Theorem expaddzaplem 10653

Description: Lemma for expaddzap 10654. (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 10628	. . . 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 9293	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN0 )
7		expcl 10628	. . . 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 9438	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. ZZ )
11		expap0i 10642	. . . 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 8813	. 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 8048	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. CC )
16	15	negnegd 8321	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u -u M = M )
17		nnnegz 9320	. . . . . . . . . 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 2271	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. ZZ )
20	2	nn0zd 9437	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. ZZ )
21	19, 20	zaddcld 9443	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( M + N ) e. ZZ )
22		expclzap 10635	. . . . . . 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 8815	. . . 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 10652	. . . . . . 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 9440	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( M + N ) e. CC )
34	33, 15	negsubd 8336	. . . . . . . . 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 9295	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. CC )
36	15, 35	pncan2d 8332	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) - M ) = N )
37	34, 36	eqtrd 2226	. . . . . . . 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 5934	. . . . . 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 2228	. . . . 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 5933	. . . 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 2228	. . 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 10619	. . . . 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 9441	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u ( M + N ) e. ZZ )
50		expclzap 10635	. . . . . . . . . 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 10642	. . . . . . . . . 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 8815	. . . . . . 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 10652	. . . . . . . . . 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 8344	. . . . . . . . . . . . 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 5933	. . . . . . . . . . . 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 8317	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. CC )
64	63, 35	npcand 8334	. . . . . . . . . . . 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 2226	. . . . . . . . . . 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 5934	. . . . . . . . 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 2228	. . . . . . . 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 5933	. . . . . . 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 2228	. . . . . 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 5934	. . . . 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 8826	. . . . . 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 2226	. . . 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 2226	. . 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 9332	. . . . 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 10619	. . . 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 5933	. 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 2236	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 2164   class class class wbr 4029  (class class class)co 5918   CCcc 7870   RRcr 7871   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   - cmin 8190   -ucneg 8191   # cap 8600   / cdiv 8691   NNcn 8982   NN0cn0 9240   ZZcz 9317   ^cexp 10609

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-seqfrec 10519  df-exp 10610

This theorem is referenced by:  expaddzap  10654