Theorem expaddzap 10368

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 9092	. . 3 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2		elznn0nn 9092	. . . 4 |- ( M e. ZZ <-> ( M e. NN0 \/ ( M e. RR /\ -u M e. NN ) ) )
3		expadd 10366	. . . . . . . 8 |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
4	3	3expia 1184	. . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> ( N e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
5	4	adantlr 469	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ M e. NN0 ) -> ( N e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
6		expaddzaplem 10367	. . . . . . 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 1184	. . . . . 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 787	. . . . 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 10367	. . . . . . . . 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 984	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> M e. NN0 )
11	10	nn0cnd 9056	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> M e. CC )
12		simp2l 1008	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> N e. RR )
13	12	recnd 7818	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> N e. CC )
14	11, 13	addcomd 7937	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( M + N ) = ( N + M ) )
15	14	oveq2d 5798	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ ( M + N ) ) = ( A ^ ( N + M ) ) )
16		simp1l 1006	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> A e. CC )
17		expcl 10342	. . . . . . . . . . 11 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC )
18	16, 10, 17	syl2anc 409	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ M ) e. CC )
19		simp1r 1007	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> A # 0 )
20	13	negnegd 8088	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u -u N = N )
21		simp2r 1009	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u N e. NN )
22	21	nnnn0d 9054	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u N e. NN0 )
23		nn0negz 9112	. . . . . . . . . . . . 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 2218	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> N e. ZZ )
26		expclzap 10349	. . . . . . . . . . 11 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) e. CC )
27	16, 19, 25, 26	syl3anc 1217	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ N ) e. CC )
28	18, 27	mulcomd 7811	. . . . . . . . 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 2183	. . . . . . . 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 1184	. . . . . . 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 1008	. . . . . . . . . . . . . . 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 7818	. . . . . . . . . . . . . 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 1010	. . . . . . . . . . . . . . 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 7818	. . . . . . . . . . . . . 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 8110	. . . . . . . . . . . . 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 5798	. . . . . . . . . . . 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 1006	. . . . . . . . . . . . 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 1009	. . . . . . . . . . . . . 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 9054	. . . . . . . . . . . . 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 1011	. . . . . . . . . . . . . 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 9054	. . . . . . . . . . . . 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 10366	. . . . . . . . . . . . 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 1217	. . . . . . . . . . . 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 2173	. . . . . . . . . . 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 5798	. . . . . . . . . 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 8896	. . . . . . . . . . 11 |- ( 1 x. 1 ) = 1
48	47	oveq1i 5792	. . . . . . . . . 10 |- ( ( 1 x. 1 ) / ( ( A ^ -u M ) x. ( A ^ -u N ) ) ) = ( 1 / ( ( A ^ -u M ) x. ( A ^ -u N ) ) )
49	46, 48	eqtr4di 2191	. . . . . . . . 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 10342	. . . . . . . . . . 11 |- ( ( A e. CC /\ -u M e. NN0 ) -> ( A ^ -u M ) e. CC )
51	38, 40, 50	syl2anc 409	. . . . . . . . . 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 1007	. . . . . . . . . . 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 9195	. . . . . . . . . . 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 10356	. . . . . . . . . . 11 |- ( ( A e. CC /\ A # 0 /\ -u M e. ZZ ) -> ( A ^ -u M ) # 0 )
55	38, 52, 53, 54	syl3anc 1217	. . . . . . . . . 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 10342	. . . . . . . . . . 11 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( A ^ -u N ) e. CC )
57	38, 42, 56	syl2anc 409	. . . . . . . . . 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 9195	. . . . . . . . . . 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 10356	. . . . . . . . . . 11 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
60	38, 52, 58, 59	syl3anc 1217	. . . . . . . . . 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 7737	. . . . . . . . . . 11 |- 1 e. CC
62		divmuldivap 8496	. . . . . . . . . . 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 433	. . . . . . . . . 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 1218	. . . . . . . . 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 2176	. . . . . . . 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 7809	. . . . . . . . 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 9058	. . . . . . . . . 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 2217	. . . . . . . . 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 10333	. . . . . . . . 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 1218	. . . . . . . 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 10333	. . . . . . . . . 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 1218	. . . . . . . . 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 10333	. . . . . . . . . 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 1218	. . . . . . . . 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 5800	. . . . . . . 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 2183	. . . . . . 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 1184	. . . . . 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 787	. . . . 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 707	. . . 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 377	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 698   /\ w3a 963   = wceq 1332   e. wcel 1481   class class class wbr 3937  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645   + caddc 7647   x. cmul 7649   -ucneg 7958   # cap 8367   / cdiv 8456   NNcn 8744   NN0cn0 9001   ZZcz 9078   ^cexp 10323

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-seqfrec 10250  df-exp 10324

This theorem is referenced by:  m1expeven  10371  expsubap  10372  expp1zap  10373