Theorem expaddzaplem 10674

Description: Lemma for expaddzap 10675. (Contributed by Jim Kingdon, 10-Jun-2020.)

Assertion

Ref	Expression
expaddzaplem	|- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )

Proof of Theorem expaddzaplem

Step	Hyp	Ref	Expression
1		simp1l 1023	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> A e. CC )
2		simp3 1001	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. NN0 )
3		expcl 10649	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
4	1, 2, 3	syl2anc 411	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ N ) e. CC )
5		simp2r 1026	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN )
6	5	nnnn0d 9302	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN0 )
7		expcl 10649	. . . 4 |- ( ( A e. CC /\ -u M e. NN0 ) -> ( A ^ -u M ) e. CC )
8	1, 6, 7	syl2anc 411	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u M ) e. CC )
9		simp1r 1024	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> A # 0 )
10	5	nnzd 9447	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. ZZ )
11		expap0i 10663	. . . 4 |- ( ( A e. CC /\ A # 0 /\ -u M e. ZZ ) -> ( A ^ -u M ) # 0 )
12	1, 9, 10, 11	syl3anc 1249	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u M ) # 0 )
13	4, 8, 12	divrecap2d 8821	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( A ^ N ) / ( A ^ -u M ) ) = ( ( 1 / ( A ^ -u M ) ) x. ( A ^ N ) ) )
14		simp2l 1025	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. RR )
15	14	recnd 8055	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. CC )
16	15	negnegd 8328	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u -u M = M )
17		nnnegz 9329	. . . . . . . . . 10 |- ( -u M e. NN -> -u -u M e. ZZ )
18	5, 17	syl 14	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u -u M e. ZZ )
19	16, 18	eqeltrrd 2274	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. ZZ )
20	2	nn0zd 9446	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. ZZ )
21	19, 20	zaddcld 9452	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( M + N ) e. ZZ )
22		expclzap 10656	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ ( M + N ) e. ZZ ) -> ( A ^ ( M + N ) ) e. CC )
23	1, 9, 21, 22	syl3anc 1249	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) e. CC )
24	23	adantr 276	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( A ^ ( M + N ) ) e. CC )
25	8	adantr 276	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( A ^ -u M ) e. CC )
26	12	adantr 276	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( A ^ -u M ) # 0 )
27	24, 25, 26	divcanap4d 8823	. . . 4 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( ( ( A ^ ( M + N ) ) x. ( A ^ -u M ) ) / ( A ^ -u M ) ) = ( A ^ ( M + N ) ) )
28	1	adantr 276	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> A e. CC )
29		simpr 110	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( M + N ) e. NN0 )
30	6	adantr 276	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> -u M e. NN0 )
31		expadd 10673	. . . . . . 7 |- ( ( A e. CC /\ ( M + N ) e. NN0 /\ -u M e. NN0 ) -> ( A ^ ( ( M + N ) + -u M ) ) = ( ( A ^ ( M + N ) ) x. ( A ^ -u M ) ) )
32	28, 29, 30, 31	syl3anc 1249	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( A ^ ( ( M + N ) + -u M ) ) = ( ( A ^ ( M + N ) ) x. ( A ^ -u M ) ) )
33	21	zcnd 9449	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( M + N ) e. CC )
34	33, 15	negsubd 8343	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) + -u M ) = ( ( M + N ) - M ) )
35	2	nn0cnd 9304	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. CC )
36	15, 35	pncan2d 8339	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) - M ) = N )
37	34, 36	eqtrd 2229	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) + -u M ) = N )
38	37	adantr 276	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( ( M + N ) + -u M ) = N )
39	38	oveq2d 5938	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( A ^ ( ( M + N ) + -u M ) ) = ( A ^ N ) )
40	32, 39	eqtr3d 2231	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( ( A ^ ( M + N ) ) x. ( A ^ -u M ) ) = ( A ^ N ) )
41	40	oveq1d 5937	. . . 4 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( ( ( A ^ ( M + N ) ) x. ( A ^ -u M ) ) / ( A ^ -u M ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
42	27, 41	eqtr3d 2231	. . 3 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
43	1	adantr 276	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> A e. CC )
44	9	adantr 276	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> A # 0 )
45	33	adantr 276	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( M + N ) e. CC )
46		simpr 110	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> -u ( M + N ) e. NN0 )
