expmulzap - Intuitionistic Logic Explorer

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Theorem expmulzap 10656

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 9331	. . 3 $\/$ $/\$
2		elznn0nn 9331	. . . 4 $\/$ $/\$
3		expmul 10655	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1207	. . . . . . 7 $/\$
5	4	adantlr 477	. . . . . 6 $/\$ # $/\$
6		simp2l 1025	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
7	6	recnd 8048	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
8		simp3 1001	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
9	8	nn0cnd 9295	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
10	7, 9	mulneg1d 8430	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
11	10	oveq2d 5934	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
12		simp1l 1023	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
13		simp2r 1026	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
14	13	nnnn0d 9293	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
15		expmul 10655	. . . . . . . . . . . 12 $/\$ $/\$
16	12, 14, 8, 15	syl3anc 1249	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17	11, 16	eqtr3d 2228	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
18	17	oveq2d 5934	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
19		expcl 10628	. . . . . . . . . . 11 $/\$
20	12, 14, 19	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21		simp1r 1024	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ #
22	13	nnzd 9438	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
23		expap0i 10642	. . . . . . . . . . 11 $/\$ # $/\$ #
24	12, 21, 22, 23	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
25	8	nn0zd 9437	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
26		exprecap 10651	. . . . . . . . . 10 $/\$ # $/\$
27	20, 24, 25, 26	syl3anc 1249	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
28	18, 27	eqtr4d 2229	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
29	7, 9	mulcld 8040	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
30	14, 8	nn0mulcld 9298	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
31	10, 30	eqeltrrd 2271	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
32		expineg2 10619	. . . . . . . . 9 $/\$ # $/\$ $/\$
33	12, 21, 29, 31, 32	syl22anc 1250	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
34		expineg2 10619	. . . . . . . . . 10 $/\$ # $/\$ $/\$
35	12, 21, 7, 14, 34	syl22anc 1250	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
36	35	oveq1d 5933	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
37	28, 33, 36	3eqtr4d 2236	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
38	37	3expia 1207	. . . . . 6 $/\$ # $/\$ $/\$
39	5, 38	jaodan 798	. . . . 5 $/\$ # $/\$ $\/$ $/\$
40		simp2 1000	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
41	40	nn0cnd 9295	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
42		simp3l 1027	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
43	42	recnd 8048	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
44	41, 43	mulneg2d 8431	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
45	44	oveq2d 5934	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
46		simp1l 1023	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
47		simp3r 1028	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
48	47	nnnn0d 9293	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
49		expmul 10655	. . . . . . . . . . 11 $/\$ $/\$
50	46, 40, 48, 49	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
51	45, 50	eqtr3d 2228	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
52	51	oveq2d 5934	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
53		simp1r 1024	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
54	41, 43	mulcld 8040	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
55	40, 48	nn0mulcld 9298	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
56	44, 55	eqeltrrd 2271	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
57	46, 53, 54, 56, 32	syl22anc 1250	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
58		expcl 10628	. . . . . . . . . 10 $/\$
59	46, 40, 58	syl2anc 411	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
60	40	nn0zd 9437	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
61		expap0i 10642	. . . . . . . . . 10 $/\$ # $/\$ #
62	46, 53, 60, 61	syl3anc 1249	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
63		expineg2 10619	. . . . . . . . 9 $/\$ # $/\$ $/\$
64	59, 62, 43, 48, 63	syl22anc 1250	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
65	52, 57, 64	3eqtr4d 2236	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
66	65	3expia 1207	. . . . . 6 $/\$ # $/\$ $/\$
67		simp1l 1023	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
68		simp1r 1024	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
69		simp2l 1025	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
70	69	recnd 8048	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
71		simp2r 1026	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
72	71	nnnn0d 9293	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
73	67, 68, 70, 72, 34	syl22anc 1250	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
74	73	oveq1d 5933	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
75	67, 72, 19	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
76	71	nnzd 9438	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
77	67, 68, 76, 23	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
78	75, 77	recclapd 8800	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
79	75, 77	recap0d 8801	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$ #
80		simp3l 1027	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
81	80	recnd 8048	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
82		simp3r 1028	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
83	82	nnnn0d 9293	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
84		expineg2 10619	. . . . . . . . 9 $/\$ # $/\$ $/\$
85	78, 79, 81, 83, 84	syl22anc 1250	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
86	82	nnzd 9438	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
87		exprecap 10651	. . . . . . . . . . 11 $/\$ # $/\$
88	75, 77, 86, 87	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
89	88	oveq2d 5934	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
90		expcl 10628	. . . . . . . . . . 11 $/\$
91	75, 83, 90	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
92		expap0i 10642	. . . . . . . . . . 11 $/\$ # $/\$ #
93	75, 77, 86, 92	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
94	91, 93	recrecapd 8804	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
95		expmul 10655	. . . . . . . . . . 11 $/\$ $/\$
96	67, 72, 83, 95	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
97	70, 81	mul2negd 8432	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
98	97	oveq2d 5934	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
99	96, 98	eqtr3d 2228	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
100	89, 94, 99	3eqtrd 2230	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
101	74, 85, 100	3eqtrrd 2231	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
102	101	3expia 1207	. . . . . 6 $/\$ # $/\$ $/\$ $/\$
103	66, 102	jaodan 798	. . . . 5 $/\$ # $/\$ $\/$ $/\$ $/\$
104	39, 103	jaod 718	. . . 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 709 $/\$ w3a 980

wceq 1364

wcel 2164 class class class wbr 4029 (class class class)co 5918

cc 7870

cr 7871

cc0 7872

c1 7873

cmul 7877

cneg 8191 # cap 8600

cdiv 8691

cn 8982

cn0 9240

cz 9317

cexp 10609

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-n0 9241 df-z 9318 df-uz 9593 df-seqfrec 10519 df-exp 10610

This theorem is referenced by: iexpcyc 10715 lgseisenlem1 15186 lgseisenlem4 15189 lgsquadlem1 15191 m1lgs 15192

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