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Theorem expaddzap 10908

Description: Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)

**Assertion**
Ref	Expression
expaddzap	$/\$ # $/\$ $/\$

**Proof of Theorem expaddzap**
Step	Hyp	Ref	Expression
1		elznn0nn 9554	. . 3 $\/$ $/\$
2		elznn0nn 9554	. . . 4 $\/$ $/\$
3		expadd 10906	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1232	. . . . . . 7 $/\$
5	4	adantlr 477	. . . . . 6 $/\$ # $/\$
6		expaddzaplem 10907	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
7	6	3expia 1232	. . . . . 6 $/\$ # $/\$ $/\$
8	5, 7	jaodan 805	. . . . 5 $/\$ # $/\$ $\/$ $/\$
9		expaddzaplem 10907	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
10		simp3 1026	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
11	10	nn0cnd 9518	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
12		simp2l 1050	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
13	12	recnd 8267	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14	11, 13	addcomd 8389	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
15	14	oveq2d 6044	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
16		simp1l 1048	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		expcl 10882	. . . . . . . . . . 11 $/\$
18	16, 10, 17	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
19		simp1r 1049	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ #
20	13	negnegd 8540	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
21		simp2r 1051	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
22	21	nnnn0d 9516	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
23		nn0negz 9574	. . . . . . . . . . . . 13
24	22, 23	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
25	20, 24	eqeltrrd 2309	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
26		expclzap 10889	. . . . . . . . . . 11 $/\$ # $/\$
27	16, 19, 25, 26	syl3anc 1274	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
28	18, 27	mulcomd 8260	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
29	9, 15, 28	3eqtr4d 2274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
30	29	3expia 1232	. . . . . . 7 $/\$ # $/\$ $/\$
31	30	impancom 260	. . . . . 6 $/\$ # $/\$ $/\$
32		simp2l 1050	. . . . . . . . . . . . . . 15 $/\$ # $/\$ $/\$ $/\$ $/\$
33	32	recnd 8267	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
34		simp3l 1052	. . . . . . . . . . . . . . 15 $/\$ # $/\$ $/\$ $/\$ $/\$
35	34	recnd 8267	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
36	33, 35	negdid 8562	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
37	36	oveq2d 6044	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$ $/\$
38		simp1l 1048	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
39		simp2r 1051	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
40	39	nnnn0d 9516	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
41		simp3r 1053	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
42	41	nnnn0d 9516	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
43		expadd 10906	. . . . . . . . . . . . 13 $/\$ $/\$
44	38, 40, 42, 43	syl3anc 1274	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$ $/\$
45	37, 44	eqtrd 2264	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	oveq2d 6044	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
47		1t1e1 9355	. . . . . . . . . . 11
48	47	oveq1i 6038	. . . . . . . . . 10
49	46, 48	eqtr4di 2282	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
50		expcl 10882	. . . . . . . . . . 11 $/\$
51	38, 40, 50	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
52		simp1r 1049	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$ #
53	40	nn0zd 9661	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
54		expap0i 10896	. . . . . . . . . . 11 $/\$ # $/\$ #
55	38, 52, 53, 54	syl3anc 1274	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
56		expcl 10882	. . . . . . . . . . 11 $/\$
57	38, 42, 56	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
58	42	nn0zd 9661	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
59		expap0i 10896	. . . . . . . . . . 11 $/\$ # $/\$ #
60	38, 52, 58, 59	syl3anc 1274	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
61		ax-1cn 8185	. . . . . . . . . . 11
62		divmuldivap 8951	. . . . . . . . . . 11 $/\$ $/\$ $/\$ # $/\$ $/\$ #
63	61, 61, 62	mpanl12 436	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
64	51, 55, 57, 60, 63	syl22anc 1275	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
65	49, 64	eqtr4d 2267	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
66	33, 35	addcld 8258	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
67	40, 42	nn0addcld 9520	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
68	36, 67	eqeltrd 2308	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
69		expineg2 10873	. . . . . . . . 9 $/\$ # $/\$ $/\$
70	38, 52, 66, 68, 69	syl22anc 1275	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
71		expineg2 10873	. . . . . . . . . 10 $/\$ # $/\$ $/\$
72	38, 52, 33, 40, 71	syl22anc 1275	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
73		expineg2 10873	. . . . . . . . . 10 $/\$ # $/\$ $/\$
74	38, 52, 35, 42, 73	syl22anc 1275	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
75	72, 74	oveq12d 6046	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
76	65, 70, 75	3eqtr4d 2274	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
77	76	3expia 1232	. . . . . 6 $/\$ # $/\$ $/\$ $/\$
78	31, 77	jaodan 805	. . . . 5 $/\$ # $/\$ $\/$ $/\$ $/\$
79	8, 78	jaod 725	. . . 4 $/\$ # $/\$ $\/$ $/\$ $\/$ $/\$
80	2, 79	sylan2b 287	. . 3 $/\$ # $/\$ $\/$ $/\$
81	1, 80	biimtrid 152	. 2 $/\$ # $/\$
82	81	impr 379	1 $/\$ # $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 716 $/\$ w3a 1005

wceq 1398

wcel 2202 class class class wbr 4093 (class class class)co 6028

cc 8090

cr 8091

cc0 8092

c1 8093

caddc 8095

cmul 8097

cneg 8410 # cap 8820

cdiv 8911

cn 9202

cn0 9461

cz 9540

cexp 10863

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-precex 8202 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-apti 8207 ax-pre-ltadd 8208 ax-pre-mulgt0 8209 ax-pre-mulext 8210

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-sub 8411 df-neg 8412 df-reap 8814 df-ap 8821 df-div 8912 df-inn 9203 df-n0 9462 df-z 9541 df-uz 9817 df-seqfrec 10773 df-exp 10864

This theorem is referenced by: m1expeven 10911 expsubap 10912 expp1zap 10913 pcaddlem 12992 expghmap 14703 lgseisenlem4 15892 lgsquadlem1 15896 lgsquad2lem1 15900

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