expmulzap - Intuitionistic Logic Explorer

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Theorem expmulzap 10694

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 9357	. . 3 $\/$ $/\$
2		elznn0nn 9357	. . . 4 $\/$ $/\$
3		expmul 10693	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1207	. . . . . . 7 $/\$
5	4	adantlr 477	. . . . . 6 $/\$ # $/\$
6		simp2l 1025	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
7	6	recnd 8072	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
8		simp3 1001	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
9	8	nn0cnd 9321	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
10	7, 9	mulneg1d 8454	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
11	10	oveq2d 5941	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
12		simp1l 1023	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
13		simp2r 1026	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
14	13	nnnn0d 9319	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
15		expmul 10693	. . . . . . . . . . . 12 $/\$ $/\$
16	12, 14, 8, 15	syl3anc 1249	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17	11, 16	eqtr3d 2231	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
18	17	oveq2d 5941	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
19		expcl 10666	. . . . . . . . . . 11 $/\$
20	12, 14, 19	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21		simp1r 1024	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ #
22	13	nnzd 9464	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
23		expap0i 10680	. . . . . . . . . . 11 $/\$ # $/\$ #
24	12, 21, 22, 23	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
25	8	nn0zd 9463	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
26		exprecap 10689	. . . . . . . . . 10 $/\$ # $/\$
27	20, 24, 25, 26	syl3anc 1249	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
28	18, 27	eqtr4d 2232	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
29	7, 9	mulcld 8064	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
30	14, 8	nn0mulcld 9324	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
31	10, 30	eqeltrrd 2274	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
32		expineg2 10657	. . . . . . . . 9 $/\$ # $/\$ $/\$
33	12, 21, 29, 31, 32	syl22anc 1250	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
34		expineg2 10657	. . . . . . . . . 10 $/\$ # $/\$ $/\$
35	12, 21, 7, 14, 34	syl22anc 1250	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
36	35	oveq1d 5940	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
37	28, 33, 36	3eqtr4d 2239	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
38	37	3expia 1207	. . . . . 6 $/\$ # $/\$ $/\$
39	5, 38	jaodan 798	. . . . 5 $/\$ # $/\$ $\/$ $/\$
40		simp2 1000	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
41	40	nn0cnd 9321	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
42		simp3l 1027	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
43	42	recnd 8072	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
44	41, 43	mulneg2d 8455	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
45	44	oveq2d 5941	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
46		simp1l 1023	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
47		simp3r 1028	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
48	47	nnnn0d 9319	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
49		expmul 10693	. . . . . . . . . . 11 $/\$ $/\$
50	46, 40, 48, 49	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
51	45, 50	eqtr3d 2231	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
52	51	oveq2d 5941	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
53		simp1r 1024	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
54	41, 43	mulcld 8064	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
55	40, 48	nn0mulcld 9324	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
56	44, 55	eqeltrrd 2274	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
57	46, 53, 54, 56, 32	syl22anc 1250	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
58		expcl 10666	. . . . . . . . . 10 $/\$
59	46, 40, 58	syl2anc 411	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
60	40	nn0zd 9463	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
61		expap0i 10680	. . . . . . . . . 10 $/\$ # $/\$ #
62	46, 53, 60, 61	syl3anc 1249	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
63		expineg2 10657	. . . . . . . . 9 $/\$ # $/\$ $/\$
64	59, 62, 43, 48, 63	syl22anc 1250	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
65	52, 57, 64	3eqtr4d 2239	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
66	65	3expia 1207	. . . . . 6 $/\$ # $/\$ $/\$
67		simp1l 1023	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
68		simp1r 1024	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
69		simp2l 1025	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
70	69	recnd 8072	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
71		simp2r 1026	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
72	71	nnnn0d 9319	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
73	67, 68, 70, 72, 34	syl22anc 1250	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
74	73	oveq1d 5940	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
75	67, 72, 19	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
76	71	nnzd 9464	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
77	67, 68, 76, 23	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
78	75, 77	recclapd 8825	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
79	75, 77	recap0d 8826	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$ #
80		simp3l 1027	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
81	80	recnd 8072	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
82		simp3r 1028	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
83	82	nnnn0d 9319	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
84		expineg2 10657	. . . . . . . . 9 $/\$ # $/\$ $/\$
85	78, 79, 81, 83, 84	syl22anc 1250	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
86	82	nnzd 9464	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
87		exprecap 10689	. . . . . . . . . . 11 $/\$ # $/\$
88	75, 77, 86, 87	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
89	88	oveq2d 5941	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
90		expcl 10666	. . . . . . . . . . 11 $/\$
91	75, 83, 90	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
92		expap0i 10680	. . . . . . . . . . 11 $/\$ # $/\$ #
93	75, 77, 86, 92	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
94	91, 93	recrecapd 8829	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
95		expmul 10693	. . . . . . . . . . 11 $/\$ $/\$
96	67, 72, 83, 95	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
97	70, 81	mul2negd 8456	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
98	97	oveq2d 5941	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
99	96, 98	eqtr3d 2231	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
100	89, 94, 99	3eqtrd 2233	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
101	74, 85, 100	3eqtrrd 2234	. . . . . . 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 2167 class class class wbr 4034 (class class class)co 5925

cc 7894

cr 7895

cc0 7896

c1 7897

cmul 7901

cneg 8215 # cap 8625

cdiv 8716

cn 9007

cn0 9266

cz 9343

cexp 10647

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-n0 9267 df-z 9344 df-uz 9619 df-seqfrec 10557 df-exp 10648

This theorem is referenced by: iexpcyc 10753 lgseisenlem1 15395 lgseisenlem4 15398 lgsquadlem1 15402 lgsquad2lem1 15406 m1lgs 15410

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