1l 962	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
47		simp3r 967	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
48	47	nnnn0d 8341	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
49		expmul 9521	. . . . . . . . . . 11 $/\$ $/\$
50	46, 40, 48, 49	syl3anc 1169	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
51	45, 50	eqtr3d 2115	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
52	51	oveq2d 5548	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
53		simp1r 963	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
54	41, 43	mulcld 7139	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
55	40, 48	nn0mulcld 8346	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
56	44, 55	eqeltrrd 2156	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
57	46, 53, 54, 56, 32	syl22anc 1170	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
58		expcl 9494	. . . . . . . . . 10 $/\$
59	46, 40, 58	syl2anc 403	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
60	40	nn0zd 8467	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
61		expap0i 9508	. . . . . . . . . 10 $/\$ # $/\$ #
62	46, 53, 60, 61	syl3anc 1169	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
63		expineg2 9485	. . . . . . . . 9 $/\$ # $/\$ $/\$
64	59, 62, 43, 48, 63	syl22anc 1170	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
65	52, 57, 64	3eqtr4d 2123	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
66	65	3expia 1140	. . . . . 6 $/\$ # $/\$ $/\$
67		simp1l 962	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
68		simp1r 963	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
69		simp2l 964	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
70	69	recnd 7147	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
71		simp2r 965	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
72	71	nnnn0d 8341	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
73	67, 68, 70, 72, 34	syl22anc 1170	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
74	73	oveq1d 5547	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
75	67, 72, 19	syl2anc 403	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
76	71	nnzd 8468	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
77	67, 68, 76, 23	syl3anc 1169	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
78	75, 77	recclapd 7869	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
79	75, 77	recap0d 7870	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$ #
80		simp3l 966	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
81	80	recnd 7147	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
82		simp3r 967	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
83	82	nnnn0d 8341	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
84		expineg2 9485	. . . . . . . . 9 $/\$ # $/\$ $/\$
85	78, 79, 81, 83, 84	syl22anc 1170	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
86	82	nnzd 8468	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
87		exprecap 9517	. . . . . . . . . . 11 $/\$ # $/\$
88	75, 77, 86, 87	syl3anc 1169	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
89	88	oveq2d 5548	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
90		expcl 9494	. . . . . . . . . . 11 $/\$
91	75, 83, 90	syl2anc 403	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
92		expap0i 9508	. . . . . . . . . . 11 $/\$ # $/\$ #
93	75, 77, 86, 92	syl3anc 1169	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
94	91, 93	recrecapd 7873	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
95		expmul 9521	. . . . . . . . . . 11 $/\$ $/\$
96	67, 72, 83, 95	syl3anc 1169	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
97	70, 81	mul2negd 7517	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
98	97	oveq2d 5548	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
99	96, 98	eqtr3d 2115	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
100	89, 94, 99	3eqtrd 2117	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
101	74, 85, 100	3eqtrrd 2118	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
102	101	3expia 1140	. . . . . 6 $/\$ # $/\$ $/\$ $/\$
103	66, 102	jaodan 743	. . . . 5 $/\$ # $/\$ $\/$ $/\$ $/\$
104	39, 103	jaod 669	. . . 4 $/\$ # $/\$ $\/$ $/\$ $\/$ $/\$
105	2, 104	sylan2b 281	. . 3 $/\$ # $/\$ $\/$ $/\$
106	1, 105	syl5bi 150	. 2 $/\$ # $/\$
107	106	impr 371	1 $/\$ # $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102 $\/$ wo 661 $/\$ w3a 919

wceq 1284

wcel 1433 class class class wbr 3785 (class class class)co 5532

cc 6979

cr 6980

cc0 6981

c1 6982

cmul 6986

cneg 7280 # cap 7681

cdiv 7760

cn 8039

cn0 8288

cz 8351

cexp 9475

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-iseq 9432 df-iexp 9476

This theorem is referenced by: iexpcyc 9579

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