47		expineg2 10640	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( ( M + N ) e. CC /\ -u ( M + N ) e. NN0 ) ) -> ( A ^ ( M + N ) ) = ( 1 / ( A ^ -u ( M + N ) ) ) )
48	43, 44, 45, 46, 47	syl22anc 1250	. . . 4 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ ( M + N ) ) = ( 1 / ( A ^ -u ( M + N ) ) ) )
49	21	znegcld 9450	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u ( M + N ) e. ZZ )
50		expclzap 10656	. . . . . . . . . 10 |- ( ( A e. CC /\ A # 0 /\ -u ( M + N ) e. ZZ ) -> ( A ^ -u ( M + N ) ) e. CC )
51	1, 9, 49, 50	syl3anc 1249	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u ( M + N ) ) e. CC )
52	51	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ -u ( M + N ) ) e. CC )
53	4	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ N ) e. CC )
54		expap0i 10663	. . . . . . . . . 10 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) # 0 )
55	1, 9, 20, 54	syl3anc 1249	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ N ) # 0 )
56	55	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ N ) # 0 )
57	52, 53, 56	divcanap4d 8823	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( ( ( A ^ -u ( M + N ) ) x. ( A ^ N ) ) / ( A ^ N ) ) = ( A ^ -u ( M + N ) ) )
58	2	adantr 276	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> N e. NN0 )
59		expadd 10673	. . . . . . . . . 10 |- ( ( A e. CC /\ -u ( M + N ) e. NN0 /\ N e. NN0 ) -> ( A ^ ( -u ( M + N ) + N ) ) = ( ( A ^ -u ( M + N ) ) x. ( A ^ N ) ) )
60	43, 46, 58, 59	syl3anc 1249	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ ( -u ( M + N ) + N ) ) = ( ( A ^ -u ( M + N ) ) x. ( A ^ N ) ) )
61	15, 35	negdi2d 8351	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u ( M + N ) = ( -u M - N ) )
62	61	oveq1d 5937	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( -u ( M + N ) + N ) = ( ( -u M - N ) + N ) )
63	15	negcld 8324	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. CC )
64	63, 35	npcand 8341	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( -u M - N ) + N ) = -u M )
65	62, 64	eqtrd 2229	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( -u ( M + N ) + N ) = -u M )
66	65	adantr 276	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( -u ( M + N ) + N ) = -u M )
67	66	oveq2d 5938	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ ( -u ( M + N ) + N ) ) = ( A ^ -u M ) )
68	60, 67	eqtr3d 2231	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( ( A ^ -u ( M + N ) ) x. ( A ^ N ) ) = ( A ^ -u M ) )
69	68	oveq1d 5937	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( ( ( A ^ -u ( M + N ) ) x. ( A ^ N ) ) / ( A ^ N ) ) = ( ( A ^ -u M ) / ( A ^ N ) ) )
70	57, 69	eqtr3d 2231	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ -u ( M + N ) ) = ( ( A ^ -u M ) / ( A ^ N ) ) )
71	70	oveq2d 5938	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( 1 / ( A ^ -u ( M + N ) ) ) = ( 1 / ( ( A ^ -u M ) / ( A ^ N ) ) ) )
72	8, 4, 12, 55	recdivapd 8834	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( 1 / ( ( A ^ -u M ) / ( A ^ N ) ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
73	72	adantr 276	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( 1 / ( ( A ^ -u M ) / ( A ^ N ) ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
74	71, 73	eqtrd 2229	. . . 4 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( 1 / ( A ^ -u ( M + N ) ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
75	48, 74	eqtrd 2229	. . 3 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
76		elznn0 9341	. . . . 5 |- ( ( M + N ) e. ZZ <-> ( ( M + N ) e. RR /\ ( ( M + N ) e. NN0 \/ -u ( M + N ) e. NN0 ) ) )
77	76	simprbi 275	. . . 4 |- ( ( M + N ) e. ZZ -> ( ( M + N ) e. NN0 \/ -u ( M + N ) e. NN0 ) )
78	21, 77	syl 14	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) e. NN0 \/ -u ( M + N ) e. NN0 ) )
79	42, 75, 78	mpjaodan 799	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
80		expineg2 10640	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. CC /\ -u M e. NN0 ) ) -> ( A ^ M ) = ( 1 / ( A ^ -u M ) ) )
81	1, 9, 15, 6, 80	syl22anc 1250	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ M ) = ( 1 / ( A ^ -u M ) ) )
82	81	oveq1d 5937	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( A ^ M ) x. ( A ^ N ) ) = ( ( 1 / ( A ^ -u M ) ) x. ( A ^ N ) ) )
83	13, 79, 82	3eqtr4d 2239	1 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   class class class wbr 4033  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   - cmin 8197   -ucneg 8198   # cap 8608   / cdiv 8699   NNcn 8990   NN0cn0 9249   ZZcz 9326   ^cexp 10630

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-seqfrec 10540  df-exp 10631

This theorem is referenced by:  expaddzap  10675