Theorem expaddzap 10337

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 9068	. . 3 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2		elznn0nn 9068	. . . 4 |- ( M e. ZZ <-> ( M e. NN0 \/ ( M e. RR /\ -u M e. NN ) ) )
3		expadd 10335	. . . . . . . 8 |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
4	3	3expia 1183	. . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> ( N e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
5	4	adantlr 468	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ M e. NN0 ) -> ( N e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
6		expaddzaplem 10336	. . . . . . 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 1183	. . . . . 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 786	. . . . 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 10336	. . . . . . . . 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 983	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> M e. NN0 )
11	10	nn0cnd 9032	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> M e. CC )
12		simp2l 1007	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> N e. RR )
13	12	recnd 7794	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> N e. CC )
14	11, 13	addcomd 7913	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( M + N ) = ( N + M ) )
15	14	oveq2d 5790	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ ( M + N ) ) = ( A ^ ( N + M ) ) )
16		simp1l 1005	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> A e. CC )
17		expcl 10311	. . . . . . . . . . 11 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC )
18	16, 10, 17	syl2anc 408	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ M ) e. CC )
19		simp1r 1006	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> A # 0 )
20	13	negnegd 8064	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u -u N = N )
21		simp2r 1008	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u N e. NN )
22	21	nnnn0d 9030	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u N e. NN0 )
23		nn0negz 9088	. . . . . . . . . . . . 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 2217	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> N e. ZZ )
26		expclzap 10318	. . . . . . . . . . 11 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) e. CC )
27	16, 19, 25, 26	syl3anc 1216	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ N ) e. CC )
28	18, 27	mulcomd 7787	. . . . . . . . 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 2182	. . . . . . . 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 1183	. . . . . . 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 258	. . . . . 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 1007	. . . . . . . . . . . . . . 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 7794	. . . . . . . . . . . . . 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 1009	. . . . . . . . . . . . . . 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 7794	. . . . . . . . . . . . . 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 8086	. . . . . . . . . . . . 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 5790	. . . . . . . . . . . 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 1005	. . . . . . . . . . . . 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 1008	. . . . . . . . . . . . . 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 9030	. . . . . . . . . . . . 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 1010	. . . . . . . . . . . . . 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 9030	. . . . . . . . . . . . 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 10335	. . . . . . . . . . . . 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 1216	. . . . . . . . . . . 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 2172	. . . . . . . . . . 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 5790	. . . . . . . . . 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 8872	. . . . . . . . . . 11 |- ( 1 x. 1 ) = 1
48	47	oveq1i 5784	. . . . . . . . . 10 |- ( ( 1 x. 1 ) / ( ( A ^ -u M ) x. ( A ^ -u N ) ) ) = ( 1 / ( ( A ^ -u M ) x. ( A ^ -u N ) ) )
49	46, 48	syl6eqr 2190	. . . . . . . . 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 10311	. . . . . . . . . . 11 |- ( ( A e. CC /\ -u M e. NN0 ) -> ( A ^ -u M ) e. CC )
51	38, 40, 50	syl2anc 408	. . . . . . . . . 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 1006	. . . . . . . . . . 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 9171	. . . . . . . . . . 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 10325	. . . . . . . . . . 11 |- ( ( A e. CC /\ A # 0 /\ -u M e. ZZ ) -> ( A ^ -u M ) # 0 )
55	38, 52, 53, 54	syl3anc 1216	. . . . . . . . . 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 10311	. . . . . . . . . . 11 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( A ^ -u N ) e. CC )
57	38, 42, 56	syl2anc 408	. . . . . . . . . 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 9171	. . . . . . . . . . 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 10325	. . . . . . . . . . 11 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
60	38, 52, 58, 59	syl3anc 1216	. . . . . . . . . 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 7713	. . . . . . . . . . 11 |- 1 e. CC
62		divmuldivap 8472	. . . . . . . . . . 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 432	. . . . . . . . . 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 1217	. . . . . . . . 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 2175	. . . . . . . 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 7785	. . . . . . . . 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 9034	. . . . . . . . . 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 2216	. . . . . . . . 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 10302	. . . . . . . . 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 1217	. . . . . . . 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 10302	. . . . . . . . . 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 1217	. . . . . . . . 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 10302	. . . . . . . . . 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 1217	. . . . . . . . 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 5792	. . . . . . . 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 2182	. . . . . . 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 1183	. . . . . 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 786	. . . . 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 706	. . . 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 285	. . 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 151	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ M e. ZZ ) -> ( N e. ZZ -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
82	81	impr 376	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 103   \/ wo 697   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   -ucneg 7934   # cap 8343   / cdiv 8432   NNcn 8720   NN0cn0 8977   ZZcz 9054   ^cexp 10292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-seqfrec 10219  df-exp 10293

This theorem is referenced by:  m1expeven  10340  expsubap  10341  expp1zap  10342