expmulzap - Intuitionistic Logic Explorer

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Theorem expmulzap 10566

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 9267	. . 3 $\/$ $/\$
2		elznn0nn 9267	. . . 4 $\/$ $/\$
3		expmul 10565	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1205	. . . . . . 7 $/\$
5	4	adantlr 477	. . . . . 6 $/\$ # $/\$
6		simp2l 1023	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
7	6	recnd 7986	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
8		simp3 999	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
9	8	nn0cnd 9231	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
10	7, 9	mulneg1d 8368	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
11	10	oveq2d 5891	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
12		simp1l 1021	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
13		simp2r 1024	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
14	13	nnnn0d 9229	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
15		expmul 10565	. . . . . . . . . . . 12 $/\$ $/\$
16	12, 14, 8, 15	syl3anc 1238	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17	11, 16	eqtr3d 2212	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
18	17	oveq2d 5891	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
19		expcl 10538	. . . . . . . . . . 11 $/\$
20	12, 14, 19	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21		simp1r 1022	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ #
22	13	nnzd 9374	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
23		expap0i 10552	. . . . . . . . . . 11 $/\$ # $/\$ #
24	12, 21, 22, 23	syl3anc 1238	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
25	8	nn0zd 9373	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
26		exprecap 10561	. . . . . . . . . 10 $/\$ # $/\$
27	20, 24, 25, 26	syl3anc 1238	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
28	18, 27	eqtr4d 2213	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
29	7, 9	mulcld 7978	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
30	14, 8	nn0mulcld 9234	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
31	10, 30	eqeltrrd 2255	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
32		expineg2 10529	. . . . . . . . 9 $/\$ # $/\$ $/\$
33	12, 21, 29, 31, 32	syl22anc 1239	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
34		expineg2 10529	. . . . . . . . . 10 $/\$ # $/\$ $/\$
35	12, 21, 7, 14, 34	syl22anc 1239	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
36	35	oveq1d 5890	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
37	28, 33, 36	3eqtr4d 2220	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
38	37	3expia 1205	. . . . . 6 $/\$ # $/\$ $/\$
39	5, 38	jaodan 797	. . . . 5 $/\$ # $/\$ $\/$ $/\$
40		simp2 998	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
41	40	nn0cnd 9231	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
42		simp3l 1025	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
43	42	recnd 7986	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
44	41, 43	mulneg2d 8369	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
45	44	oveq2d 5891	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
46		simp1l 1021	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
47		simp3r 1026	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
48	47	nnnn0d 9229	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
49		expmul 10565	. . . . . . . . . . 11 $/\$ $/\$
50	46, 40, 48, 49	syl3anc 1238	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
51	45, 50	eqtr3d 2212	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
52	51	oveq2d 5891	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
53		simp1r 1022	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
54	41, 43	mulcld 7978	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
55	40, 48	nn0mulcld 9234	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
56	44, 55	eqeltrrd 2255	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
57	46, 53, 54, 56, 32	syl22anc 1239	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
58		expcl 10538	. . . . . . . . . 10 $/\$
59	46, 40, 58	syl2anc 411	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
60	40	nn0zd 9373	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
61		expap0i 10552	. . . . . . . . . 10 $/\$ # $/\$ #
62	46, 53, 60, 61	syl3anc 1238	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
63		expineg2 10529	. . . . . . . . 9 $/\$ # $/\$ $/\$
64	59, 62, 43, 48, 63	syl22anc 1239	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
65	52, 57, 64	3eqtr4d 2220	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
66	65	3expia 1205	. . . . . 6 $/\$ # $/\$ $/\$
67		simp1l 1021	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
68		simp1r 1022	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
69		simp2l 1023	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
70	69	recnd 7986	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
71		simp2r 1024	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
72	71	nnnn0d 9229	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
73	67, 68, 70, 72, 34	syl22anc 1239	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
74	73	oveq1d 5890	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
75	67, 72, 19	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
76	71	nnzd 9374	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
77	67, 68, 76, 23	syl3anc 1238	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
78	75, 77	recclapd 8738	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
79	75, 77	recap0d 8739	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$ #
80		simp3l 1025	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
81	80	recnd 7986	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
82		simp3r 1026	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
83	82	nnnn0d 9229	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
84		expineg2 10529	. . . . . . . . 9 $/\$ # $/\$ $/\$
85	78, 79, 81, 83, 84	syl22anc 1239	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
86	82	nnzd 9374	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
87		exprecap 10561	. . . . . . . . . . 11 $/\$ # $/\$
88	75, 77, 86, 87	syl3anc 1238	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
89	88	oveq2d 5891	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
90		expcl 10538	. . . . . . . . . . 11 $/\$
91	75, 83, 90	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
92		expap0i 10552	. . . . . . . . . . 11 $/\$ # $/\$ #
93	75, 77, 86, 92	syl3anc 1238	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
94	91, 93	recrecapd 8742	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
95		expmul 10565	. . . . . . . . . . 11 $/\$ $/\$
96	67, 72, 83, 95	syl3anc 1238	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
97	70, 81	mul2negd 8370	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
98	97	oveq2d 5891	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
99	96, 98	eqtr3d 2212	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
100	89, 94, 99	3eqtrd 2214	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
101	74, 85, 100	3eqtrrd 2215	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
102	101	3expia 1205	. . . . . 6 $/\$ # $/\$ $/\$ $/\$
103	66, 102	jaodan 797	. . . . 5 $/\$ # $/\$ $\/$ $/\$ $/\$
104	39, 103	jaod 717	. . . 4 $/\$ # $/\$ $\/$ $/\$ $\/$ $/\$
105	2, 104	sylan2b 287	. . 3 $/\$ # $/\$ $\/$ $/\$
106	1, 105	biimtrid 152	. 2 $/\$ # $/\$
107	106	impr 379	1 $/\$ # $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 708 $/\$ w3a 978

wceq 1353

wcel 2148 class class class wbr 4004 (class class class)co 5875

cc 7809

cr 7810

cc0 7811

c1 7812

cmul 7816

cneg 8129 # cap 8538

cdiv 8629

cn 8919

cn0 9176

cz 9253

cexp 10519

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-n0 9177 df-z 9254 df-uz 9529 df-seqfrec 10446 df-exp 10520

This theorem is referenced by: iexpcyc 10625 lgseisenlem1 14453 m1lgs 14455

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