Theorem expaddzap 8933

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

Assertion

Ref	Expression
expaddzap	|- (((A e. CC /\ A # 0) /\ ( M e. ZZ /\ N e. ZZ)) -> (A ^( M + N)) = ((A ^ M) x. (A ^ N)))

Proof of Theorem expaddzap

Step	Hyp	Ref	Expression
1		elznn0nn 8015	. . 3 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN)))
2		elznn0nn 8015	. . . 4 |- ( M e. ZZ <-> ( M e. NN0 \/ ( M e. RR /\ -u M e. NN)))
3		expadd 8931	. . . . . . . 8 |- ((A e. CC /\ M e. NN0 /\ N e. NN0) -> (A ^( M + N)) = ((A ^ M) x. (A ^ N)))
4	3	3expia 1105	. . . . . . 7 |- ((A e. CC /\ M e. NN0) -> ( N e. NN0 -> (A ^( M + N)) = ((A ^ M) x. (A ^ N))))
5	4	adantlr 446	. . . . . 6 |- (((A e. CC /\ A # 0) /\ M e. NN0) -> ( N e. NN0 -> (A ^( M + N)) = ((A ^ M) x. (A ^ N))))
6		expaddzaplem 8932	. . . . . . 7 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ N e. NN0) -> (A ^( M + N)) = ((A ^ M) x. (A ^ N)))
7	6	3expia 1105	. . . . . 6 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN)) -> ( N e. NN0 -> (A ^( M + N)) = ((A ^ M) x. (A ^ N))))
8	5, 7	jaodan 709	. . . . 5 |- (((A e. CC /\ A # 0) /\ ( M e. NN0 \/ ( M e. RR /\ -u M e. NN))) -> ( N e. NN0 -> (A ^( M + N)) = ((A ^ M) x. (A ^ N))))
9		expaddzaplem 8932	. . . . . . . . 9 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> (A ^( N + M)) = ((A ^ N) x. (A ^ M)))
10		simp3 905	. . . . . . . . . . . 12 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> M e. NN0)
11	10	nn0cnd 7993	. . . . . . . . . . 11 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> M e. CC)
12		simp2l 929	. . . . . . . . . . . 12 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> N e. RR)
13	12	recnd 6831	. . . . . . . . . . 11 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> N e. CC)
14	11, 13	addcomd 6941	. . . . . . . . . 10 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> ( M + N) = ( N + M))
15	14	oveq2d 5471	. . . . . . . . 9 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> (A ^( M + N)) = (A ^( N + M)))
16		simp1l 927	. . . . . . . . . . 11 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> A e. CC)
17		expcl 8907	. . . . . . . . . . 11 |- ((A e. CC /\ M e. NN0) -> (A ^ M) e. CC)
18	16, 10, 17	syl2anc 391	. . . . . . . . . 10 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> (A ^ M) e. CC)
19		simp1r 928	. . . . . . . . . . 11 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> A # 0)
20	13	negnegd 7089	. . . . . . . . . . . 12 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> -u -u N = N)
21		simp2r 930	. . . . . . . . . . . . . 14 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> -u N e. NN)
22	21	nnnn0d 7991	. . . . . . . . . . . . 13 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> -u N e. NN0)
23		nn0negz 8035	. . . . . . . . . . . . 13 |- ( -u N e. NN0 -> -u -u N e. ZZ)
24	22, 23	syl 14	. . . . . . . . . . . 12 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> -u -u N e. ZZ)
25	20, 24	eqeltrrd 2112	. . . . . . . . . . 11 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> N e. ZZ)
26		expclzap 8914	. . . . . . . . . . 11 |- ((A e. CC /\ A # 0 /\ N e. ZZ) -> (A ^ N) e. CC)
27	16, 19, 25, 26	syl3anc 1134	. . . . . . . . . 10 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> (A ^ N) e. CC)
28	18, 27	mulcomd 6826	. . . . . . . . 9 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> ((A ^ M) x. (A ^ N)) = ((A ^ N) x. (A ^ M)))
29	9, 15, 28	3eqtr4d 2079	. . . . . . . 8 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN) /\ M e. NN0) -> (A ^( M + N)) = ((A ^ M) x. (A ^ N)))
30	29	3expia 1105	. . . . . . 7 |- (((A e. CC /\ A # 0) /\ ( N e. RR /\ -u N e. NN)) -> ( M e. NN0 -> (A ^( M + N)) = ((A ^ M) x. (A ^ N))))
31	30	impancom 247	. . . . . 6 |- (((A e. CC /\ A # 0) /\ M e. NN0) -> (( N e. RR /\ -u N e. NN) -> (A ^( M + N)) = ((A ^ M) x. (A ^ N))))
32		simp2l 929	. . . . . . . . . . . . . . 15 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> M e. RR)
33	32	recnd 6831	. . . . . . . . . . . . . 14 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> M e. CC)
34		simp3l 931	. . . . . . . . . . . . . . 15 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> N e. RR)
35	34	recnd 6831	. . . . . . . . . . . . . 14 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> N e. CC)
36	33, 35	negdid 7111	. . . . . . . . . . . . 13 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> -u( M + N) = ( -u M + -u N))
37	36	oveq2d 5471	. . . . . . . . . . . 12 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> (A ^ -u( M + N)) = (A ^( -u M + -u N)))
38		simp1l 927	. . . . . . . . . . . . 13 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> A e. CC)
39		simp2r 930	. . . . . . . . . . . . . 14 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> -u M e. NN)
40	39	nnnn0d 7991	. . . . . . . . . . . . 13 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> -u M e. NN0)
41		simp3r 932	. . . . . . . . . . . . . 14 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> -u N e. NN)
42	41	nnnn0d 7991	. . . . . . . . . . . . 13 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> -u N e. NN0)
43		expadd 8931	. . . . . . . . . . . . 13 |- ((A e. CC /\ -u M e. NN0 /\ -u N e. NN0) -> (A ^( -u M + -u N)) = ((A ^ -u M) x. (A ^ -u N)))
