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Theorem expaddzap 10566

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 9269	. . 3 $\/$ $/\$
2		elznn0nn 9269	. . . 4 $\/$ $/\$
3		expadd 10564	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1205	. . . . . . 7 $/\$
5	4	adantlr 477	. . . . . 6 $/\$ # $/\$
6		expaddzaplem 10565	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
7	6	3expia 1205	. . . . . 6 $/\$ # $/\$ $/\$
8	5, 7	jaodan 797	. . . . 5 $/\$ # $/\$ $\/$ $/\$
9		expaddzaplem 10565	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
10		simp3 999	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
11	10	nn0cnd 9233	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
12		simp2l 1023	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
13	12	recnd 7988	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14	11, 13	addcomd 8110	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
15	14	oveq2d 5893	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
16		simp1l 1021	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		expcl 10540	. . . . . . . . . . 11 $/\$
18	16, 10, 17	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
19		simp1r 1022	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ #
20	13	negnegd 8261	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
21		simp2r 1024	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
22	21	nnnn0d 9231	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
23		nn0negz 9289	. . . . . . . . . . . . 13
24	22, 23	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
25	20, 24	eqeltrrd 2255	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
26		expclzap 10547	. . . . . . . . . . 11 $/\$ # $/\$
27	16, 19, 25, 26	syl3anc 1238	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
28	18, 27	mulcomd 7981	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
29	9, 15, 28	3eqtr4d 2220	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
30	29	3expia 1205	. . . . . . 7 $/\$ # $/\$ $/\$
31	30	impancom 260	. . . . . 6 $/\$ # $/\$ $/\$
32		simp2l 1023	. . . . . . . . . . . . . . 15 $/\$ # $/\$ $/\$ $/\$ $/\$
33	32	recnd 7988	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
34		simp3l 1025	. . . . . . . . . . . . . . 15 $/\$ # $/\$ $/\$ $/\$ $/\$
35	34	recnd 7988	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
36	33, 35	negdid 8283	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
37	36	oveq2d 5893	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$ $/\$
38		simp1l 1021	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
39		simp2r 1024	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
40	39	nnnn0d 9231	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
41		simp3r 1026	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
42	41	nnnn0d 9231	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
43		expadd 10564	. . . . . . . . . . . . 13 $/\$ $/\$
44	38, 40, 42, 43	syl3anc 1238	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$ $/\$
45	37, 44	eqtrd 2210	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	oveq2d 5893	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
47		1t1e1 9073	. . . . . . . . . . 11
48	47	oveq1i 5887	. . . . . . . . . 10
49	46, 48	eqtr4di 2228	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
50		expcl 10540	. . . . . . . . . . 11 $/\$
51	38, 40, 50	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
52		simp1r 1022	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$ #
53	40	nn0zd 9375	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
54		expap0i 10554	. . . . . . . . . . 11 $/\$ # $/\$ #
55	38, 52, 53, 54	syl3anc 1238	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
56		expcl 10540	. . . . . . . . . . 11 $/\$
57	38, 42, 56	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
58	42	nn0zd 9375	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
59		expap0i 10554	. . . . . . . . . . 11 $/\$ # $/\$ #
60	38, 52, 58, 59	syl3anc 1238	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
61		ax-1cn 7906	. . . . . . . . . . 11
62		divmuldivap 8671	. . . . . . . . . . 11 $/\$ $/\$ $/\$ # $/\$ $/\$ #
63	61, 61, 62	mpanl12 436	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
64	51, 55, 57, 60, 63	syl22anc 1239	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
65	49, 64	eqtr4d 2213	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
66	33, 35	addcld 7979	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
67	40, 42	nn0addcld 9235	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
68	36, 67	eqeltrd 2254	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
69		expineg2 10531	. . . . . . . . 9 $/\$ # $/\$ $/\$
70	38, 52, 66, 68, 69	syl22anc 1239	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
71		expineg2 10531	. . . . . . . . . 10 $/\$ # $/\$ $/\$
72	38, 52, 33, 40, 71	syl22anc 1239	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
73		expineg2 10531	. . . . . . . . . 10 $/\$ # $/\$ $/\$
74	38, 52, 35, 42, 73	syl22anc 1239	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
75	72, 74	oveq12d 5895	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
76	65, 70, 75	3eqtr4d 2220	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
77	76	3expia 1205	. . . . . 6 $/\$ # $/\$ $/\$ $/\$
78	31, 77	jaodan 797	. . . . 5 $/\$ # $/\$ $\/$ $/\$ $/\$
79	8, 78	jaod 717	. . . 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 708 $/\$ w3a 978

wceq 1353

wcel 2148 class class class wbr 4005 (class class class)co 5877

cc 7811

cr 7812

cc0 7813

c1 7814

caddc 7816

cmul 7818

cneg 8131 # cap 8540

cdiv 8631

cn 8921

cn0 9178

cz 9255

cexp 10521

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-n0 9179 df-z 9256 df-uz 9531 df-seqfrec 10448 df-exp 10522

This theorem is referenced by: m1expeven 10569 expsubap 10570 expp1zap 10571 pcaddlem 12340

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