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Theorem expaddzap 10998

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 9637	. . 3 $\/$ $/\$
2		elznn0nn 9637	. . . 4 $\/$ $/\$
3		expadd 10996	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1236	. . . . . . 7 $/\$
5	4	adantlr 481	. . . . . 6 $/\$ # $/\$
6		expaddzaplem 10997	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
7	6	3expia 1236	. . . . . 6 $/\$ # $/\$ $/\$
8	5, 7	jaodan 809	. . . . 5 $/\$ # $/\$ $\/$ $/\$
9		expaddzaplem 10997	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
10		simp3 1030	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
11	10	nn0cnd 9601	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
12		simp2l 1054	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
13	12	recnd 8344	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14	11, 13	addcomd 8467	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
15	14	oveq2d 6091	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
16		simp1l 1052	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		expcl 10972	. . . . . . . . . . 11 $/\$
18	16, 10, 17	syl2anc 415	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
19		simp1r 1053	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ #
20	13	negnegd 8618	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
21		simp2r 1055	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
22	21	nnnn0d 9599	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
23		nn0negz 9657	. . . . . . . . . . . . 13
24	22, 23	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
25	20, 24	eqeltrrd 2316	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
26		expclzap 10979	. . . . . . . . . . 11 $/\$ # $/\$
27	16, 19, 25, 26	syl3anc 1278	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
28	18, 27	mulcomd 8337	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
29	9, 15, 28	3eqtr4d 2281	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
30	29	3expia 1236	. . . . . . 7 $/\$ # $/\$ $/\$
31	30	impancom 260	. . . . . 6 $/\$ # $/\$ $/\$
32		simp2l 1054	. . . . . . . . . . . . . . 15 $/\$ # $/\$ $/\$ $/\$ $/\$
33	32	recnd 8344	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
34		simp3l 1056	. . . . . . . . . . . . . . 15 $/\$ # $/\$ $/\$ $/\$ $/\$
35	34	recnd 8344	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
36	33, 35	negdid 8640	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
37	36	oveq2d 6091	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$ $/\$
38		simp1l 1052	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
39		simp2r 1055	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
40	39	nnnn0d 9599	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
41		simp3r 1057	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
42	41	nnnn0d 9599	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
43		expadd 10996	. . . . . . . . . . . . 13 $/\$ $/\$
44	38, 40, 42, 43	syl3anc 1278	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$ $/\$
45	37, 44	eqtrd 2271	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	oveq2d 6091	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
47		1t1e1 9436	. . . . . . . . . . 11
48	47	oveq1i 6085	. . . . . . . . . 10
49	46, 48	eqtr4di 2289	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
50		expcl 10972	. . . . . . . . . . 11 $/\$
51	38, 40, 50	syl2anc 415	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
52		simp1r 1053	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$ #
53	40	nn0zd 9745	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
54		expap0i 10986	. . . . . . . . . . 11 $/\$ # $/\$ #
55	38, 52, 53, 54	syl3anc 1278	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
56		expcl 10972	. . . . . . . . . . 11 $/\$
57	38, 42, 56	syl2anc 415	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
58	42	nn0zd 9745	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
59		expap0i 10986	. . . . . . . . . . 11 $/\$ # $/\$ #
60	38, 52, 58, 59	syl3anc 1278	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
61		ax-1cn 8262	. . . . . . . . . . 11
62		divmuldivap 9032	. . . . . . . . . . 11 $/\$ $/\$ $/\$ # $/\$ $/\$ #
63	61, 61, 62	mpanl12 440	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
64	51, 55, 57, 60, 63	syl22anc 1279	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
65	49, 64	eqtr4d 2274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
66	33, 35	addcld 8335	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
67	40, 42	nn0addcld 9603	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
68	36, 67	eqeltrd 2315	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
69		expineg2 10963	. . . . . . . . 9 $/\$ # $/\$ $/\$
70	38, 52, 66, 68, 69	syl22anc 1279	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
71		expineg2 10963	. . . . . . . . . 10 $/\$ # $/\$ $/\$
72	38, 52, 33, 40, 71	syl22anc 1279	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
73		expineg2 10963	. . . . . . . . . 10 $/\$ # $/\$ $/\$
74	38, 52, 35, 42, 73	syl22anc 1279	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
75	72, 74	oveq12d 6093	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
76	65, 70, 75	3eqtr4d 2281	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
77	76	3expia 1236	. . . . . 6 $/\$ # $/\$ $/\$ $/\$
78	31, 77	jaodan 809	. . . . 5 $/\$ # $/\$ $\/$ $/\$ $/\$
79	8, 78	jaod 729	. . . 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 720 $/\$ w3a 1009

wceq 1402

wcel 2209 class class class wbr 4125 (class class class)co 6075

cc 8167

cr 8168

cc0 8169

c1 8170

caddc 8172

cmul 8174

cneg 8488 # cap 8899

cdiv 8992

cn 9283

cn0 9542

cz 9623

cexp 10953

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-frec 6652 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-n0 9543 df-z 9624 df-uz 9901 df-seqfrec 10863 df-exp 10954

This theorem is referenced by: m1expeven 11001 expsubap 11002 expp1zap 11003 pcaddlem 13096 expghmap 14914 lgseisenlem4 16106 lgsquadlem1 16110 lgsquad2lem1 16114

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