44	38, 40, 42, 43	syl3anc 1134	. . . . . . . . . . . 12 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> (A ^( -u M + -u N)) = ((A ^ -u M) x. (A ^ -u N)))
45	37, 44	eqtrd 2069	. . . . . . . . . . 11 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> (A ^ -u( M + N)) = ((A ^ -u M) x. (A ^ -u N)))
46	45	oveq2d 5471	. . . . . . . . . 10 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> ( 1 / (A ^ -u( M + N))) = ( 1 / ((A ^ -u M) x. (A ^ -u N))))
47		1t1e1 7825	. . . . . . . . . . 11 |- ( 1 x. 1) = 1
48	47	oveq1i 5465	. . . . . . . . . 10 |- (( 1 x. 1) / ((A ^ -u M) x. (A ^ -u N))) = ( 1 / ((A ^ -u M) x. (A ^ -u N)))
49	46, 48	syl6eqr 2087	. . . . . . . . 9 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> ( 1 / (A ^ -u( M + N))) = (( 1 x. 1) / ((A ^ -u M) x. (A ^ -u N))))
50		expcl 8907	. . . . . . . . . . 11 |- ((A e. CC /\ -u M e. NN0) -> (A ^ -u M) e. CC)
51	38, 40, 50	syl2anc 391	. . . . . . . . . 10 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> (A ^ -u M) e. CC)
52		simp1r 928	. . . . . . . . . . 11 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> A # 0)
53	40	nn0zd 8114	. . . . . . . . . . 11 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> -u M e. ZZ)
54		expap0i 8921	. . . . . . . . . . 11 |- ((A e. CC /\ A # 0 /\ -u M e. ZZ) -> (A ^ -u M) # 0)
55	38, 52, 53, 54	syl3anc 1134	. . . . . . . . . 10 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> (A ^ -u M) # 0)
56		expcl 8907	. . . . . . . . . . 11 |- ((A e. CC /\ -u N e. NN0) -> (A ^ -u N) e. CC)
57	38, 42, 56	syl2anc 391	. . . . . . . . . 10 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> (A ^ -u N) e. CC)
58	42	nn0zd 8114	. . . . . . . . . . 11 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> -u N e. ZZ)
59		expap0i 8921	. . . . . . . . . . 11 |- ((A e. CC /\ A # 0 /\ -u N e. ZZ) -> (A ^ -u N) # 0)
60	38, 52, 58, 59	syl3anc 1134	. . . . . . . . . 10 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> (A ^ -u N) # 0)
61		ax-1cn 6756	. . . . . . . . . . 11 |- 1 e. CC
62		divmuldivap 7450	. . . . . . . . . . 11 |- ((( 1 e. CC /\ 1 e. CC) /\ (((A ^ -u M) e. CC /\ (A ^ -u M) # 0) /\ ((A ^ -u N) e. CC /\ (A ^ -u N) # 0))) -> (( 1 / (A ^ -u M)) x. ( 1 / (A ^ -u N))) = (( 1 x. 1) / ((A ^ -u M) x. (A ^ -u N))))
63	61, 61, 62	mpanl12 412	. . . . . . . . . 10 |- ((((A ^ -u M) e. CC /\ (A ^ -u M) # 0) /\ ((A ^ -u N) e. CC /\ (A ^ -u N) # 0)) -> (( 1 / (A ^ -u M)) x. ( 1 / (A ^ -u N))) = (( 1 x. 1) / ((A ^ -u M) x. (A ^ -u N))))
64	51, 55, 57, 60, 63	syl22anc 1135	. . . . . . . . 9 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> (( 1 / (A ^ -u M)) x. ( 1 / (A ^ -u N))) = (( 1 x. 1) / ((A ^ -u M) x. (A ^ -u N))))
65	49, 64	eqtr4d 2072	. . . . . . . 8 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> ( 1 / (A ^ -u( M + N))) = (( 1 / (A ^ -u M)) x. ( 1 / (A ^ -u N))))
66	33, 35	addcld 6824	. . . . . . . . 9 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> ( M + N) e. CC)
67	40, 42	nn0addcld 7995	. . . . . . . . . 10 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> ( -u M + -u N) e. NN0)
68	36, 67	eqeltrd 2111	. . . . . . . . 9 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> -u( M + N) e. NN0)
69		expineg2 8898	. . . . . . . . 9 |- (((A e. CC /\ A # 0) /\ (( M + N) e. CC /\ -u( M + N) e. NN0)) -> (A ^( M + N)) = ( 1 / (A ^ -u( M + N))))
70	38, 52, 66, 68, 69	syl22anc 1135	. . . . . . . 8 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> (A ^( M + N)) = ( 1 / (A ^ -u( M + N))))
71		expineg2 8898	. . . . . . . . . 10 |- (((A e. CC /\ A # 0) /\ ( M e. CC /\ -u M e. NN0)) -> (A ^ M) = ( 1 / (A ^ -u M)))
72	38, 52, 33, 40, 71	syl22anc 1135	. . . . . . . . 9 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> (A ^ M) = ( 1 / (A ^ -u M)))
73		expineg2 8898	. . . . . . . . . 10 |- (((A e. CC /\ A # 0) /\ ( N e. CC /\ -u N e. NN0)) -> (A ^ N) = ( 1 / (A ^ -u N)))
74	38, 52, 35, 42, 73	syl22anc 1135	. . . . . . . . 9 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> (A ^ N) = ( 1 / (A ^ -u N)))
75	72, 74	oveq12d 5473	. . . . . . . 8 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> ((A ^ M) x. (A ^ N)) = (( 1 / (A ^ -u M)) x. ( 1 / (A ^ -u N))))
76	65, 70, 75	3eqtr4d 2079	. . . . . . 7 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN) /\ ( N e. RR /\ -u N e. NN)) -> (A ^( M + N)) = ((A ^ M) x. (A ^ N)))
77	76	3expia 1105	. . . . . 6 |- (((A e. CC /\ A # 0) /\ ( M e. RR /\ -u M e. NN)) -> (( N e. RR /\ -u N e. NN) -> (A ^( M + N)) = ((A ^ M) x. (A ^ N))))
78	31, 77	jaodan 709	. . . . 5 |- (((A e. CC /\ A # 0) /\ ( M e. NN0 \/ ( M e. RR /\ -u M e. NN))) -> (( N e. RR /\ -u N e. NN) -> (A ^( M + N)) = ((A ^ M) x. (A ^ N))))
79	8, 78	jaod 636	. . . 4 |- (((A e. CC /\ A # 0) /\ ( M e. NN0 \/ ( M e. RR /\ -u M e. NN))) -> (( N e. NN0 \/ ( N e. RR /\ -u N e. NN)) -> (A ^( M + N)) = ((A ^ M) x. (A ^ N))))
80	2, 79	sylan2b 271	. . 3 |- (((A e. CC /\ A # 0) /\ M e. ZZ) -> (( N e. NN0 \/ ( N e. RR /\ -u N e. NN)) -> (A ^( M + N)) = ((A ^ M) x. (A ^ N))))
81	1, 80	syl5bi 141	. 2 |- (((A e. CC /\ A # 0) /\ M e. ZZ) -> ( N e. ZZ -> (A ^( M + N)) = ((A ^ M) x. (A ^ N))))
82	81	impr 361	1 |- (((A e. CC /\ A # 0) /\ ( M e. ZZ /\ N e. ZZ)) -> (A ^( M + N)) = ((A ^ M) x. (A ^ N)))

