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Theorem expaddzap 10945

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 9591	. . 3 $\/$ $/\$
2		elznn0nn 9591	. . . 4 $\/$ $/\$
3		expadd 10943	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1232	. . . . . . 7 $/\$
5	4	adantlr 477	. . . . . 6 $/\$ # $/\$
6		expaddzaplem 10944	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
7	6	3expia 1232	. . . . . 6 $/\$ # $/\$ $/\$
8	5, 7	jaodan 805	. . . . 5 $/\$ # $/\$ $\/$ $/\$
9		expaddzaplem 10944	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
10		simp3 1026	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
11	10	nn0cnd 9555	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
12		simp2l 1050	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
13	12	recnd 8302	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14	11, 13	addcomd 8424	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
15	14	oveq2d 6066	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
16		simp1l 1048	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		expcl 10919	. . . . . . . . . . 11 $/\$
18	16, 10, 17	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
19		simp1r 1049	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ #
20	13	negnegd 8575	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
21		simp2r 1051	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
22	21	nnnn0d 9553	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
23		nn0negz 9611	. . . . . . . . . . . . 13
24	22, 23	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
25	20, 24	eqeltrrd 2310	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
26		expclzap 10926	. . . . . . . . . . 11 $/\$ # $/\$
27	16, 19, 25, 26	syl3anc 1274	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
28	18, 27	mulcomd 8295	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
29	9, 15, 28	3eqtr4d 2275	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
30	29	3expia 1232	. . . . . . 7 $/\$ # $/\$ $/\$
31	30	impancom 260	. . . . . 6 $/\$ # $/\$ $/\$
32		simp2l 1050	. . . . . . . . . . . . . . 15 $/\$ # $/\$ $/\$ $/\$ $/\$
33	32	recnd 8302	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
34		simp3l 1052	. . . . . . . . . . . . . . 15 $/\$ # $/\$ $/\$ $/\$ $/\$
35	34	recnd 8302	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
36	33, 35	negdid 8597	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
37	36	oveq2d 6066	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$ $/\$
38		simp1l 1048	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
39		simp2r 1051	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
40	39	nnnn0d 9553	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
41		simp3r 1053	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
42	41	nnnn0d 9553	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
43		expadd 10943	. . . . . . . . . . . . 13 $/\$ $/\$
44	38, 40, 42, 43	syl3anc 1274	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$ $/\$
45	37, 44	eqtrd 2265	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	oveq2d 6066	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
47		1t1e1 9390	. . . . . . . . . . 11
48	47	oveq1i 6060	. . . . . . . . . 10
49	46, 48	eqtr4di 2283	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
50		expcl 10919	. . . . . . . . . . 11 $/\$
51	38, 40, 50	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
52		simp1r 1049	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$ #
53	40	nn0zd 9698	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
54		expap0i 10933	. . . . . . . . . . 11 $/\$ # $/\$ #
55	38, 52, 53, 54	syl3anc 1274	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
56		expcl 10919	. . . . . . . . . . 11 $/\$
57	38, 42, 56	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
58	42	nn0zd 9698	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
59		expap0i 10933	. . . . . . . . . . 11 $/\$ # $/\$ #
60	38, 52, 58, 59	syl3anc 1274	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
61		ax-1cn 8220	. . . . . . . . . . 11
62		divmuldivap 8986	. . . . . . . . . . 11 $/\$ $/\$ $/\$ # $/\$ $/\$ #
63	61, 61, 62	mpanl12 436	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
64	51, 55, 57, 60, 63	syl22anc 1275	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
65	49, 64	eqtr4d 2268	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
66	33, 35	addcld 8293	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
67	40, 42	nn0addcld 9557	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
68	36, 67	eqeltrd 2309	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
69		expineg2 10910	. . . . . . . . 9 $/\$ # $/\$ $/\$
70	38, 52, 66, 68, 69	syl22anc 1275	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
71		expineg2 10910	. . . . . . . . . 10 $/\$ # $/\$ $/\$
72	38, 52, 33, 40, 71	syl22anc 1275	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
73		expineg2 10910	. . . . . . . . . 10 $/\$ # $/\$ $/\$
74	38, 52, 35, 42, 73	syl22anc 1275	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
75	72, 74	oveq12d 6068	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
76	65, 70, 75	3eqtr4d 2275	. . . . . . 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 2203 class class class wbr 4109 (class class class)co 6050

cc 8125

cr 8126

cc0 8127

c1 8128

caddc 8130

cmul 8132

cneg 8445 # cap 8855

cdiv 8946

cn 9237

cn0 9496

cz 9577

cexp 10900

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 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245

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 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-frec 6622 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-n0 9497 df-z 9578 df-uz 9854 df-seqfrec 10810 df-exp 10901

This theorem is referenced by: m1expeven 10948 expsubap 10949 expp1zap 10950 pcaddlem 13037 expghmap 14755 lgseisenlem4 15946 lgsquadlem1 15950 lgsquad2lem1 15954

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