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 628   /\ w3a 884   = wceq 1242   e. wcel 1390   class class class wbr 3755  (class class class)co 5455   CCcc 6689   RRcr 6690   0cc0 6691   1c1 6692   + caddc 6694   x. cmul 6696   -ucneg 6960   # cap 7345   / cdiv 7413   NNcn 7675   NN0cn0 7937   ZZcz 8001   ^cexp 8888

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6754  ax-resscn 6755  ax-1cn 6756  ax-1re 6757  ax-icn 6758  ax-addcl 6759  ax-addrcl 6760  ax-mulcl 6761  ax-mulrcl 6762  ax-addcom 6763  ax-mulcom 6764  ax-addass 6765  ax-mulass 6766  ax-distr 6767  ax-i2m1 6768  ax-1rid 6770  ax-0id 6771  ax-rnegex 6772  ax-precex 6773  ax-cnre 6774  ax-pre-ltirr 6775  ax-pre-ltwlin 6776  ax-pre-lttrn 6777  ax-pre-apti 6778  ax-pre-ltadd 6779  ax-pre-mulgt0 6780  ax-pre-mulext 6781

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-if 3326  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-frec 5918  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-iltp 6452  df-enr 6634  df-nr 6635  df-ltr 6638  df-0r 6639  df-1r 6640  df-0 6698  df-1 6699  df-r 6701  df-lt 6704  df-pnf 6839  df-mnf 6840  df-xr 6841  df-ltxr 6842  df-le 6843  df-sub 6961  df-neg 6962  df-reap 7339  df-ap 7346  df-div 7414  df-inn 7676  df-n0 7938  df-z 8002  df-uz 8230  df-iseq 8873  df-iexp 8889

This theorem is referenced by:  expsubap  8936  expp1zap  8937